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TLMaths - Core Maths Level 3 Certificate

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This post may contain Amazon affiliate links. As an Amazon Associate, I earn a small commission from qualifying purchases. This comes at no cost to you. Thanks for your support of this blog.

This year, I'm structuring things a bit differently in Algebra 1. Some changes have been for the better, and some haven't. I've already started making a list of things to keep and change for next year. Today, I want to blog about one of those things worth keeping. It's the type of thing that makes me wonder, "Why have I never done this before?!?"

Yesterday, I posted about a Ghosts in the Graveyard game I made to review independent and dependent variables. To be honest, I have always HATED teaching independent and dependent variables. They were in the standards, so I taught them. But, I never took the time to connect independent and dependent variables to the other concepts we were learning. No wonder I felt like it was a pointless thing to teach!

This year, I decided to teach finding rate of change directly after teaching independent and dependent variables. It was a beautiful thing to watch it unfold! Every time I gave my students a table or set of points, we classified the variables as dependent or independent before moving on to finding rate of change.

I defined rate of change for my students as the change in the dependent variable divided by the change of the independent variable. This is the first time I have ever taught rate of change independently from slope, and I LOVE it! My students are finding the slope, but they don't know that yet. Most of my 9th graders come in my room with a history of either loving slope or hating slope from their middle school years. By focusing only on rate of change and not slope, I'm giving them each a fresh start.

Here are the notes we ended up taking on calculating rate of change from a table or set of points. We made a small book to glue in our notes. You can find the instructions for folding this book here.

Another BIG change I made this year was making students write the units next to the arrows they drew to calculate the change. My students are struggling so much less this year with remembering their units in their final answers. Yay!

You can download the file for this practice book here.

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Calculating Rate of Change from a Table or Set of Points | Math = Love

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This post may contain Amazon affiliate links. As an Amazon Associate, I earn a small commission from qualifying purchases. This comes at no cost to you. Thanks for your support of this blog. This year, I'm structuring things a bit differently in Algebra 1. Some changes have been for the better, and some haven't.

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t is a quiet suburban street in Auckland: plush lawns, manicured hedges, orderly picket fences, stately wooden villas. Until you come upon the rabbits. So many rabbits: scattered across the lawn, sunbathing, consorting under deck chairs, crunching piles of cauliflower leaves. Grey, black, spotted, tan, white with shining pink eyes.

Last month there were about 400, although even that estimate was probably conservative. In recent weeks, their ranks have been thinned. The rabbits' domain is the lawn of an otherwise nondescript, slightly run-down villa in the neighbourhood of Mount Eden. In the seven years they have spent quietly colonising this yard, it has become some of the country's most coveted and unaffordable real estate. House prices have risen 70%. Oblivious to it all, the rabbits find themselves on the front lines of a battle for Auckland suburbia. Now, they are the subject of a court order - hundreds of rabbits must go.

The story starts simply. In August 2014, residents Elaine Cowlin and Dylan Lewis bought four rabbits from a local pet shop. They didn't de-sex them. The rabbits began to breed. "We just love rabbits," Cowlin tells the Guardian. The property has two houses - Lewis lives in one, providing full time rabbit care in exchange for free rent.

Elaine Cowlin with one of her cherished rabbits. Photograph: Beka Hope

"The numbers were in check for maybe the first four, five years," says Lewis. "Then they got out of hand."

"It gets quite explosive after some while," Cowlin muses, looking out over the garden. "Not a long while either.'

Rabbit populations are a textbook example of the exponential growth curve - without predators, their population grows by a factor of around 3.5 each year. If Cowlin's 400 were left alone for three years, they would outnumber the human population of Mount Eden - a sprawling suburb home to around 14,700 people. While they're relatively common as pets, wild rabbits are considered pests in New Zealand. According to Newsroom, they cost New Zealand over $50m in lost agricultural production, and another $25m in pest control each year.

Now, for the vast majority of these bunnies, the end is nigh. In June, after years of complaints from neighbours, the environment court ruled that all but 16 of the rabbits must go "by any means necessary". The pair have until 2 August to get rid of the rabbits themselves. So far, around 200 have been euthanised, and others taken to animal shelters. Lewis says there are 53 left - an estimate which seems extremely optimistic, judging by the number hopping past him. He plans to take the remainder to shelters.

For the neighbours, the rabbit house has become a nightmare - a chaos agent descended on the prosperous safety of suburbia, leaving a trail of faeces, dead rabbits, potholes and a distinct odour. They have laid a multitude of complaints with council. In recent years, tension have run high: Lewis says the local cats are decapitating his pets, and so he started putting dead rabbits on neighbour's property in protest. He received a trespass notice, which prohibits him returning to the property.

"I'm actually trespassed from [that] house because I kept putting dead rabbits on the property," Lewis says, offhand. "I got upset," he says.

One resident who asks not to be named said: "It's not about neighbours at war. It's about human rights". They point out where the driveway is strewn with rabbit droppings: "E. coli everywhere."

The bevy of rabbits makes it hard to sell houses, the resident says. They also suspect that there are far more rabbits than are immediately visible: "For every rabbit above ground there are six below."

Over the fence, a carefully maintained lawn is pitted with holes. A single bunny stares, unapologetic, from the grass.

Now 82, Cowlin first moved into the street when she was 24 and newly married. They bought the first rabbits a few years after her husband passed away. Now, she says she'll never leave the property - the house has too many memories.

The rabbits have caused concerns that property prices in the area are being affected. Photograph: Beka Hope

"It's nothing like it used to be," she says. "When we first moved into the street there were constant knockings on the door. Now, people in this street are more short term. They're often here for their children to go to certain schools. Everything's quicker."

Quicker, and vastly more expensive. Auckland's suburbs are now one of the world's least affordable places to live. The average house price in Mount Eden is $1.9m, up nearly 18% in the past 12 months. "Things have changed so much ... not just in this street, all over Auckland," Cowlin says. "There's an atmosphere of distrust and suspicion, people valuing their property much more than anything else."

One of the villas across the road sold for $3.62m in February. In such a place, a multitude of fluffy animals let loose is not just an eccentricity - it represents a potential threat to the value of a street packed with multi-million-dollar assets.

Cowlin is sad to see them go. But perhaps there is an element of relief, too. She says she has burned through much of her savings, feeding her rabbit companions. "It's a total commitment," she says. But why rabbits?

"They're free spirited," Lewis says. "Yes," nods Cowlin. "They've got their own minds."

As she talks, more than 100 pigeons stud the powerline over the property, like some kind of omen.

"Yeah," Lewis says. "We feed the pigeons as well."

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'They got out of hand': Auckland street wrestles with fate of 400 pet rabbits

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t is a quiet suburban street in Auckland: plush lawns, manicured hedges, orderly picket fences, stately wooden villas. Until you come upon the rabbits. So many rabbits: scattered across the lawn, sunbathing, consorting under deck chairs, crunching piles of cauliflower leaves. Grey, black, spotted, tan, white with shining pink eyes.

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Also use for estimation

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Quibans 92 - Hoarding toilet paper

From the Daily Telegraph, near the start of the coronavirus pandemic:

Why the UK won't run out of toilet roll

13 March 2020
As panic over the coronavirus outbreak continues to spread, empty shelves where toilet roll once would have been has become a common sight in many supermarkets. Consumers have been stockpiling the bathroom staple, sparking fears that supermarkets could soon be caught short. However, there might not be any need to panic.

The UK exported $129m (£103m) of toilet roll in 2019 and exports more than it imports.

"As the world's 11th-largest exporter of toilet roll, at least if our supply routes shut down, we will be able to keep calm, stop exporting, and carry on," Dr Rebecca Harding of data firm Coriolis Technologies said.

Ireland accounted for $90.4m or ##% of all UK toilet roll exports in 2019 and is dependent on the UK for 80pc of all toilet roll. Dr Harding said the Irish reliance on the UK for toilet roll is evidence of the importance of borderless trade on the island of Ireland.

1. What is the exchange rate?

2. What percentage of UK exports of toilet roll go to Ireland?

3. How much does Ireland spend on toilet roll per year?

4. The population of Ireland is 4.83 million.How much does each person spend on toilet roll per week?

5. Four sheets of toilet paper cost about 1p.How many sheets is that per person per day?

1. £1 = $1.2524

2. 70%

3. 90.4 million divided by 0.8 = $113 million

4. 45 cents per week

5. 45 cents is about 36p per week.Divide by 7 to get the cost per day and multiply by 4 to find the number.20.5 sheets per day.

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Quibans 92 - Hoarding toilet paper From the Daily Telegraph, near the start of the coronavirus pandemic: Why the UK won't run out of toilet roll 13 March 2020 As panic over the coronavirus outbreak continues to spread, empty shelves where toilet roll once would have been has become a common sight in many supermarkets.

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use for spearman's rank, correlation

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Relationship between film time and rating

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Relationship between film time and rating

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Relationship between film length and average IMDB rating, using only film data. [OC]

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1.7k votes, 142 comments. 15.5m members in the dataisbeautiful community. A place to share and discuss visual representations of data: Graphs ...

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An isolated island is ruled by a dictator. Every family on the island has exactly two children. Each child is equally likely to be a boy or a girl. The dictator has decreed that the first girl (if any) born in the family must bear the name Mary Ann (the name of the beloved mother-in-law of the dictator). You are told that a randomly chosen family that is unknown to you has a girl named Mary Ann. What is the probability that this family has two girls?

The dictator has passed away and his son Little Joe has taken over. The wife of the new dictator bears the name Mary Ann and she is not happy with the fact that there are so many Mary Anns on the island. Little Joe goes some way to meeting the dissatisfaction of his wife and changes the rules. The name of a first girl born in a family is determined by the roll of a die and becomes Mary Ann only if the die shows a six. If a second girl is born in the family and the first girl is not named Mary Ann, then the name of the second girl is determined by a coin flip and becomes Mary Ann only if the coin shows heads. What is now the probability that a randomly chosen family has two girls when you are told that this family has a girl named Mary Ann?
Win the car, dodge the goat
In a game show, a father and his daughter are standing for three closed doors, behind which, a car, the key to the car, and a goat are hidden in random order. Each of them may open up to two doors, one at a time, and this should happen outside the sight of the other. The daughter is given the task of finding the car, the father must find the key. Only if both are successful, they get to keep the car. Father and daughter are allowed to discuss a strategy before the game starts. What is an optimal strategy and what is the probability of winning the car?
The last chocolate
There are ten chocolates in a Christmas tree, two of which are white and the other eight are dark. You take chocolates from the tree, one at a time and at random, and eat them until you pick a chocolate of the other color. You hang this chocolate back and start again with the leftover chocolates. What are the chances that the last chocolate you eat is white?
How to pick random numbers?
Pete plays the following game against one opponent. Two random numbers are chosen between 0 and 1 and each of the two players receives one of these numbers. Each player can choose to discard his number and receive a new random number between 0 and 1. This choice is made without knowing the other player's number or whether the other player chose to replace his number. After each player has had an opportunity to replace his number, the numbers are compared and the player with the higher number wins. What strategy should Pete follow to ensure he will win at least 50% of the time, regardless of his opponent's strategy?
Who takes the penalty kicks?
The coach of a soccer team doesn't want to see any arguing about penalty kicks of his team. He selects the three basic players Ryan, Mike and Luke, and places them at a round table. Next the coach takes a seat between Ryan and Luke with Ryan on his right side and Luke on his left side. The coach then tosses a fair coin. If it turns up heads, then the player on the left of the coach - Luke - will do the second toss of the coin; otherwise, Ryan will do the second toss. If Luke performs the second toss, then Mike will have the next turn if heads appears; otherwise, the coach is again up. Similarly, if Ryan performs the second toss, then the coach will have the next turn if heads appears; otherwise Mike will do the honors. The process of tossing the coin and passing it to the left or to the right depending on whether it turns up heads or tails is followed until there is just one player left who has not been the recipient of the coin. Then the game is over and that player becomes the penalty taker. What is for each player the chance to become the penalty taker? Bonus question: what is the answer if five basic players of the team are at the round table and the coach has again Ryan and Luke on his right and left side?

In the last puzzle any resemblance to an existing situation or existing persons is purely coincidental.
A malicious Santa Claus
Saint Nicholas and Santa Claus lived together in relative peace for a long time in the liberal city of Amsterdam. But one day, Saint Nicholas and his staff members are unexpectedly told by the powers-that-be of the city board that they are no longer allowed to put dice and card games in their packages, although they have always made an excellent and highly appreciated choice from the wide range of educational games for the December holiday season. From now on, the giving of these games is reserved for the ruthless, power-hungry Santa Claus who falsely arranged this behind the scenes with the aim of attracting extra reindeer for his own ends, using the heavier workload as an argument. This evil turn of events leads to a sharp reaction from a respected emeritus Saint Nicholas who argues that the general interest of a good educational package for young adults has been subordinated to Santa's flat merchant spirit, for whom only turnover counts and quality does not matter. Slightly shocked by this fierce reaction, the chairman of the Union for Educational Games of Chance on a Calvinistic basis promises - after consultation with Santa - to offer the Saint Nicholas group an opportunity to retain the gift right to give dice and card games. Without being informed in advance, the ten staff members of the Saint Nicholas group are convened and each placed in a separate room. Each of them is given a fair coin in their hands and must decide within five minutes whether or not to toss it, whereby consultation between the staff members is not possible. The Saint Nicholas group only gets the right to give dice and card games back if at least one of the ten staff members tosses the coin and each of the tossed coins gives heads. However, the staff members of the Saint Nicholas group are pre-eminently trained in logical thinking and decision making, which is also necessary if you have to organize dangerous trips over steep roofs for years. It is therefore immediately clear to them that they must each act according to the same decision rule. What is this decision rule and what are the chances that they will regain the gift right to dice and card games? Let me reveal that randomization is the keyword to the solution. The probability that the ten staff members of the Saint Nicholas group will be successful in reclaiming the gift right for dice and card games is 26.3%, where Santa Claus and his clique estimated the probability as negligible. Even if a hundred and not ten staffers of Saint Nicholas had been subjected to the test, the chance of success would still have been a little over 25%!

Please feel free to send an e-mail to Henk Tijms if you want to know the solutions. His email can be found on his personal website.

Henk Tijms is Honorary Fellow of Tinbergen Institute and emeritus professor Operations Research at the former department Econometrics and Operations Research of Vrije Universiteit Amsterdam. He has written several books on applied probability and stochastic optimization, including the bestselling books "Surprises in Probability - Seventeen Short Stories" and "Basic Probability, What Every Math Student Should Know". Homepage: personal.vu.nl/h.c.tijms/.

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Probability Puzzles with a Crazy Santa Claus

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For your enjoyment and instruction, this column contains six challenging probability puzzles of varying difficulty with sometimes surprising solutions. Some of those puzzles are adaptations of puzzles by honorary fellow Henk Tijms that previously appeared in The New York Times, The Guardian and FiveThirtyEight.

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An interactive grouped frequency table. Click the tiles to hide or reveal their values.

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Desmos activity using percentage change, quartiles, box plots

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COVID-19 Tier Map Scotland COVID Polling Nowcast Current Parliament General Elections EU Elections Local Elections Council By-Elections Councils Target Seats Hypothetical Maps Historic GE Maps

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In this activity, students apply (and deepen) their knowledge of linear inequalities in two variables to "collect" as many points in the coordinate plane as they can. This activity may take more than one class period. We recommend having students log in so that they can continue their work at a later time.

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questions on sequences, trig, optimisation..

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Review of Linear Equations - Review of Systems of Equations Factoring by Completing the Square - Parts 1, 2, 3 (Review) Factoring by Completing the Square - Part 4 (Review) Applications of Linear Equations The Pythagorean Theorem Remainder Theorem Nonlinear Systems Arithmetic Sequences Exercises in Symmetry Functions - Definition Graphical Solutions Quadratic Inequalities Updated!

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Practise using the sigma notation to find the sum of various number series

This is level 1: write out the terms of the series defined by the sigma expression. The first question has been done as an example. You will be awarded a trophy if you get at least 5 correct and you do this activity online.

© Transum Mathematics 1997-2020
Scan the QR code below to visit the online version of this activity.

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Description of Levels

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Level 1 - Write out the terms of the series defined by the sigma expression

Level 2 - Determine whether a series is arithmetic or geometric

Level 3 - Evaluate the sums of the series

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

Sequences - A list of resources including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don't yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Example

You may also want to use a calculator to check your working. See Calculator Workout skill ##.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don't yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Practise using the sigma notation to find the sum of various number series This is level 1: write out the terms of the series defined by the sigma expression. The first question has been done as an example. You will be awarded a trophy if you get at least 5 correct and you do this activity online.

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local and national stats

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The profile brings together data from several sources. Details about these and related terminology are given in the definitions section.
All figures are the most recent available.
Under Universal Credit a broader span of claimants are required to look for work than under Jobseeker's Allowance. As Universal Credit Full Service is rolled out in particular areas, the number of people recorded as being on the Claimant Count is therefore likely to rise.

Source: ONS Annual Civil Service Employment Survey

Note: Percentages based on % of total jobs in area that are civil service jobs

Source: ONS Annual Civil Service Employment Survey

Note: Percentages based on % of Total Civil Service Jobs made up of each category

Resident Population

The estimated population of an area includes all those usually resident in the area, whatever their nationality. HM Forces stationed outside the United Kingdom are excluded but foreign forces stationed here are included. Students are taken to be resident at their term-time address.

Labour Supply

Labour supply consists of people who are employed, as well as those people defined as unemployed or economically inactive, who can be considered to be potential labour supply. Information in this section relates to the characteristics of people living in an area.

Most labour supply data comes from the Annual Population Survey (APS). The APS is the largest regular household survey in the United Kingdom. It includes data from the Labour Force Survey (LFS), plus further sample boosts in England, Wales and Scotland. The survey includes data from a sample of around 256,000 people aged 16 and over.

As APS estimates are based on samples, they are subject to sampling variability. This means that if another sample for the same period were drawn, a different estimate might be produced. In general, the larger the number of people in a sample, the smaller the variation between estimates. Estimates for smaller areas such as local authorities are therefore less reliable than those for larger areas such as regions. When the sample size is too small to produce reliable estimates, the estimates are replaced with a #.

Economically Active

Economically active

People who are either in employment or unemployed.

Economic activity rate

People, who are economically active, expressed as a percentage of all people.

In employment

People who did some paid work in the reference week (whether as an employee or self employed); those who had a job that they were temporarily away from (eg, on holiday); those on government-supported training and employment programmes; and those doing unpaid family work.

Employment rate

The number of people in employment expressed as a percentage of all people aged 16-64.

Employees and self employed

The division between employees and self employed is based on survey respondents' own assessment of their employment status. The percentage show the number in each category as a percentage of all people aged 16-64. The sum of employees and self employed will not equal the in employment figure due to the inclusion of those on government-supported training and employment programmes, and those doing unpaid family work in the latter.

Unemployed

Refers to people without a job who were available to start work in the two weeks following their interview and who had either looked for work in the four weeks prior to interview or were waiting to start a job they had already obtained.

Model-based unemployed

As unemployed form a small percentage of the population, the APS unemployed estimates within local authorities are based on very small samples so for many areas would be unreliable. To overcome this ONS has developed a statistical model that provides better estimates of total unemployed for unitary authorities and local authority districts (unemployment estimates for counties are direct survey estimates). Model-based estimates are not produced for male or female unemployed.

The model-based estimate improves on the APS estimate by borrowing strength from the Claimant Count to produce an estimate that is more precise (i.e. has a smaller confidence interval). The number of people measured by the Claimant Count is not itself a measure of unemployment but is strongly correlated with unemployment, and, as it is an administrative count, is known without sampling error. The gain in precision is greatest for areas with smaller sample sizes.

Unemployment rate

Unemployed as a percentage of the economically active population.

Economically inactive

Economically inactive

People who are neither in employment nor unemployed. This group includes, for example, all those who were looking after a home or retired.

Wanting a job

People not in employment who want a job but are not classed as unemployed because they have either not sought work in the last four weeks or are not available to start work.

Not wanting a job

People who are neither in employment nor unemployed and who do not want a job.

Workless Households

Households

A household is defined as a single person, or a group of people living at the same address who have the address as their only or main residence and either share one main meal a day or share living accommodation (or both). For the purposes of this table, estimates only include those households where at least 1 person is aged 16 to 64.

Workless households

Households where no-one aged 16 or over is in employment. These members may be unemployed or economically inactive. Economically inactive members may be unavailable to work because of family commitments, retirement or study, or unable to work through sickness or disability.

Children

Children refers to all children under 16.

Occupation

Occupations are classified according to the Standard Occupation Classification 2010. Descriptions of the job titles included in each code are available in the SOC manuals.

Qualifications

Qualifications data are only be available from the APS for calendar year periods, for example, Jan to Dec 2005. The variables show the total number of people who are qualified at a particular level and above, so data in this table are not additive. Separate figures for each NVQ level are available in the full Annual Population Survey data set (Query data).

The trade apprenticeships are split 50/50 between NVQ level 2 and 3. This follows ONS policy for presenting qualifications data in publications. Separate counts for trade apprenticeships can be obtained from the full APS data set (Query data).

No qualifications

No formal qualifications held.

Other qualifications

includes foreign qualifications and some professional qualifications.

NVQ 1 equivalent

e.g. fewer than 5 GCSEs at grades A-C, foundation GNVQ, NVQ 1, intermediate 1 national qualification (Scotland) or equivalent.

NVQ 2 equivalent

e.g. 5 or more GCSEs at grades A-C, intermediate GNVQ, NVQ 2, intermediate 2 national qualification (Scotland) or equivalent.

NVQ 3 equivalent

e.g. 2 or more A levels, advanced GNVQ, NVQ 3, 2 or more higher or advanced higher national qualifications (Scotland) or equivalent.

NVQ 4 equivalent and above

e.g. HND, Degree and Higher Degree level qualifications or equivalent.

Earnings by Residence

The figures show the median earnings in pounds for employees living in the area who are on adults rates of pay and whose pay was not affected by absence. Figures for earnings come from the Annual Survey of Hours and Earnings (ASHE). The ASHE is based on a 1 per cent sample of employees, information on whose earnings and hours is obtained from employers. The survey does not cover self-employed. Information relates to a pay period in April.

The earnings information collected relates to gross pay before tax, national insurance or other deductions, and excludes payments in kind. It is restricted to earnings relating to the survey pay period and so excludes payments of arrears from another period made during the survey period; any payments due as a result of a pay settlement but not yet paid at the time of the survey will also be excluded.

Out-Of-Work Benefits

Claimant Count (Experimental Statistics)

The Claimant Count is the number of people claiming benefit principally for the reason of being unemployed. This is measured by combining the number of people claiming Jobseeker's Allowance (JSA) and National Insurance credits with the number of people receiving Universal Credit principally for the reason of being unemployed. Claimants declare that they are out of work, capable of, available for and actively seeking work during the week in which the claim is made.

The measure of the number of people receiving Universal Credit principally for the reason of being unemployed is still being developed by the Department for Work and Pensions. Consequently this component of the total Claimant Count does not yet correctly reflect the target population of unemployed claimants and is subject to revisions. For this reason the Claimant Count is currently designated as Experimental Statistics.

The Claimant Count is mostly derived from DWP administrative systems. For various reasons, e.g. a claimant's National Insurance number is not known, a small number of claims have to be dealt with manually. These clerical claims do not have as much detail as the computerised claims and therefore, whilst part of the claimant count by sex table, cannot be included the age breakdown.

Rates by age

Unemployment benefits normally only apply to people aged 18 years and over. They can only be claimed by 16 and 17 year olds in exceptional circumstances. Consequently the counts for this age group are typically very low.

DWP Working-Age Client Group

From August 2017 DWP discontinued this dataset when they changed the way they publish their benefit statistics. The last period of data is the November 2016 figures published in May 2017.

The number of working-age people who are claiming one or more main DWP benefits. The main benefits are: bereavement benefit, carer's allowance, disability living allowance, ESA and incapacity benefit, severe disablement allowance, income support, jobseeker's allowance, and widow's benefit. The age at which women reach State Pension age is gradually increasing from 60 to 65 between April 2010 and April 2020. Throughout this period, only women below State Pension age are counted as working age benefit claimants."

The total count is broken down by statistical groups. These categorise each person according to the main reason why they are claiming benefit. Each client is classified to a single group.

Benefits are arranged hierarchically and claimants are assigned to a group according to the top most benefit they receive. Thus a person who is a lone parent and receives Incapacity Benefit would be classified as incapacity benefits. Consequently, the group lone parent will not contain all lone parents as some will be included in the incapacity benefits group and Job seekers groups.

Main out-of-work benefits consists of the groups: job seekers, ESA and incapacity benefits, lone parents and others on income related benefits.

These groups have been chosen to best represent a count of all those benefit recipients who cannot be in full-time employment as part of their condition of entitlement. Those claiming solely Bereavement Benefits or Disability Living Allowance (DLA) are not included as these are not out-of-work or income based benefits. DLA is paid to those needing help with personal care. These people can, and some will, be in full-time employment. If DLA claimants are also in receipt of JSA, IS, ESA or Incapacity Benefits in addition to DLA they will be counted under the relevant statistical group. In addition, we exclude those claiming solely carer's benefits or claiming carer's benefits alongside income support, as DWP does not pursue active labour market policies for this group. Carers benefits are paid to those with full time caring responsibilities. The group entitled to Carer's benefits alongside Income Support (IS) includes around 86,000 claimants and has been stable over time.

This Nomis series is different to that published in the Office for National Statistics (ONS) Labour Market Bulletin. The Nomis series uses DWP Jobseeker's Allowance numbers, whilst the Labour Market Bulletin uses the Claimant Count, using different methods, coverage and reference periods

Labour Demand

Labour demand includes jobs available within the area.

Jobs Density

The level of jobs per resident aged 16-64. For example, a job density of 1.0 would mean that there is one job for every resident aged 16-64.

The total number of jobs is a workplace-based measure and comprises employee jobs, self-employed, government-supported trainees and HM Forces. The number of residents aged 16-64 figures used to calculate jobs densities are based on the relevant mid-year population estimates.

Employee Jobs

The number of jobs held by employees. Employee jobs excludes self-employed, government-supported trainees and HM Forces, so this count will be smaller than the total jobs figure shown in the Jobs density table. The information comes from the Business Register and Employment Survey (BRES) - an employer survey conducted in September of each year. The BRES records a job at the location of an employee's workplace (rather than at the location of the business's main office).

Full-time and part-time:

In the BRES, part-time employees are those working for 30 or fewer hours per week.

Note

All figures exclude farm-based agriculture

Earnings by Place of Work

The figures show the median earnings in pounds for employees working in the area who are on adults rates of pay and whose pay was not affected by absence. Figures for earnings come from the Annual Survey of Hours and Earnings (ASHE). The ASHE is based on a 1 per cent sample of employees, information on whose earnings and hours is obtained from employers. The survey does not cover self-employed. In 2004 information related to the pay period which included 21 April.

The earnings information collected relates to gross pay before tax, national insurance or other deductions, and excludes payments in kind. It is restricted to earnings relating to the survey pay period and so excludes payments of arrears from another period made during the survey period; any payments due as a result of a pay settlement but not yet paid at the time of the survey will also be excluded.

Annual Civil Service Employment Survey

The Annual Civil Service Employment Survey (ACSES) is based on a census of civil service departments on 31 March. ACSES counts all home Civil Service employees. It excludes the Northern Ireland Civil Service, other Crown servants and employees of the wider public sector. There are home Civil Service employees based in Northern Ireland and Overseas. Headcount statistics are based on the number of employees with an employment contract who are being paid by the organisation. Employees can be permanent, on a fixed-term contract or employed on a temporary basis. The self-employed, contract workers and agency workers are excluded. Employees not on the payroll and not being paid during the reference period are also excluded, for example, those on unpaid maternity leave, unpaid sick absence and career breaks. Full-time employees are those who are contracted to work 37 hours per week (36 hours per week in London for employees employed prior to 2013. Employees in London substantively promoted since 2013, or who have joined the Civil Service subsequent to this date, are now contracted to work 37 hours per week if on a full-time basis). Part-time employees are those who work less than the normal contracted hours.

Note ACSES data is currenly only available for local authorities in England and Wales.

UK Business Counts

The data contained in the table are compiled from an extract taken from the Inter-Departmental Business Register (IDBR) recording the position of units as at March of the reference year. The IDBR contains information on VAT traders and PAYE employers in a statistical register which provides the basis for the Office for National Statistics to conduct surveys of businesses.

The table presents analysis of businesses at both Enterprise and Local Unit level. An Enterprise is the smallest combination of legal units (generally based on VAT and/or PAYE records) which has a certain degree of autonomy within an Enterprise Group. An individual site (for example a factory or shop) in an enterprise is called a local unit.

The employment information on the IDBR is drawn mainly from the Business Register Employment Survey (BRES). Because this is based on a sample of enterprises, estimates from previous returns and from other ONS surveys have also been used. For the smallest units, either PAYE jobs or employment imputed from VAT turnover is used.

Estimates in the table are rounded to prevent disclosure.

Under the terms of the Open Government Licence (OGL) and UK Government Licensing Framework (launched 30 September 2010), anyone wishing to use or re-use ONS material, whether commercially or privately, may do so freely without a specific application for a licence, subject to the conditions of the OGL and the Framework.

These new arrangements replace the previous Click-Use and Value Added Licences. For further information, go to the links above, phone 020 8876 3444 or email psi@nationalarchives.gov.uk

Users should include a source accreditation to ONS:
Source: Office for National Statistics
Copyright of a third party

The permission to reproduce Crown protected material does not extend to any material on this site which is identified as being authored by a third party. Authorisation to reproduce such material must be obtained from the copyright holders concerned.

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Labour Market Profile - Nomis - Official Labour Market Statistics

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The profile brings together data from several sources. Details about these and related terminology are given in the definitions section. All figures are the most recent available. Under Universal Credit a broader span of claimants are required to look for work than under Jobseeker's Allowance.

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Distilling the world’s data, information & knowledge into beautiful infographics & visualizations

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Monty Hall revisited - Timmy, Tammy and the goat

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Monty Hall revisited - Timmy, Tammy and the goat

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systematic listing strategies

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probability - timmy, tammy, goat and car - Monty Hall twist.pptx

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In the past week, the 30th anniversary of the Challenger shuttle disaster has been marked with tributes for the sacrifice of the crew. In the investigation that came after the tragedy, the brilliant physicist Richard Feynman identified a culture at Nasa where risk was not understood, writes mathematician Dr John Moriarty.

The Challenger was lost because one small part - an O-ring seal - failed during a launch in cold weather. The possibility of this part failing had been predicted long before, but Nasa managers chose to ignore the concerns.

The issue of "safety factors" was at the heart of the problem.

Suppose I tell you that a bridge you are about to drive over has a "safety factor of 2". Would you be reassured and put your foot down?

Perhaps you might hit the brake instead, and ask: "A factor of 2 with respect to what"? If the bridge was built twice as strong as is necessary for normal traffic, this sounds fine.

But if this "factor of 2" means something else - that only half of the essential structural components of the bridge will break under normal traffic conditions, you might opt for an alternative crossing.

Before Challenger's final mission - listed as 51-L - it was known that the O-ring seals on a previous flight - 51-C - had eroded to a depth of one-third of their radius. Instead of regarding this little-understood problem as an unacceptable risk, it was interpreted as a "a safety factor of three." Should the interpretation have been more pessimistic?

Pessimists, typically, are not popular people. The rest of us don't want to hear about bad things when they might not even happen.

Risk is about things that haven't actually happened. We can study the past of course, but - as the financial services industry constantly reminds us - past performance is not necessarily a guide to the future.

The thing is, the pessimists are actually right. To prove this we just need a monkey and a typewriter. Given enough time and paper, it's technically possible for a monkey to type out the complete works of Shakespeare. They simply need to press the keys in the right order.

The probability of this is very small of course, but it is not zero. It is possible to prove mathematically that if something - anything - is possible, and if it is tried repeatedly and independently, it is guaranteed to happen eventually.

Not that it is possible, nor even likely - it is guaranteed.

Richard Feynman, the great American physicist, wrote a detailed report on risk following the Space Shuttle Challenger disaster, entitled Personal Observations on the Reliability of the Shuttle.

His closing phrase was powerful: "Nature cannot be fooled."

One may argue whether the abstracted world of mathematics really corresponds to nature. When it comes to risk, though, we have just proved a long-term version of Murphy's Law - what can go wrong, will go wrong eventually. Just give it time.

Put like this, risk seems to be a depressing business. Properly understood though, risk is a useful tool - it can be good. If you've ever bought a standby theatre ticket, for example, you have traded a cheaper price for the risk of missing your preferred show and time.

Accepting risk can make our goals achievable. Of course nobody thought that taking humankind into space would be risk-free. The mathematical understanding that we cannot stop bad things happening leads us naturally to a much more important and helpful question - how much risk are we willing to accept?

Much of the reasoning about risk at Nasa effectively took the form that, if disaster hadn't happened yet, it probably wouldn't happen next time either

This is the question asked of us by peddlers of investments. We must think hypothetically - if the value of our shiny new investment was to suddenly halve, they ask, would we be able to sleep at night? When it comes to investments we can retrospectively compare the performance of our chosen instrument relative with other choices we could have made. In general, though, such comparisons are not available.

If the cautionary notes of the pessimists help us to avoid bad things, can they really claim credit because of what didn't happen? How can we possibly prove that being more careful was worthwhile, or worth the extra inconvenience or expense?

According to Feynman, there may have been a shortage of such pessimists at the management levels of Nasa prior to the Challenger tragedy.

In his account, much of the reasoning about risk at Nasa effectively took the form that, if disaster hadn't happened yet, it probably wouldn't happen next time either.

As he points out, we only have to think of a game of Russian roulette to see the problem with that reasoning. Instead Feynman recommended looking for warning signs. In Challenger's case the O-rings were known to corrode and this warning was arguably not given sufficient weight.

Another key criticism was that much of the Shuttle was designed "top down", that is in near-final form, rather than design evolving incrementally - part by part - as engineering usually does.

Richard Feynman 1918-88

Feynman worked on the Manhattan Project, the WW2 project to develop the atomic bomb

He won the 1965 Nobel Prize for physics for his work on quantum physics and inventing Feynman diagrams

Later in his career he made pioneering developments in parallel and quantum computing

The argument here is that Nasa introduced too much innovation all at the same time, thus obscuring problems with individual new components. The implication is that as a result some warning signs were not seen at all.

In the context of the final years of the space race between the US and USSR for supremacy in spaceflight capability, the proper investigation of all warnings was inevitably balanced against speed of progress.

From the perspective of shuttle disasters, a main failure was therefore arguably to address the key question of how much risk was acceptable.

According to Feynman's report this issue was clouded by sharp differences between the risk assessments of engineers "on the ground" and those of managers. While some engineers put the chances of disaster around one in 100, some managers thought them to be closer to one in 100,000.

Given that the space shuttle Columbia was also lost 17 years later, it would appear that the engineers may have been closer to the truth. It stands to reason that the engineers at least had a more informed view of potential warning signs and their implications for risk.

With predictions about the future being so difficult, there is a good argument that nobody ever has an exact picture of risk.

Statistician George Box once said: "All models are wrong but some are useful."

All models are wrong because they merely approximate the truth - this is why events such as flooding, accidents at nuclear reactors, and financial crises seem to happen much more often than they should.

The idea of a good risk model is not to be correct, but to be useful.

Usually we cannot show a claim about future risks to be true or false. And since the event we care about will either happen or not happen, even hindsight may not allow us to properly judge such claims. For this reason some suggest that underplayed models of risk can be very useful for public relations or political purposes.

But as the Challenger disaster showed, this is not real usefulness. To be useful, models of risk must be honest.

They should not simply look backwards and say that nothing bad has happened as yet. They should take into account warnings like climate change, financial products that many trade but few understand, or corroded O-rings.

We may not want to listen to the pessimists, but we should at least talk to the engineers.
John Moriarty is a reader in mathematics at Queen Mary University of London
The Space Shuttle programme

US manned launch vehicle programme, which ran between 1981 and 2011

NASA's space shuttle fleet - Columbia, Challenger, Discovery, Atlantis and Endeavour - flew 135 missions, and helped construct the International Space Station and inspired generations

As well as Challenger, Columbia disintegrated on 1 February 2003 with the loss of seven crew members

The final space shuttle mission (pictured above) took place in July 2011; the remaining shuttles are now on display in various sites across the US

Nasa: The Space Shuttle BBC iWonder: What makes space travel so dangerous? Subscribe to the BBC News Magazine's email newsletter to get articles sent to your inbox.

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Challenger and the misunderstanding of risk

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In the past week, the 30th anniversary of the Challenger shuttle disaster has been marked with tributes for the sacrifice of the crew. In the investigation that came after the tragedy, the brilliant physicist Richard Feynman identified a culture at Nasa where risk was not understood, writes mathematician Dr John Moriarty.

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Integral Edexcel Core Maths resources, plus a bit of other stuff

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Online learning resources, videos to encourage students to take Core Maths...

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BBC Radio 4 - More or Less: Behind the Stats - Downloads

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Podcast downloads for More or Less: Behind the Stats

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Good intro to Core Maths

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Switzerland leads the Big Mac index as of July 2023. The index is a way of measuring whether currencies are over- or undervalued against each other.

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Minimum wage, Boohoo and founder becoming millionaire

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Covid-19 has found its way into the neglected cracks in our economies. In the US, Europe and Asia, poor working conditions in care homes, meat plants and factories have helped to spread the virus. This week, a campaign group blamed clothing sweatshops in the English city of Leicester for contributing to a local surge in cases.

The story of Leicester's garment district is worth knowing, because it reveals something important about how the British economy evolved in the decade between the financial crisis and this pandemic. It also demonstrates what it will take for the country to "build back better", as UK Prime Minister Boris Johnson promised to do this week.

In 2018, I investigated Leicester's clothing industry: a bizarre microeconomy where £4 to £4.50 an hour was the going rate for sewing machinists in many factories (£3 for packers). Tiny sweatshops were crammed into crumbling old buildings and legally compliant factories using expensive machines were being outcompeted by illegally underpaid humans.

It might sound Victorian but the sector is embedded in the 21st-century economy. The big buyers of Leicester's clothes are online "fast fashion" retailers, which have thrived thanks to the proximity and speed of their UK suppliers. Boohoo, which sources about 40 per cent of its clothing in the UK, much of it in Leicester, prospered in lockdown by switching nimbly to producing leisurewear for the homebound, while rivals were left with shipping containers from Asia full of summer party dresses.

Boohoo's co-founder Mahmud Kamani has become a billionaire and is set for a £50m bonus if Boohoo's share price continues to rise. The campaign group Labour Behind the Label has alleged that the some of company's suppliers contributed to the spread of coronavirus; Boohoo says it adhered to all government guidance and that its supply chain adheres to all labour laws, though it does not publish a list of its suppliers.

Leicester's sweatshops, which comprise part but not all of the city's garment sector, were an open secret even before my investigation began. In fact, a senior Whitehall official first told me about them. A local official in Leicester warned me in 2018 that, if I published my story, I would cause mass unemployment for people with no other options. In fact, nothing changed. After publication, I was invited to testify at a parliamentary select committee hearing into the costs of online fast fashion. The government rejected every one of the committee's recommendations.

The bill for that inaction has now come due. As well as the feared harm to public health, Leicester's economy will suffer from a prolonged lockdown and fall further behind more prosperous parts of the UK. So much for "levelling up". In truth, there was always an economic cost to allowing labour exploitation to flourish. The garment factories are a microcosm of Britain's productivity problem. Job creation boomed after the financial crisis and ministers did not worry initially about their quality. They pushed up the minimum wage but left it badly under-enforced, while cutting back health and safety inspections to free up "entrepreneurialism".

Leicester shows that when human labour is too cheap, it doesn't make sense to invest in machines and technology. The UK's low-paid sectors are 30 per cent less productive on average than the same sectors in Belgium, France, Germany and the Netherlands.

As unemployment rises in the months ahead, it will be vital to focus on jobs. But to prioritise quantity while turning a blind eye to quality would be to repeat past mistakes. This week Mr Johnson promised a "New Deal" for Britain to close the "yawning gaps between the best and the rest" in productivity, skills and regional inequality. Leicester is the place to start.

In a world of ever-faster fashion, there is a genuine business case for UK retailers to source a proportion of their clothes at home. Leicester's workers are skilled garment makers and the city's compliant factories (which have invested in some machines) are not prohibitively expensive. If the government devotes enough money and political will to properly enforcing its own labour laws, it can set a level playing field that ensures decent, if low-paid, jobs in a community that needs them. A big push on free local training, especially in English, would give others the chance to move into different work.

Two years ago, I spoke to workers on £4 an hour who had concluded the government did not think they were worth the protection of the law. "The government knows what's happening in Leicester", one said flatly, "and that's it. They don't do anything." This week, Mr Johnson promised to create "a government that is powerful and determined and that puts its arms around people". The fate of Leicester's clothing workers will be the test of whether he meant it.
Sarah O'Connor's 2018 investigation into labour exploitation in Leicester's dark factories is now free to read

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Leicester's dark factories show up a diseased system | Free to read

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Covid-19 has found its way into the neglected cracks in our economies. In the US, Europe and Asia, poor working conditions in care homes, meat plants and factories have helped to spread the virus. This week, a campaign group blamed clothing sweatshops in the English city of Leicester for contributing to a local surge in cases.

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Watch featured video from Africa.

Watch featured video from North America

Watch featured video from South America

Watch featured video from Europe

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Watch featured video from Oceania

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Videos about the 100 People Project, participating schools and global issues that teachers can use to supplement global education curricula.

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Should you trust your memory? The fascinating research of psychologist Elizabeth Loftus shows how unreliable memory can be.

Many criminal court cases rest on eye-witness accounts of what happened. There's a problem though. Elizabeth Loftus - one of the world's most influential psychologists - has shown in numerous experiments that memory is not nearly as reliable as we once believed. It is easy to alter memories. It's even quite easy to implant entirely false memories - making people believe they remember something which never occurred.

Presented by David Edmonds

(Photo: Brain and eraser, Photo credit: Shutterstock)

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BBC World Service - The Big Idea, Memory wars

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Should you trust your memory? The fascinating research of psychologist Elizabeth Loftus shows how unreliable memory can be. Many criminal court cases rest on eye-witness accounts of what happened. There's a problem though. Elizabeth Loftus - one of the world's most influential psychologists - has shown in numerous experiments that memory is not nearly as reliable as we once believed.

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With David Spiegelhalter

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How can we best communicate the risks and the evidence on the most pressing issues of the day - from genetics and nutrition, to climate change and immigration? David Spiegelhalter is joined by the world's top experts to tackle urgent, practical challenges which affect us all.

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Learning from mistakes - using the Fukushima disaster, Eyam etc...

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My podcast, telling true stories about mistakes and what we should learn from them.

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Risk-related stats and covid

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Winton Centre for Risk and Evidence Communication

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Research and data to make progress against the world's largest problems

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the maths behind the fastest person on earth

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the maths behind the fastest person on earth

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percentage change, wind speed as a factor...

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Usain Bolt: breaking records. PA

Who is the fastest man on Earth? Usain Bolt, right? Wrong. The unpopular answer is, in fact, Justin Gatlin. In 2011, he ran the 100 metres in 9.45 seconds, the fastest time a human has ever run that distance, smashing Usain Bolt's best time by a massive 0.13 seconds.

At the time, the controversial US athlete - twice banned from competing for drug-related offences - was being pushed along the track by a huge 20 metre-per-second tail wind (the limit for a time to be ratified as a record is +2m/s). The wind was generated by a number of giant fans as part of a Japanese game show, so the "record" didn't count. Nevertheless, no-one has ever been recorded to run faster from a stationary start on the flat - although some sub-world record times have also been clocked for people running down hill.

Wind assistance improves athletes' performances only in these short sprinting events and some of the field events, such as long jump, triple jump, which require sprints in a single direction. For most other track athletes, wind is anathema.

You wind some, you lose some

When Roger Banister looked out of the window on the morning of May 6 1954, for example, he almost decided to postpone his attempt to break the four-minute mile record because it was too windy. Just before the event, however, the winds dropped enough for him to want to take part and the rest is history. He knew what mathematics can prove: that if you have to do at least one lap of the track, then wind, no matter how light, will always slow you down.

Consider a wind blowing up the home straight of a standard 400 metre athletics track. When Mo Farah runs with the wind, his speed is increased by a set amount, but when he runs against it on the back straight the wind decreases his speed by the same amount. It seems reasonable that these two effects might balance each other out giving Farah the same lap time as if there were no wind at all. Strangely, however, it doesn't work out that way.

The reason, in essence, is that because you run faster down the home straight, you derive the benefit of being pushed by the wind for only a short time. When you run down the back straight, however, you spend longer going slower, fighting against the wind. The difference in the time you spend being assisted by the wind versus the time you spend battling it ensures your lap time always slows. Imagine the extreme example of a wind so strong that it doubles your speed down the home straight. On the back straight, however, it would reduce your speed to zero meaning that you'd never finish the race.

PA

So wind and other adverse weather conditions can act as a leveller, adding to the uncertainty about the results of individual races. Surely, though, whatever the weather, one thing we can be confident about is that the world's fastest sprinter will always be a man? Well, perhaps not. Researchers from the University of Oxford found that, although 100 metres' times for both men and women have been decreasing linearly over the years, the women's time was decreasing at a much faster rate than than the men's. The team concluded that, if trends continue as they have over the last 90 years, women could be dominating the 100 metres by 2156.

However, sports scientists have been critical of the findings and have suggested that increased participation and training opportunities for women over the same time period have led to the artificially rapid reduction in their 100 metre times in comparison to the men's. They argue that now men and women are on a roughly level playing field the decrease in women's 100 metre times will begin to slow to a rate comparable with that of the men. Critics also cite fundamental difference in men's and women's physiology, including oxygen carrying capacity and body fat levels, which suggest that the "fastest person on earth" will never be a woman.

The fastest race

In fact, although billed as the race for the fastest person on Earth, it's questionable whether the 100 metres always produces the fastest performance. When Bolt set two new world records for the 100 metres and 200 metres at the 2008 Bejing Olympics in the bird's nest stadium, his 200 metres took him 19.30 seconds, less than twice the time of his 9.69 second 100 metres. This means that on average, he was running faster in the longer event.

Part of this increase, however, is due to not having to react and accelerate up to speed in the second 100 metres of the 200 metre race. Based on average speed over the whole race, the title of "fastest person on earth" has switched back and forth between 100 and 200 metre runners since the records began.

This effect is even more exaggerated in the 4x100 metre relay in which all but one of the runners begins their 100-metre stretch from a running start. In the last leg, the "anchor" doesn't even have to worry about passing on the baton at the other end so can achieve some incredibly quick times. Several sub-nine second times have been recorded in this leg of the race including Bolt's electronically-timed 8.65 seconds in 2014.

Despite this, the fastest human footspeed was recorded between 60 and 80 metres in Bolt's world record 9.58-second 100 metres in Berlin. He was clocked at 44.64kph or 27.8mph.

So despite Gatlin's "record", the official "fastest man on Earth" title still rests with Bolt - at least for now.

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The maths behind the fastest person on Earth (and no it's not Usain Bolt)

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Who is the fastest man on Earth? Usain Bolt, right? Wrong. The unpopular answer is, in fact, Justin Gatlin. In 2011, he ran the 100 metres in 9.45 seconds, the fastest time a human has ever run that distance, smashing Usain Bolt's best time by a massive 0.13 seconds.

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David McCandless: The beauty of data visualization

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David McCandless turns complex data sets (like worldwide military spending, media buzz, Facebook status updates) into beautiful, simple diagrams that tease out unseen patterns and connections. Good design, he suggests, is the best way to navigate information glut -- and it may just change the way we see the world.

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Australia: How dangerous is climate change?

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Infographic To share with students Australia fires

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https://varndean.adobeconnect.com/_a35016157/p533vwpysfx3/

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getting maximum marks for data analysis questions

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Comparing Data: Starter activities for Core Maths

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Resources on histograms, correlation coefficients, standard deviation, bar charts...

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PROBLEM SOLVING
1 Mathematical Modelling

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The Critical Maths project aimed to build on the ideas of Professor Sir Timothy Gowers to develop a suitable mathematics-based curriculum for post-16 students who had succeeded at GCSE but were not intending to take A level Mathematics.

Background

Research published by the Nuffield Foundation in 2010 showed that levels of participation in maths education post-16 in England were significantly lower than in other developed countries, and reports from the Advisory Committee for Mathematics Education in 2011 highlighted the need for more young people to learn maths to a higher level in order to meet the needs of industry and higher education.

To increase the level of participation, a new curriculum was needed, suitable for students who have achieved a standard pass in GCSE Mathematics.

Funding

The DfE funded MEI to investigate how Professor Sir Timothy Gowers's ideas might inform a curriculum that could become the basis of a new course for students who do not currently study maths post-16.

Activities

Our work included the following aspects:

Consulting with higher education, employer organisations, learned societies and professional bodies across a range of disciplines and an advisory group of experts to develop a suitable curriculum. The curriculum was based on students engaging with realistic problems and developing skills of analysing problems and thinking flexibly to solve them. Careful consideration was given to the kinds of problems which would be appropriate for the students to work on, with a particular emphasis on the kinds of situations which might come up in a conversation with friends.

Developing teaching and learning resources which were trialled in schools and colleges. The purpose of this trialling was to refine the resources to ensure that they worked as intended and were engaging to the target audience.

Developing and trialling a suitable assessment methodology to enable the curriculum to be assessed in a way that encourages the development of transferable mathematical and statistical problem-solving skills. Such problem-solving skills are highly valued by higher education and employers.

Developing a professional development programme to introduce teachers to teaching starting from a context and steering discussions to maximise learning. These approaches were also being used in Japan and in the Netherlands. Professor Gowers stressed the importance of this in his blog.

Access to the classroom-trialled resources produced through this work is available through a guest-login to Integral resources:

"The main point is one I've basically made already: the discussions should start from the real-life problem rather than starting from the mathematics. Pupils should not feel that the question is an excuse to force some mathematics on them: they should be interested in the question and should feel the need for the mathematics, the need arising because one can give much better answers if one models the situation mathematically and analyses the model." Professor Sir Timothy Gowers, Gowers's Weblog

Impact

This curriculum development work, together with the Integrating Mathematical Problem Solving project findings was made available to inform awarding bodies' development of the new Core Maths qualifications, designed for students who have passed GCSE Mathematics but who do not intend to pursue a strongly maths-related discipline in higher education or employment. Core Maths qualifications were taken by over 12,000 students in 2021. The potential cohort is over 250,000 and the long-term aim is that over 100,000 students will take Core Maths qualifications each year.

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Critical Maths: A maths-based thinking curriculum for Level 3 (2012-2015) - MEI

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The Critical Maths project aimed to build on the ideas of Professor Sir Timothy Gowers to develop a suitable mathematics-based curriculum for post-16 students who had succeeded at GCSE but were not intending to take A level Mathematics.

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Looking for ideas for teaching Core Maths? On this page, we have collected together some tasks which we hope you will find useful for teaching different aspects of Core Maths. We have grouped them under the headings from the Core Maths Support Programme.

This page is a work in progress, so please get in touch if you have any feedback or would like to suggest an NRICH task to add to any of the sections.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Tax NI and student loan calculator 2018-2019 bands.xlsx

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April 16, 2021

In week 16 (week ending 23 April) the number of deaths registered in England and Wales was 5.3% below the five-year average (556 fewer deaths). This is the 7th consecutive week there have been fewer deaths than expected. Of the registered deaths registered 260 mentioned COVID-19 in week 16.

See ONS update

Florence Nightingale used a form of radial plots to display data from the Crimean War that demonstrated the death rate amongst soldiers could be reduced by better hygiene practices.

When time-series data is plotted as a normal x-y plot with the time variable along the x-axis it is difficult to identify seasonality or other factors that may influence the data.

Plotting the weekly numbers of deaths using this radial form demonstrates the variability during the first months of the year - For example, deaths from "Aussie flu" in the first weeks of 2018 can be observed. The increase in the number of deaths from the first week in April 2020 can clearly be seen.

Plots up to Dec 25 provided by Irene M. Stratton, Senior Statistician, Gloucestershire Retinal Research Group.
Plots thereafter provided by CEBM
Acknowledgements: Jason Oke and Carl Heneghan commented on earlier versions of this graph

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COVID-19 - Florence Nightingale Diagrams of Deaths in England & Wales - The Centre for Evidence-Based Medicine

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April 16, 2021 In week 16 (week ending 23 April) the number of deaths registered in England and Wales was 5.3% below the five-year average (556 fewer deaths). This is the 7th consecutive week there have been fewer deaths than expected. Of the registered deaths registered 260 mentioned COVID-19 in week 16.

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legacy A level questions - use first bit

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graphical intro

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Here's an idea for introducing exponential functions through the medium of digital bacteria.

Watch this video

In the spirit of Mr Meyer's; Three Act Tasks ask "what's the first question that comes to mind after watching this video?"

A few of many potential questions;

Is this true?
How can this possibly be true?
How many bacterium would there be after 1 hour, 2 hours... 3 hours?
How many bacterium would there be after a week, a month, a year?
Can we write an equation linking number of bacterium to time?
Do bacteria really multiply like this?
If they do then why don't bacteria cover the entire surface of the earth?
Well we can certainly use GeoGebra to help tackle a few of these.

This sketch will enable you and your students to study some different hypothetical models of reproduction;

Starting with the green bacteria, play the animation; a few questions/tasks;

Sketch a graph of the number of bacterium against time (on mini-whiteboards or paper).
Can you write down an equation linking the number of bacterium to time?
How many bacterium do you predict there would be after 28 days?

If students want to re-examine some specific points in time, you (or they) can just drag the time slider back a bit. Once students have had a go at sketching you can then check the green box in the Graphics 2 window to plot up the number of bacteria. Then if you select the Graphics 2 window by clicking in it you can type a proposed equation into the input bar to validate it.

Uncheck the green boxes and repeat the same sort of thing for the hypothetical blue, pink and red bacteria. One point to note is that the petri dish can start to get pretty crowded and the red bacteria start to overlay each other. If all of the bacteria are shown at once, the other coloured bacteria are set-up to sit on top of the red bacteria so that they stand out - you can zoom in to take a closer look at just how dense the red bacteria are as " t" increases.

If students haven't come across exponential functions before they will probably need some help with the last equation. A nice hint to give is;

Ok so we need a more powerful expression but you need no more than the symbols you have already used, how can you rearrange them:

...$2^x$ I here you say, is that going to be different to $x^2$...why?

You might chose to get them to plot these in Desmos or using the GeoGebra Chrome app to make a detailed comparison.

I like the way that both the Petri dish and the graphs really demonstrate how powerful exponential functions are. Once students have had a go at sketching each of the graphs individually, try getting them to put them all on one sketch. You can use the slider in the Graphics 2 panel to change the aspect ratio so that students can see just how insignificant the other functions become relative to the exponential function as "t" increases.

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Here's an idea for introducing exponential functions through the medium of digital bacteria. Watch this video In the spirit of Mr Meyer's; Three Act Tasks ask "what's the first question that comes to mind after watching this video?" A few of many potential questions; Is this true? How can this possibly be true?

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Statistics Blog > Misleading Statistics Examples

In my previous post, I wrote about misleading graphs in real-life. Misleading statistics examples ranged from Fox News' coverage of politics to The Times newspaper's claim that it beat the competition with slightly distorted graphs about circulation. Graphs aren't the only way to distort statistics. In fact, stats are very easy to distort because most people don't understand stats - even "experts!" See Even Physicians Don't Understand Statistics for a run-down on how doctors frequently misread stats about your odds of cancer. Misleading statistics examples are abundant in advertising and in the news. Here are some of the most famous misleading statistics examples...and the most distorted.

Misleading Statistics Examples in Advertising.

80% of dentists will recommend everything.

Anyone remember Colgate's claim that 80% of dentists recommended the brand? You won't be seeing that slogan again, at least not in the UK. Consumers were led to believe that 80% of dentists recommended Colgate while 20% recommended other brands. It turns out that when dentists were surveyed, they could choose several brands - not just one. So other brands could be just as popular as Colgate. This completely misleading statistic was banned by the Advertising Standards Authority.

If it sounds to good to be true...

In 2009 and 2010, Reebok made the following claims about its EasyTone and RunTone shoes: Lab tests "proved" that the shoes work "your hamstrings and calves up to 11% harder and tone your butt up to 28% more than regular sneakers ... just by walking!". The figures turned out to be complete garbage. The FTC stated that Reebok needed to pay a settlement of $25 million for deceptive advertising.

In the News..."

"70 cents of every dollar spent on food stamps goes to bureaucrats" Rep. Michelle Bachmann (R-Minn) (2013)

That quote, from Republican Michelle Bachmann, is completely false. The actual figure is one third of one percent! You can find the real figure here at USDA.gov.

Perhaps the most famous case ever of misleading statistics in the news is the case of Sally Clark, who was convicted of murdering her children. She was freed after it was found the statistics used in her murder trial were completely wrong.

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Misleading Statistics Examples in Advertising and The News - Statistics How To

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Statistics Blog > Misleading Statistics Examples In my previous post, I wrote about misleading graphs in real-life. Misleading statistics examples ranged from Fox News' coverage of politics to The Times newspaper's claim that it beat the competition with slightly distorted graphs about circulation. Graphs aren't the only way to distort statistics.

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Different way of exploring why we need frequency density

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How Florence Nightingale beat Donald Trump's tweets

Last week was the 200th anniversary of the birth of Florence Nightingale. She was of course best known for her work as a nurse during the Crimean War, but she was also admired for her pioneering use of statistics to fight disease.

The diagram that she's most famous for is known as a 'coxcomb', so-called because its jagged appearance is a bit like the comb-like crest on a cockerel's head.

The coxcomb example below left shows the number of deaths in the Crimea and their cause, from April 1854 to March 1855. What stands out is the huge surge in deaths over the winter, from December to February. Even more significant is the fact that the orange area that represents deaths from wounds is dwarfed by the grey area representing death from disease.

Talking of orange...in 2018,the New York Times borrowed the coxcomb idea to produce the chart above right. It shows how often Donald Trump tweeted at different times of the day during the first year of his Presidency. The standout message is that Trump goes into Twitter over-drive soon after he wakes up (when his minders are asleep?), and hardly tweets at all during the middle of the day.

But the New York Times diagram is misleading. Between 7am and 8am, Trump tweeted four times as often as he did at noon. However, the area of the 7am segment is sixteen times bigger than the noon segment, giving us an exaggerated impression of the magnitude of Trump's early morning Twitter storm.

What about Florence Nightingale - are her diagrams just as misleading? No! The genius of Nightingale's coxcomb diagram is that the radial scales were not linear. Instead, she used a square root scale, so that a wedge that's twice as far from the centre represents a number that's four times as big. This means that the areas on the chart are in proportion to the numbers that they represent.

And that is one reason why Florence Nightingale deserved her place as a Fellow of the Royal Statistical Society.

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Last week was the 200th anniversary of the birth of Florence Nightingale. She was of course best known for her work as a nurse during the Crimean War, but she was also admired for her pioneering use of statistics to fight disease.

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Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.

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mathematics in context (level 3 core maths)

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Core Maths Mathematics in Context (Level 3 Core Maths)

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Our Mathematics in Context qualification enables students to improve their knowledge of maths and understand how to apply it in the real world.

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comparing different boards

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GCSE and A level exam questions rewritten with a twist on environmental issues.

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Welcome to the Maths for Planet Earth website. This site is for students and teachers of mathematics.

Our aim is to integrate climate change into the school curriculum, beyond the usual suspects of geography and environmental science.

The world needs brainy mathematicians to help tackle climate change.

All of these questions closely follow example problems from GCSE and A level past exams.

Please use the menus at the top of the site to find problems according to course type, and subject area

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Maths for Planet Earth - Climate Based Maths Questions for Students and Teachers

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Welcome to the Maths for Planet Earth website. This site is for students and teachers of mathematics. Our aim is to integrate climate change into the school curriculum, beyond the usual suspects of geography and environmental science. The world needs brainy mathematicians to help tackle climate change.

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﻿ ﻿ The purpose of this site is to share ideas from the This participatory action research project ran from June 2013 to July 2014 and involved a group of five teacher researchers and myself. Its aim was to develop teaching approaches and ideas which address issues of social justice in the secondary mathematics classroom.
Teaching Maths for Social Justice project.

Some of the ideas from the project have been included in a book published by the ATM.

​Dr Pete Wright, UCL Institute of Education

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Sharing ideas from the Teaching Maths for Social Justice project.

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ill Thursday 13 August 2020 be remembered as a pivotal moment in democracy's relationship with digital technology? Because of the coronavirus outbreak, A-level and GCSE examinations had to be cancelled, leaving education authorities with a choice: give the kids the grades that had been predicted by their teachers, or use an algorithm. They went with the latter.

The outcome was that more than one-third of results in England (35.6%) were downgraded by one grade from the mark issued by teachers. This meant that a lot of pupils didn't get the grades they needed to get to their university of choice. More ominously, the proportion of private-school students receiving A and A* was more than twice as high as the proportion of students at comprehensive schools, underscoring the gross inequality in the British education system.

What happened next was predictable but significant. A lot of teenagers, realising that their life chances had just been screwed by a piece of computer code, took to the streets. "Fuck the algorithm" became a popular slogan. And, in due course, the government caved in and reversed the results - though not before a lot of emotional distress and administrative chaos had been caused. And then Boris Johnson blamed the fiasco on " a mutant algorithm " which, true to form, was a lie. No mutation was involved. The algorithm did what it said on the tin. The only mutation was in the behaviour of the humans affected by its calculations: they revolted against what it did.

And that was a genuine first - the only time I can recall when an algorithmic decision had been challenged in public protests that were powerful enough to prompt a government climbdown. In a world increasingly - and invisibly - regulated by computer code, this uprising might look like a promising precedent. But there are several good reasons, alas, for believing that it might instead be a blip. The nature of algorithms is changing, for one thing; their penetration into everyday life has deepened; and whereas the Ofqual algorithm's grades affected the life chances of an entire generation of young people, the impact of the dominant algorithms in our unregulated future will be felt by isolated individuals in private, making collective responses less likely.

According to the Shorter Oxford Dictionary, the word "algorithm" - meaning "a procedure or set of rules for calculation or problem-solving, now esp with a computer" - dates from the early 19th century, but it's only comparatively recently that it has penetrated everyday discourse. Programming is basically a process of creating new algorithms or adapting existing ones. The title of the first volume, published in 1968, of Donald Knuth's magisterial five-volume The Art of Computer Programming, for example, is "Fundamental Algorithms". So in one way the increasing prevalence of algorithms nowadays simply reflects the ubiquity of computers in our daily lives, especially given that anyone who carries a smartphone is also carrying a small computer.

The Ofqual algorithm that caused the exams furore was a classic example of the genre, in that it was deterministic and intelligible. It was a program designed to do a specific task: to calculate standardised grades for pupils based on information a) from teachers and b) about schools in the absence of actual examination results. It was deterministic in the sense that it did only one thing, and the logic that it implemented - and the kinds of output it would produce - could be understood and predicted by any competent technical expert who was allowed to inspect the code. (In that context, it's interesting that the Royal Statistical Society offered to help with the algorithm but withdrew because it regarded the non-disclosure agreement it would have had to sign as unduly restrictive.)

Classic algorithms are still everywhere in commerce and government (there's one currently causing grief for Boris Johnson because it's recommending allowing more new housing development in Tory constituencies than Labour ones). But they are no longer where the action is.

Since the early 1990s - and the rise of the web in particular - computer scientists (and their employers) have become obsessed with a new genre of algorithms that enable machines to learn from data. The growth of the internet - and the intensive surveillance of users that became an integral part of its dominant business model - started to produce torrents of behavioural data that could be used to train these new kinds of algorithm. Thus was born machine-learning (ML) technology, often referred to as "AI", though this is misleading - ML is basically ingenious algorithms plus big data.

Machine-learning systems are 'uninterpretable'. Which should, in principle, limit their domains of application

Machine-learning algorithms are radically different from their classical forebears. The latter take some input and some logic specified by the programmer and then process the input to produce the output. ML algorithms do not depend on rules defined by human programmers. Instead, they process data in raw form - for example text, emails, documents, social media content, images, voice and video. And instead of being programmed to perform a particular task they are programmed to learn to perform the task. More often than not, the task is to make a prediction or to classify something.

This has the implication that ML systems can produce outputs that their creators could not have envisaged. Which in turn means that they are "uninterpretable" - their effectiveness is limited by the machines' current inability to explain their decisions and actions to human users. They are therefore unsuitable if the need is to understand relationships or causality; they mostly work well where one only needs predictions. Which should, in principle, limit their domains of application - though at the moment, scandalously, it doesn't.

Illustration by Dom McKenzie.

Machine-learning is the tech sensation du jour and the tech giants are deploying it in all their operations. When the Google boss, Sundar Pichai, declares that Google plans to have "AI everywhere", what he means is "ML everywhere". For corporations like his, the attractions of the technology are many and varied. After all, in the past decade, machine learning has enabled self-driving cars, practical speech recognition, more powerful web search, even an improved understanding of the human genome. And lots more.

Because of its ability to make predictions based on observations of past behaviour, ML technology is already so pervasive that most of us encounter it dozens of times a day without realising it. When Netflix or Amazon tell you about interesting movies or goods, that's ML being deployed as a "recommendation engine". When Google suggests other search terms you might consider, or Gmail suggests how the sentence you're composing might end, that's ML at work. When you find unexpected but possibly interesting posts in your Facebook newsfeed, they're there because the ML algorithm that "curates" the feed has learned about your preferences and interests. Likewise for your Twitter feed. When you suddenly wonder how you've managed to spend half an hour scrolling through your Instagram feed, the reason may be that the ML algorithm that curates it knows the kinds of images that grab you.

As we click and click, we are carried along by the exciting sensation of uncovering more secrets and deeper truths

The tech companies extol these services as unqualified public goods. What could possibly be wrong with a technology that learns what its users want and provides it? And at no charge? Quite a lot, as it happens. Take recommendation engines. When you watch a YouTube video you see a list of other videos that might interest you down the right-hand side of the screen. That list has been curated by a machine-learning algorithm that has learned what has interested you in the past, and also knows how long you've spent during those previous viewings (using time spent as a proxy for level of interest). Nobody outside YouTube knows exactly what criteria the algorithm is using to choose recommended videos, but because it's basically an advertising company, one criterion will definitely be: "maximise the amount of time a viewer spends on the site".

In recent years there has been much debate about the impact of such a maximisation strategy. In particular, does it push certain kinds of user towards increasingly extremist content? The answer seems to be that it can. "What we are witnessing," says Zeynep Tufe kci, a prominent internet scholar, "is the computational exploitation of a natural human desire: to look 'behind the curtain', to dig deeper into something that engages us. As we click and click, we are carried along by the exciting sensation of uncovering more secrets and deeper truths. YouTube leads viewers down a rabbit hole of extremism, while Google racks up the ad sales."

What we have also discovered since 2016 is that the micro-targeting enabled by ML algorithms deployed by social media companies has weakened or undermined some of the institutions on which a functioning democracy depends. It has, for example, produced a polluted public sphere in which mis- and disinformation compete with more accurate news. And it has created digital echo-chambers and led people to viral conspiracy theories such as Qanon and malicious content orchestrated by foreign powers and domestic ideologues.

The side-effects of machine-learning within the walled gardens of online platforms are problematic enough, but they become positively pathological when the technology is used in the offline world by companies, government, local authorities, police forces, health services and other public bodies to make decisions that affect the lives of citizens. Who should get what universal benefits? Whose insurance premiums should be heavily weighted? Who should be denied entry to the UK? Whose hip or cancer operation should be fast-tracked? Who should get a loan or a mortgage? Who should be stopped and searched? Whose children should get a place in which primary school? Who should get bail or parole, and who should be denied them? The list of such decisions for which machine-learning solutions are now routinely touted is endless. And the rationale is always the same: more efficient and prompt service; judgments by impartial algorithms rather than prejudiced, tired or fallible humans; value for money in the public sector; and so on.

The overriding problem with this rosy tech "solutionism" is the inescapable, intrinsic flaws of the technology. The way its judgments reflect the biases in the data-sets on which ML systems are trained, for example - which can make the technology an amplifier of inequality, racism or poverty. And on top of that there's its radical inexplicability. If a conventional old-style algorithm denies you a bank loan, its reasoning can be explained by examination of the rules embodied in its computer code. But when a machine-learning algorithm makes a decision, the logic behind its reasoning can be impenetrable, even to the programmer who built the system. So by incorporating ML into our public governance we are effectively laying the foundations of what the legal scholar Frank Pasquale warned against in his 2016 book The Black Box Society.

In theory, the EU's General Data Protection Regulation (GDPR) gives people a right to be given an explanation for an output of an algorithm - though some legal experts are dubious about the practical usefulness of such a "right". Even if it did turn out to be useful, though, the bottom line is that injustices inflicted by a ML system will be experienced by individuals rather than by communities. The one thing machine learning does well is "personalisation". This means that public protests against the personalised inhumanity of the technology are much less likely - which is why last month's demonstrations against the output of the Ofqual algorithm could be a one-off.

In the end the question we have to ask is: why is the Gadarene rush of the tech industry (and its boosters within government) to deploy machine-learning technology - and particularly its facial-recognition capabilities - not a major public policy issue?

The explanation is that for several decades ruling elites in liberal democracies have been mesmerised by what one can only call "tech exceptionalism" - ie the idea that the companies that dominate the industry are somehow different from older kinds of monopolies, and should therefore be exempt from the critical scrutiny that consolidated corporate power would normally attract.

The only consolation is that recent developments in the US and the EU suggest that perhaps this hypnotic regulatory trance may be coming to an end. To hasten our recovery, therefore, a thought experiment might be helpful.

Imagine what it would be like if we gave the pharmaceutical industry the leeway that we currently grant to tech companies. Any smart biochemist working for, say, AstraZeneca, could come up with a strikingly interesting new molecule for, say, curing Alzheimer's. She would then run it past her boss, present the dramatic results of preliminary experiments to a lab seminar after which the company would put it on the market. You only have to think of the Thalidomide scandal to realise why we don't allow that kind of thing. Yet it is exactly what the tech companies are able to do with algorithms that turn out to have serious downsides for society.

What that analogy suggests is that we are still at the stage with tech companies that societies were in the era of patent medicines and snake oil. Or, to put it in a historical frame, we are somewhere between 1906, when the Pure Food and Drug Act was passed by the US Congress, and 1938, the year Congress passed the Federal Food, Drug, and Cosmetic Act, which required that new drugs show safety before selling. Isn't it time we got a move on?
John Naughton chairs the advisory board of the new Minderoo Centre for Technology and Democracy at the University of Cambridge

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From viral conspiracies to exam fiascos, algorithms come with serious side effects

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ill Thursday 13 August 2020 be remembered as a pivotal moment in democracy's relationship with digital technology? Because of the coronavirus outbreak, A-level and GCSE examinations had to be cancelled, leaving education authorities with a choice: give the kids the grades that had been predicted by their teachers, or use an algorithm.

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Please be aware that resources have been published on the website in the form that they were originally supplied. This means that procedures reflect general practice and standards applicable at the time resources were produced and cannot be assumed to be acceptable today. Website users are fully responsible for ensuring that any activity, including practical work, which they carry out is in accordance with current regulations related to health and safety and that an appropriate risk assessment has been carried out.

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The collection of Nuffield Mathematics resources for teaching mathematics at L1, L2 and L3 can be found here: https://www.stem.org.uk/cxg2wb Many of the L2 and L3 resources can be used to support the teaching of the AQA Core Maths qualification L3 Certificate in Mathematical Studies.

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The Advanced Mathematics Support Programme is a government-funded initiative. It is led and delivered by MEI, with Tribal as a key partner.

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How to understand Covid-19 antibody testing in 10 steps

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How to understand Covid-19 antibody testing in 10 steps | Life and style | The Guardian

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1. Jessica Watson , GP and National Institute for Health Research doctoral research fellow 1,

2. Penny F Whiting , associate professor in clinical epidemiology 1,

3. John E Brush , professor of internal medicine 2

1. 1Centre for Academic Primary Care, Bristol Medical School, University of Bristol, Bristol, UK

2. 2Sentara Healthcare and Eastern Virginia Medical School, Norfolk, VA, USA

Interpreting the result of a test for covid-19 depends on two things: the accuracy of the test, and the pre-test probability or estimated risk of disease before testing

A positive RT-PCR test for covid-19 test has more weight than a negative test because of the test's high specificity but moderate sensitivity

A single negative covid-19 test should not be used as a rule-out in patients with strongly suggestive symptoms

Clinicians should share information with patients about the accuracy of covid-19 tests

Correspondence to J Watson Jessica.Watson{at}bristol.ac.uk

What you need to know

Across the world there is a clamour for covid-19 testing, with Tedros Adhanom Ghebreyesus, director general of the World Health Organization, encouraging countries to "test, test, test." 1 The availability of the complete genome of covid-19 early in the epidemic facilitated development of tests to detect viral RNA. 2 Multiple assays with different gene targets have been developed using reverse transcriptase polymerase chain reaction (RT-PCR). 3 These viral RNA tests use samples usually obtained from the respiratory tract by nasopharyngeal swab, to detect current infections. Serology blood tests to detect antibodies indicating past infection are being developed; these will not be considered in depth in this article.

Testing for covid-19 enables infected individuals to be identified and isolated to reduce spread, 4 allows contact tracing for exposed individuals, 5 and provides knowledge of regional and national rates of infection to inform public health interventions. However, questions remain on how to apply test results to make optimal decisions about individual patients.

Search strategy

This article was produced at speed to address an urgent need to address uncertainties in testing for covid-19. We searched Pubmed using the terms "covid", "SARS-CoV-2", "sensitivity", "specificity", "diagnosis", "test", and "PCR", and KSR evidence using terms for covid and test. This was supplemented by discussion with colleagues undertaking formal systematic reviews into covid-19 diagnosis.

How accurate are test results?

No test gives a 100% accurate result; tests need to be evaluated to determine their sensitivity and specificity, ideally by comparison with a "gold standard." The lack of such a clear-cut "gold-standard" for covid-19 testing makes evaluation of test accuracy challenging.

A systematic review of the accuracy of covid-19 tests reported false negative rates of between 2% and 29% (equating to sensitivity of 71-98%), based on negative RT-PCR tests which were positive on repeat testing. 6 The use of repeat RT-PCR testing as gold standard is likely to underestimate the true rate of false negatives, as not all patients in the included studies received repeat testing and those with clinically diagnosed covid-19 were not considered as actually having covid-19. 6

Accuracy of viral RNA swabs in clinical practice varies depending on the site and quality of sampling. In one study, sensitivity of RT-PCR in 205 patients varied, at 93% for broncho-alveolar lavage, 72% for sputum, 63% for nasal swabs, and only 32% for throat swabs. 7 Accuracy is also likely to vary depending on stage of disease 8 and degree of viral multiplication or clearance. 9 Higher sensitivities are reported depending on which gene targets are used, and whether multiple gene tests are used in combination. 3 10 Reported accuracies are much higher for in vitro studies, which measure performance of primers using coronavirus cell culture in carefully controlled conditions. 2

The lack of a clear-cut "gold-standard" is a challenge for evaluating covid-19 tests; pragmatically, clinical adjudication may be the best available "gold standard," based on repeat swabs, history, and contact with patients known to have covid-19, chest radiographs, and computed tomography scans. Inevitably this introduces some incorporation bias, where the test being evaluated forms part of the reference standard, and this would tend to inflate the measured sensitivity of these tests. 11 Disease prevalence can also affect estimates of accuracy: tests developed and evaluated in populations with high prevalence (eg, secondary care) may have lower sensitivity when applied in a lower prevalence setting (eg, primary care). 11

One community based study of 4653 close contacts of patients with covid-19 tested RT-PCR throat swabs every 48 hours during a 14 day quarantine period. Of 129 eventually diagnosed with covid-19 by RT-PCR, 92 (71.3%) had a positive test on the first throat swab, equating to a sensitivity of 71% in this lower prevalence, community setting. 12

Further evidence and independent validation of covid-19 tests are needed. 13 As current studies show marked variation and are likely to overestimate sensitivity, we will use the lower end of current estimates from systematic reviews, 6 with the approximate numbers of 70% for sensitivity and 95% for specificity for illustrative purposes.

What do clinicians need to know to understand a test result?

Sensitivity and specificity can be confusing terms that may be misunderstood 14 (see supplementary file 'Definitions and formulae for calculating measures of test accuracy'). Sensitivity is the proportion of patients with disease who have a positive test, or the true positive rate. Specificity is the proportion of patients without disease who have a negative test, or true negative rate. These terms describe the operating characteristics of a test and can be used to gauge the credibility of a test result. They can be combined to calculate likelihood ratios, which are dimensionless numbers that indicate the strength of a positive or negative test result. 15 For calculating probabilities, a likelihood ratio can be used as a multiplier to convert pre-test odds to post-test odds. Positive likelihood ratios greater than 1 are progressively stronger, with 10 representing a very strong positive test result. Negative likelihood ratios less than 1 are also progressively stronger, with 0.1 representing a very strong negative test result. In the case of the nasopharyngeal swab RNA test for covid-19, the positive likelihood ratio is about 14, which is excellent. 6 A positive covid-19 test result should be very compelling. The negative likelihood ratio is 0.3, which is a moderate result, but not nearly as compelling as a positive result because of the moderate sensitivity (about 70%) of the covid-19 test.

Interpretation of a test result depends not only on the characteristics of the test itself but also on the pre-test probability of disease. Clinicians use a heuristic (a learned mental short cut) called anchoring and adjusting to settle on a pre-test probability (called the anchor). They then adjust this probability based on additional information. This heuristic is a useful short cut but comes with the potential for bias. When people fail to estimate the pre-test probability and only respond to a piece of new information, they commit a fallacy called base-rate neglect. Another fallacy called anchoring is failing adequately to adjust one's probability estimate, given the strength of new information. Likelihood ratios can give a clinician an idea of how much to adjust their probability estimates. Clinicians intuitively use anchoring and adjusting thoughtfully to estimate pre- and post-test probabilities unconsciously in everyday clinical practice. However, faced with a new and unfamiliar disease such as covid-19, mental short cuts can be uncertain and unreliable and public narrative about the definitive nature of testing can skew perceptions.

Figure 1 shows how a clinician's thinking about a patient's probability should shift, based on either a positive or negative test result for covid-19. First, the clinician should estimate a pre-test probability, using knowledge of local rates of covid-19 infection from national 16 and regional 17 data and patients' symptoms and signs, 18 likelihood of alternative diagnoses, and history of exposure to covid-19. After choosing a pre-test probability on the x axis, one should then trace up to either the upper curve for a positive test result or the lower curve for a negative test result, then trace over to the y axis to read the estimate for post-test probability. The figure shows that the shift in the probability is asymmetric, with a positive test result having a greater impact than a negative test result, owing to the modest sensitivity and negative likelihood ratio of the RNA test.

The infographic ( fig 2) shows the outcomes when 100 people with a pre-test probability of 80% are tested for covid-19 using natural frequencies, which are generally easier to understand. Online calculators are available which allow clinicians to adjust pre-test probability, sensitivity, and specificity to estimate post-test probability 19
A 52 year old general practitioner in London develops a cough, intermittent fever, and malaise. On day 2 of his illness he receives a nasopharyngeal swab test for covid-19, which is reported as negative. His cough and low-grade fever persist but he feels systemically well enough to return to work. What should he do?
What else should clinicians consider when interpreting test results?

A single negative test result may not be informative if the pre-test probability is high
A 73 year old woman with severe chronic obstructive pulmonary disease (COPD) and a chronic cough develops acute shortness of breath and slight worsening of her non-productive cough. She reports no fever, has no known exposure to covid-19, and no recent travel. She presents to an emergency department where she is acutely short of breath. A chest radiograph shows possible infiltrates in the right upper and middle lung fields. She is admitted and placed in isolation on droplet precautions. She requires intubation for worsening respiratory distress. Initial nasopharyngeal covid-19 testing is negative. Should she remain in isolation on droplet precautions?
Pre-test probability is high in someone with typical symptoms of covid-19, an occupational risk of exposure, and working in a high prevalence region, and negative test results can therefore be misleading. Table 1 shows that for a pre-test probability of 90%, someone with a negative test has a 74% chance of having covid-19; with two negative tests this risk is still around 47%. If this doctor were to return to work and subsequently the test was confirmed as a false negative, then the decision to work would potentially have significant consequences for his patients, colleagues, and everyone with whom he came into contact. It is therefore safest for this GP with strongly suggestive symptoms to self-isolate in line with guidelines for covid-19, even though his test results are negative. This case illustrates the fallacy of base-rate neglect; it can be tempting to trust the results of an "objective" test more than one's own "subjective" clinical judgement. In general, during this pandemic, pre-test probabilities of covid-19 will be high, particularly in high prevalence secondary care settings.

A possible alternative diagnosis will reduce the pre-test probability

This patient has an alternative possible diagnosis: community-acquired pneumonia. Given her lack of other risk factors or clinical symptoms, and chest radiography findings we therefore estimate her pre-test probability at about 50%. One negative test reduces this risk to 24%, the patient therefore has an additional independently sampled nasopharyngeal swab RNA test which was negative, giving a post-test probability after two negative tests of less than 10%. She is treated with antibiotics and continues to recover.

What are the implications for practice and policy?

While positive tests for covid-19 are clinically useful, negative tests need to be interpreted with caution, taking into account the pre-test probability of disease. This has important implications for clinicians interpreting tests and policymakers designing diagnostic algorithms for covid-19. The Chinese handbook of covid-19 prevention and treatment states " if the nucleic acid test is negative at the beginning, samples should continue to be collected and tested on subsequent days." 20 False negatives carry substantial risks; patients may be moved into non-covid-19 wards leading to spread of hospital acquired covid-19 infection, 21 carers could spread infection to vulnerable dependents, and healthcare workers risk spreading covid-19 to multiple vulnerable individuals. Clear evidence-based guidelines on repeat testing are needed, to reduce the risk of false negatives.

No test is 100% accurate

If your swab test comes back positive for covid-19 then we can be very confident that you do have covid-19

However, people with covid-19 can be missed by these swab tests. If you have strong symptoms of covid-19, it is safest to self-isolate, even if the swab test does not show covid-19

RETURN TO TEXT
Clinicians should ensure that patients are counselled about the limitations of tests ( box 1). Patients with a single negative test but strongly suggestive symptoms of covid-19 should be advised to self-isolate in keeping with guidelines for suspected covid-19.

Possible phrases for explaining covid-19 testing to patients

What is the role of serology tests?

Serology tests, which detect immunoglobulins including IgG and IgM, are under development, 22 23 24 with the aim of detecting individuals who have had previous infection and therefore theoretically developed immunity. The time course and accuracy of serology tests are still under investigation, but the same principles of incorporating the test result with the clinical impression applies. False positive serology tests could cause false reassurance, behaviour change, and disease spread. If suitable accuracy can be established, the benefits of these antibody tests include establishing when healthcare workers are immune, helping to inform decisions about the lifting of lockdowns, and allowing the population to return to work. 25

The WHO message "test, test, test" 1 is important from a population perspective; low sensitivity can be accounted for when assessing burden of disease. However RT-PCR tests have limitations when used to guide decision making for individual patients. Positive tests can be useful to "rule-in" covid-19, a negative swab test cannot be considered definitive for "ruling out."

What is the protocol for covid-19 testing in your organisation?

How do you explain covid-19 test results to patients?

Reflect on a recent clinical case of suspected covid-19-what was your estimated pre-test probability? How did this alter with the results of tests?

How patients were involved in the creation of this article

Author contributorship: JW JB and PW contributed to the conception of the work, JW ran the searches and wrote the first draft of the paper with assistance from JB. PW developed the tools for fig 2. JB, JW, and PW all contributed to the revised drafts of the paper and approved the final version for submission.

Acknowledgments: The authors would like to acknowledge Jon Deeks for helpful discussions at an early point in writing this article and Richard Lehman for suggestions and comments on a draft of this article.

Competing interests The BMJ has judged that there are no disqualifying financial ties to commercial companies. The authors declare the following other interests: JB has given Grand Rounds talks on medical reasoning and has published a book The Science of the Art of Medicine: A Guide to Medical Reasoning for which he receives royalties. JW has no competing interests to declare.

Further details of The BMJ policy on financial interests are here: https://www.bmj.com/about-bmj/resources-authors/forms-policies-and-checklists/declaration-competing-interests

Funding: JW is funded by a doctoral research fellowship from the National Institute for Health Research. The views expressed in this publication are those of the authors and not necessarily those of the NHS, the National Institute for Health Research, Health Education England, or the Department of Health.

Patient consent: The cases in this article are fictitious and therefore no consent was needed.

Provenance and peer review: Commissioned, based on an idea from the author; externally peer reviewed.

Patients with covid-19 or possible covid-19 were not involved in the writing of this paper for practical reasons

Education into practice

Footnotes

This article is made freely available for use in accordance with BMJ's website terms and conditions for the duration of the covid-19 pandemic or until otherwise determined by BMJ. You may use, download and print the article for any lawful, non-commercial purpose (including text and data mining) provided that all copyright notices and trade marks are retained.

https://bmj.com/coronavirus/usage

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What you need to know Across the world there is a clamour for covid-19 testing, with Tedros Adhanom Ghebreyesus, director general of the World Health Organization, encouraging countries to "test, test, test."1 The availability of the complete genome of covid-19 early in the epidemic facilitated development of tests to detect viral RNA.2 Multiple assays with different gene targets have been developed using reverse transcriptase polymerase chain reaction (RT-PCR).3 These viral RNA tests use samples usually obtained from the respiratory tract by nasopharyngeal swab, to detect current infections.

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Test results - sensitivity and specificity

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Test results - sensitivity and specificity

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Are doctors confused by statistics? A new book by one prominent statistician says they are - and that this makes it hard for patients to make informed decisions about treatment.

In 1992, shortly after Gerd Gigerenzer moved to Chicago, he took his six-year-old daughter to the dentist. She didn't have toothache, but he thought it was about time she got acquainted with the routine of sitting in the big reclining chair and being prodded with pointy objects.

The clinic had other ideas. "The dentist wanted to X-ray her," Gigerenzer recalls. "I told first the nurse, and then him, that she had no pains and I wanted him to do a clinical examination, not an X-ray."

These words went down as well as a gulp of dental mouthwash. The dentist argued that he might miss something if he didn't perform an X-ray, and Gigerenzer would be responsible.

But the advice of the US Food and Drug Administration is not to use X-rays to screen for problems before a regular examination. Gigerenzer asked him: "Could you please tell me what's known about the potential harms of dental X-rays for children? For instance, thyroid and brain cancer? Or give me a reference so I can check the evidence?"

The dentist stared at him blankly.

Gigerenzer, director of the Harding Center for Risk Literacy in Berlin, is an expert in uncertainty and decision-making. His new book, Risk Savvy, takes aim at health professionals for not giving patients the information they need to make choices about healthcare.

But it's not just that doctors and dentists can't reel off the relevant stats for every treatment option. Even when the information is placed in front of them, Gigerenzer says, they often can't make sense of it.

In 2006 and 2007 Gigerenzer gave a series of statistics workshops to more than 1,000 practising gynaecologists, and kicked off every session with the same question:
A 50-year-old woman, no symptoms, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from you whether she has breast cancer for certain or what the chances are. Apart from the screening results, you know nothing else about this woman. How many women who test positive actually have breast cancer? What is the best answer?
Gigerenzer then supplied the assembled doctors with some data about Western women of this age to help them answer his question. (His figures were based on US studies from the 1990s, rounded up or down for simplicity - current stats from Britain's National Health Service are slightly different).

1. The probability that a woman has breast cancer is 1% ("prevalence")

2. If a woman has breast cancer, the probability that she tests positive is 90% ("sensitivity")

3. If a woman does not have breast cancer, the probability that she nevertheless tests positive is 9% ("false alarm rate")

In one session, almost half the group of 160 gynaecologists responded that the woman's chance of having cancer was nine in 10. Only 21% said that the figure was one in 10 - which is the correct answer. That's a worse result than if the doctors had been answering at random.

The fact that 90% of women with breast cancer get a positive result from a mammogram doesn't mean that 90% of women with positive results have breast cancer. The high false alarm rate, combined with the disease's prevalence of 1%, means that roughly nine out of 10 women with a worrying mammogram don't actually have breast cancer.

It's a maths puzzle many of us would struggle with. That's because, Gigerenzer says, setting probabilities out as percentages, although standard practice, is confusing. He campaigns for risks to be expressed using numbers of people instead, and if possible diagrams.

Even so, Gigerenzer says, it's surprising how few specialists understand the risk a woman with a positive mammogram result is facing - and worrying too. "We can only imagine how much anxiety those innumerate doctors instil in women," he says. Research suggests that months after a mammogram false alarm, up to a quarter of women are still affected by the process on a daily basis.

Survival rates are another source of confusion for doctors, not to mention journalists, politicians and patients. These are not, as you might assume, simply the opposite of mortality rates - the proportion of the general population who die from a disease. They describe the health outcomes of people who have been diagnosed with a disease, over a period of time - often five years from the point of diagnosis. They don't tell us about whether patients die from the disease afterwards.

Take prostate cancer. In the US, many men choose to be screened for prostate-specific antigens (PSA) which can be an indicator of the disease. In the UK, it's more common for men to get checked only after they start experiencing problems. Consequently, they are diagnosed with prostate cancer later, and are less likely to survive for five years before dying - but this doesn't mean that more men die.

Moreover, many men have "non-progressive" prostate cancer that will never kill them. While screened American men in this situation are marked as having "survived" cancer, unscreened British men aren't. These two facts explain why five-year survival rates of prostate cancer are much higher in the US than in the UK (99% rather than 81%), while the numbers of deaths every year per 100,000 men are almost the same (23 in the US, 24 in the UK).

One of the Harding Center's diagrams shows that the risk of death is the same whether men are screened for prostate cancer or not:

So when former New York mayor Rudy Giuliani declared in 2007 that someone's chance of surviving prostate cancer in the US was twice that of someone using the "socialised medicine" of Britain's National Health Service, he was wrong. And when, in 1999, there was a furore about Britain's survival rate for colon cancer (at the time 35%) being half that of the US (60%), experts again ignored the fact that that the mortality rate was about the same.

Gigerenzer's research shows just how confused doctors often are about survival and mortality rates. In a survey of 412 doctors in the US he found three-quarters mistakenly believed that higher survival rates meant more lives were saved. He also found more doctors would recommend a test to a patient on the basis of a higher survival rate, than they would on the basis of a lower mortality rate.

Unsurprisingly, patients' misconceptions about health risks are even further off the mark than doctors'. Gigerenzer and his colleagues asked over 10,000 men and women across Europe about the benefits of PSA screening and breast cancer screening respectively. Most overestimated the benefits, with respondents in the UK doing particularly badly - 99% of British men and 96% of British women overestimated the benefit of the tests. (Russians did the best, though Gigerenzer speculates that this is not because they get more good information, but because they get less misleading information.)

A quarter of British women went so far as to guess that 200 women out of every 1,000 screened have their lives saved by mammograms. But Gigerenzer says the real figure is about one woman in 1,000 - four out of every 1,000 screened women die from the disease, as opposed to five out of every 1,000 unscreened women. He says that this benefit has been represented as a "20% mortality reduction", which might explain why many women in the UK seem to think that 20% of women are saved by undergoing the procedure.

Again, the culprit is the use of percentages rather than actual numbers to represent risk and benefit. Simple factboxes - an idea from Dr Lisa Schwartz and Dr Steven Woloshin at Dartmouth Medical School - would help:

Perhaps the most notorious example of patients being misled about risk occurred in October 1995, when the UK's Committee on Safety of Medicines warned doctors that a new, third-generation oral contraceptive pill doubled the risk of thrombosis. Thousands of women came off the pill, even though the risk had merely increased from a one-in-7,000 chance of getting the disease to a two-in-7,000 chance. The following year saw an additional 13,000 abortions in the UK.

So far as doctors' understanding of statistics is concerned, Gigerenzer believes the problem is easily fixed. "It's not a problem of the medical mind. It's a problem of training at the universities, in the medical departments where young doctors are trained in everything except statistical thinking."

But there are other pressures that get in the way of clinicians and patients working together to make decisions about treatment. One trend, evident in some health systems more than others, is for doctors to practise medicine "defensively", recommending treatments that are least likely to leave them open to being sued. Gigerenzer's altercation with his daughter's dentist ended with him signing a waiver form to show that he understood the risks of her not being X-rayed. (That incident may also serve to highlight how financial incentives can be a factor, since the clinic would have earned more money if it had done the X-ray.)

A 2005 survey of 824 US hospital doctors working in fields particularly at risk of litigation, such as obstetrics and neurosurgery, revealed that 93% said they practised defensive medicine. A 2009 study of 250 Swiss doctors found that although about half thought the harms of PSA screening outweighed the benefits, 75% nevertheless recommended the procedure.

"In order to get the doctor out of the defensive medicine mood - where they protect themselves against you as a patient - it's often good to ask doctors not 'What would you recommend?' but, 'If it were your mother or your brother, what would you do?'" says Gigerenzer.

The answer may be very different. A 1993 study found that while the rate of hysterectomy among Swiss women was 16%, among female doctors and doctors' wives it was 10%.

There are three other questions Gigerenzer advises patients to ask doctors to ensure they get all the facts:

"What are the alternatives?"

"What's the benefit and what's the harm?"

"Please tell me this in terms of absolute numbers. If 100 take this medication and 100 people don't, what happens after five years?"

Once they get the answers it is up to them to make up their own mind about treatment, he says.

Find out more

Gerd Gigerenzer and Glyn Elwyn spoke to Health Check on the BBC World Service
Listen to the programme Get the Health Check podcast More from the BBC World Service
Dr Glyn Elwyn at the Dartmouth Center in the US, shares this zeal for shared decision-making, but has found that even educated patients feel uncomfortable asking their doctors too many questions or questioning their recommendations, for fear of being labelled "difficult".

He encourages patients to ask questions in a way that doesn't antagonise doctors or put them on the spot. "Framing it in such a way as, for example: 'I happen to have been doing some research. I know there is a controversy here. You may not know this immediately but could you guide me towards some reading?'"

But if a clinician dodges a question or gets angry, he says, patients should switch doctors.

As for the doctors, Elwyn recommends they come clean when they don't know something, and make use of tools like option grids - which clearly lay out treatment options and their consequences - to work through difficult decisions with patients.

"It's surprising that in the 21st Century, many still think of doctors as Gods and you don't ask God," says Gigerenzer.

"A physician is someone who can help you but also someone you need to challenge in order to get the best treatment."
Gerd Gigerenzer and Glyn Elwyn spoke to Health Check Subscribe to the BBC News Magazine's email newsletter to get articles sent to your inbox. on the BBC World Service. Listen again to the programme on the BBC iPlayer or get the Health Check podcast.

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Are doctors confused by statistics? A new book by one prominent statistician says they are - and that this makes it hard for patients to make informed decisions about treatment. In 1992, shortly after Gerd Gigerenzer moved to Chicago, he took his six-year-old daughter to the dentist.

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This collection of resources is produced by the Core Maths Support Programme to support the implementation of Core Maths. The collection contains a range of activities all designed to enable students to use and apply mathematics in unfamiliar contexts.

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Sensitivity and specificity geogebra file

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Sensitivity and specificity geogebra file

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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Working with pre-release material - Catherine Van Sarloos collection

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Working with pre-release material - Catherine Van Sarloos collection

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2. Number of people in England who had COVID-19

There is some evidence that the number of people in England testing positive has decreased in recent weeks

Our latest estimates indicate that at any given time during the two weeks from 17 May to 30 May 2020, an average of 53,000 people in England had the coronavirus (COVID-19) (95% confidence interval: 25,000 to 99,000). This equates to 0.10% (95% confidence interval: 0.05% to 0.18%) of the population in England. This estimate is based on tests performed on 19,723 people in 9,094 households.

Out of the 19,723 participants' swab tests included in this analysis, 21 individuals in 15 households tested positive for COVID-19. As a household survey, our figures do not include people staying in hospital or care homes. In these settings, rates of COVID-19 infection are likely to be different.

When analysing data for the two most recent non-overlapping 14 day periods, there is some evidence of a decrease in the proportion testing positive (Figure 1).

Figure 1: There is some evidence that the proportion of people testing positive with COVID-19 has decreased in recent weeks

Estimated percentage of the population in England who had the coronavirus (COVID-19), based on tests conducted between 3 May to 16 May 2020 and 17 May to 30 May 2020

Source: Office for National Statistics - COVID-19 Infection Survey

Download this chart

Image .csv .xls

In addition to this analysis, a more complex regression modelling approach confirms that there is a clear downward trend (Figure 2). The significance of this downward trend is shown by the credible intervals, as the lower credible interval in the first week of the study period is higher than the upper credible interval for the most recent week. This modelling is an exploratory analysis and was conducted by our research partners at the University of Oxford and the University of Manchester.

The regression model controls for age, sex and region and allows change over time by week to be anlaysed since 26 April 2020, when the study period began. More information about the methods used in the regression model is available in Section 8: Measuring the data.

Figure 2: New modelling shows the downward trend in those testing positive for COVID-19 is statistically significant

Estimated percentage of the population in England testing positive for the coronavirus (COVID-19) by week since the start of the study, 26 April 2020

Source: Office for National Statistics - COVID-19 Infection Survey

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Image .csv .xls

All estimates are subject to uncertainty, given that a sample is only part of the wider population. The 95% confidence intervals are calculated so that, if we were to repeat this study many times, with many different samples of households, then 95% of the time the confidence intervals would contain the true value that we are seeking to estimate.

More information on how our estimates compare with other sources is available in Section 8: Measuring the data.
Back to table of contents

5. Characteristics of people testing positive for COVID-19

This section includes all individuals who have ever been tested for the coronavirus (COVID-19) as part of the study. It looks at the potential risk factors associated with those who have ever tested positive for COVID-19 at any point in the study, even if they now test negative. We have conducted statistical testing to indicate whether there is any evidence of differences in infection rates for each of the characteristics provided. For information on the statistical testing, see Section 8: Measuring the data.

Previous editions of this bulletin have focused on people who tested positive over the most recent 14-day period. Our estimates for the percentage of people testing positive for COVID-19 are stable and low. Including all those who ever tested positive or never tested positive gives a larger dataset, enabling more accurate analysis of risk factors.

Over the whole study period, an estimated 0.41% (95% confidence interval: 0.33% to 0.51%) of people have ever tested positive for COVID-19. This estimate is unweighted and therefore not necessarily representative of the wider community population in England.

The estimates within this section should not be compared with the overall estimate of the percentage testing positive in England in Section 2: Number of people in England who had COVID-19, as they use different time periods and one is a weighted estimate, while the other is unweighted. In addition, the estimates in this section include everyone who has ever tested positive (including many who have now recovered) and this will always be higher than the proportion testing positive in the last 14 days of the study.

Sex and age groups

Based on test results from those who have ever tested positive over the study period (26 April to 30 May 2020), there is no evidence of differences in the proportions of men or women testing positive for COVID-19.

The black lines on Figure 3 show the confidence interval for the estimated percentage of men and women within the study that have ever been infected with COVID-19, showing the range of values overlap substantially. Statistical testing indicates that there is no evidence of different infection rates between men and women. More information on the statistical testing is available in Section 8: Measuring the data.

The range of values for the black lines on Figure 4 are relatively large across most age groups. Statistical testing indicates that there is not enough evidence to say with confidence that there is a difference in infection rates between age groups in the community.

Patient-facing healthcare or resident-facing social care workers

Of those in our study who reported working in patient-facing healthcare or resident-facing social care roles ¹, 1.87% tested positive for COVID-19 (95% confidence interval: 1.07% to 3.02%). This includes NHS professionals, such as nurses and doctors, as well as social care workers, such as nursing home or home care workers.

By comparison, the percentage of people reporting not working in these types of roles testing positive for COVID-19 was lower at 0.32% (95% confidence interval: 0.26% to 0.44%).

These estimates only include the working age population (those aged between 16 and 74 years old). In previous publications, this analysis included respondents of all ages.

Figure 5: A higher percentage of individuals who report working in patient-facing roles in health or social care tested positive for COVID-19

Estimated percentage of those ever testing positive for the coronavirus (COVID-19) in the study, by healthcare and social care workers and other individuals, England, 26 April to 30 May 2020

Source: Office for National Statistics - COVID-19 Infection Survey

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Working location

Individuals taking part in our study were asked where they are currently working. Rates of infection for COVID-19 appear higher for individuals who work outside the home compared with those who work from home. An estimated 0.72% (95% confidence interval: 0.47% to 1.05%) of individuals who reported working outside of the home tested positive for COVID-19, compared with 0.22% (95% confidence interval: 0.10% to 0.41%) of individuals who reported working at home.

Some individuals reported working both in the home and outside the home. The confidence interval for this group of individuals is large, and statistical testing indicates high uncertainty in these estimates.

Figure 6: Rates of positive tests for COVID-19 appear higher for individuals who work outside the home compared with those who work from home

Estimated percentage of those ever testing positive for the coronavirus (COVID-19) in the study, by working location, England, 26 April to 30 May 2020

Source: Office for National Statistics - COVID-19 Infection Survey

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Symptoms experienced

Individuals taking part in the COVID-19 Infection Survey were asked whether they had experienced a range of possible symptoms ² on the day that they were tested. ³ At this time, 3.62% (95% confidence interval: 2.22% to 5.53%) of people who were experiencing one or more symptoms of COVID-19 tested positive. By comparison, 0.33% (95% confidence interval: 0.25% to 0.41%) of people who were not experiencing any symptoms of COVID-19 at the time of the test tested positive.

The percentage of people testing positive for COVID-19 was higher for those who reported specifically having a cough or fever, or loss of taste or smell, on the day of testing than for any symptoms in general. Of those reporting these specific symptoms, 7.69% (95% confidence interval: 3.90% to 13.35%) also tested positive for COVID-19. This compares with an estimate of 0.36% of those who did not have a cough or fever or loss of taste or smell (95% confidence interval: 0.29% to 0.45%).

Additional analysis we have considered looks at whether people who tested positive also reported symptoms. While those with symptoms are more likely to test positive for COVID-19 than those without symptoms, this analysis shows that out of those who have ever tested positive, the percentage who reported having symptoms at the time of the test was relatively low.

Out of those people that tested positive for COVID-19 over the study period, only 22% (95% confidence interval: 14% to 32%) reported experiencing one or more of the various symptoms at the time of their test. Out of those who reported testing positive, 29% (95% confidence interval: 19% to 40%) reported experiencing symptoms at any point in the period around testing positive. This was at the time of the visit or at either the preceding or following visits.

This analysis is based on 88 individuals in the sample who tested positive for COVID-19. This is a very small denominator, meaning the confidence intervals are wide. Additionally, with such a small number of cases included in this analysis, if any of these are false-positives this would have a large effect on the results.

Notes for: Characteristics of people testing positive for COVID-19

1. We asked individuals to self-report whether they worked in patient-facing healthcare or resident-facing social care roles; where that information was missing or uncertain, we used the other information they gave us about their occupation.

2. The symptoms respondents were asked to report were: fever, muscle ache (myalgia), fatigue (weakness or tiredness), sore throat, cough, shortness of breath, headache, nausea or vomiting, abdominal pain, diarrhoea, and loss of taste or loss of smell.

3. Here we compare symptoms from the first time a person tested positive or, if they have never tested positive in the study, from their most recent test.

Back to table of contents

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Coronavirus (COVID-19) Infection Survey pilot - Office for National Statistics

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2. Number of people in England who had COVID-19 Our latest estimates indicate that at any given time during the two weeks from 17 May to 30 May 2020, an average of 53,000 people in England had the coronavirus (COVID-19) (95% confidence interval: 25,000 to 99,000).

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Explaining probability - sensitivity and specificity in testing

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Explaining probability - sensitivity and specificity in testing

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress. The tools are relevant to any curriculum that seeks to deepen students' understanding of mathematical concepts and develop their ability to apply that knowledge to non-routine problems.

More about the Math Assessment Project

100 lessons for formative assessment, some focused on developing math concepts, others on solving non-routine problems. A Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

A set of 94 exemplar summative assessment tasks to illustrate the range of performance goals required by CCSSM. The tasks come with scoring rubrics and examples of scored student work.

Complete summative test forms and rubrics designed to help teachers and students monitor their progress using a range of task types similar to the 'Tasks' section.

5 Prototype modules that encourage groups of teachers to explore the practical and pedagogical concepts behind the materials, such as formative assessment, collaborative learning and the use of unstructured problems.

The Teaching for Robust Understanding of Mathematics (TRU Math) suite is a set of tools with applications in Professional Development and research based around a framework for characterizing powerful learning environments.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress.

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Slow Reveal Graph is an instructional routine that promotes sensemaking about data. This highly engaging routine uses scaffolded visuals and discourse to help students (in K-12 and beyond) make sense of data. As more and more of the graph is revealed, students refine their interpretation and construct meaning, often in surprising ways.

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The game of Card Sharks is one of the best remembered shows to come out the Goodson-Todman stable. It ran for three years from 1978 - 1981, had a successful revival run from 1986 to 1989, and spawned several international versions, most famous of which is the British version (titled "Play Your Cards Right"), which ran for a total of 13 years.

The game itself was based on the old gambling card game of Acey-Deucey. As such, it is ripe for strategic and mathematical evaluation. While we will no doubt one day look at the proper strategy in the front game, today we're going to take a closer look at the "Money Cards" bonus round. This was one of the most exciting and lucrative bonus rounds around in the 1970's, where a contestant, if armed with enough skill and backbone, could parlay $200 into $28,800. But how should one play the Money Cards to maximize one's winnings? Should a player even play to maximize their winnings?
The Rules
Seven cards are laid out on three rows, as seen in the image to the right. An eighth card is turned face up, and the contestant wagers any or all of their bank on whether the next card will be higher or lower (Aces are high). The contestant is staked $200 at the beginning of the round, and must bet in $50 increments, with a $50 minimum bet. Guessing correctly wins the amount bet, guessing incorrectly (even tying) results in the bet being lost.

After the fourth card is turned over, play proceeds to the second row and the contestant is given another $200. Contestants keep wagering, following the same rules up until the last higher/lower decision on the top row, where the minimum bet becomes half of the player's total. In addition, the player may choose to swap the face-up card before the first, fourth, and seventh decision (the first card on each level) with a new card from the deck.

While some of the rules changed during the run of the show (the ramifications of which we will address later), this set of rules will be the basis of our discussions.
Strategy
Some of the simpler aspects of the game are easy to analyze. We know that in a deck of cards, the "8" card is the midpoint - there are 24 cards lower and 24 cards higher than it. We should thusly wager on the next card being higher when our base card is lower than 8, and lower when the base card is higher than an 8. When the card is an 8, we can count the cards we've seen (not a difficult task in the end game, as almost all cards remain visible when after they are played), and determine whether there are more higher and lower cards in the deck, and predict using this knowledge.

When it comes to choosing whether or not to switch our base card when given the opportunity, we can analyze our chance of success with every base card, as seen in the following table:

Using this data, we can determine that we have on average, a 72.4% chance of calling the next card correctly given a random base card. Therefore, it makes sense to switch our base card when our chances of winning are below that number. Using the table above, we see that we should switch on any card between a 5 and Jack.
Betting Strategy - The Naive Approach
It's clear that we should bet the minimum when we are faced with an 8 that we can't change - barring a large number of lower or higher cards already used, it's going to be a net loser, so we should minimize our losses. But what about the other 12 cards?

For our purposes, let's define a betting strategy as a series of 6 percentages, representing the percentage of your bank that you should bet depending upon your face up card. Why only 6 instead of 12? Because the chances of winning with a 2 are the same as winning with an Ace, the chances of winning with a 3 are the same as a King, and so forth. We only need 6 percentages to cover all twelve possibilities.

For example, the following would be a betting strategy: 30%|50%|70%|80%|90%|100%. These represent the percentage of your bank that you will bet, depending on the face-up card. If you are facing a 7 or a 9, you bet 30% of your bank, the left-most percentage. If you are blessed with an Ace or a 2, you bet the right-most percentage. If a betting strategy's wager falls below the minimum bet allowed, then you would instead bet the minimum.

Let's envision a theoretical contestant, All-In Adrian. He's a smart gambler, so he knows that when you're offered even money on a better than even money proposition, you should bet heavily. In this case, he's going to bet everything on every card, as long as it's not an 8. His betting strategy would be 100%|100%|100%|100%|100%|100%.

Adrian is correct, you should bet heavily when the odds are in your favor. He's found the strategy that will earn him the most money on average. But is that the necessarily the best strategy?
Betting Strategy - The Nuanced Approach
Let's look at another theoretical contestant, Nervous Nick. Nick hates gambling and hates to lose. He's going to do nothing but wager the minimum bet of $50 every time except for the last, where he'll bet the new minimum of half of whatever he has left. His strategy would be written as 0%|0%|0%|0%|0%|0%.

Something funny happens when you compare Nick and Adrian's strategies. On average, Adrian is going to win a heck of a lot more money than Nick. But if you were to look at who would be more likely to have more money after each contestant played one round, Nick would be ahead most of the time.

This speaks to the risk involved in Adrian's strategy. Sure, in the long run, he would come out ahead. But in the meantime, he's going to go bust about 70% of the time, while Nick is guaranteed to walk away from the game with some money. And, unfortunately for Adrian, there is no long run. Most contestants would only get the privilege of playing the Money Cards once or twice before losing the front game and having to leave the show. Adrian's strategy would be great if he could play the game unlimited times, but that's just not the case.
Risk
The story of Nick and Adrian shows that we can't just look at the raw value of a strategy in our analysis; we have to consider risk as well. And that's going to complicate matters. We can't find a strategy that creates a mathematically proven equilibrium between value and risk - such a strategy doesn't exist. As any economist or sociologist would tell you, different people quantify the risk/reward ratio differently. But, just because we can't provide the solution, doesn't mean we can't provide a solution.

What we want to do is analyze every possible betting strategy. We'll look at the expected value of the strategy, and to represent the risk we'll use the standard deviation that the strategy creates. The higher the standard deviation, the more volatile the possible range of values, and the higher the risk. However, analyzing every possible combination of six percentages would be extremely overwhelming. We'll limit the amount of possible strategies by doing two things:

Every percentage must be less than or equal to the next percentage in line. This makes sense - there's no reason why you would ever, say, wager 40% on a 6 but only 30% on a 5.

We'll limit our percentages to increments of 10%. We're already losing precision in our potential betting strategies by being forced to wager in $50 increments, so the precision we lose by limiting ourselves to 10 percent increments is not that much.

Those two rules took our potential number of strategies down to a manageable 8,008. We then ran a Monte Carlo simulator to determine the expected value and standard deviation of each strategy. When looking at the results, we found that we could eliminate a large number of potential strategies from consideration. To see why, we plotted each strategy's expected value and standard deviation onto a scatter chart:

The general shape of the graph forms a curve sloping upwards. In modern portfolio theory, the points on the curve form what is known as the efficient frontier. The points that form this curve are the strategies that offer the highest value for a given level of risk. Strategies that do not form part of this curve are not optimal, as they offer less return for their risk than another strategy would.

Let's take a look at an individual example involving two potential strategies: Strategy A (40%|40%|40%|50%|50%|90%) and Strategy B (0%|0%|30%|50%|60%|100%) Strategy A was determined to have a value of $1,768.28 with a deviation of $1,562.92. Strategy B has a value of $1.807.79 and a deviation of $1,496.47. Since Strategy B has both a higher value and lower risk than the Strategy A, there's never a reason to use Strategy A. Thus, we can discard Strategy A from further analysis.

Removing these inefficient strategies leaves us with a group of 647 viable strategies. Updating our scatter point chart shows us the efficient frontier in clearer detail.

Unfortunately, this is where our search for an "optimal strategy" has to end. Each of these 647 strategies is as viable as the next from a mathematical viewpoint. It only depends on how much risk you wish to undertake in the pursuit of profit.
Personal Strategies
Now, we could leave it here, and allow you to select a strategy based on your personal level of risk, but standard deviation doesn't do a very good job of measuring risk aversion. If a person says "I'm willing to accept a standard deviation of $3,000", what does that really mean? It's clear we have to reframe the question if these results are going to be worth anything.

Luckily, we can use the standard deviation of a distribution to determine how likely a given event is to occur. Given a dollar value, we can find how many standard deviations away from the mean each strategy is, and find the strategy that will give you the highest chance of achieving that dollar value by finding the strategy the lowest number of standard deviations away from the mean.

For example, say you wanted to find the strategy that gave the you highest chance of winning at least the modest sum of $500. Then you'd want to use the strategy 0%|0%|0%|20%|30%|50%, where $500 is -1.012 standard deviations away from the mean, corresponding to an estimated chance of 84.4% of making at least $500. If you wanted to shoot higher and maximize your chances of making $2,500, then the riskier strategy of 50%|90%|100%|100%|100%|100% would give you the best chance.
Rule Changes
The Money Cards underwent several rule changes over the years, each in the favor of the contestants. Two years into the show's run, the rule concerning ties changed. Where before ties were considered losses (meaning it would be possible to lose on an Ace or 2), ties were now considered pushes - no money was gained or lost if the same card came up twice. For one thing, this now meant that all strategies should end in 100% for Aces and 2's, since there was no possible way to lose money. It also would allow contestants to play more conservatively to find the best strategy for minimum values.

When the show was revived in 1986, two major rule changes happened to the Money Cards. Firstly, the amount of money given to the contestant upon reaching the second row was increased to $400, making the top prize a round $32,000. The second big change had to do with when the contestant could change cards. Now, instead of just the first card on each level being changed, you could change any card at any time, but only once per level. This complicated the strategy involving choosing when to change cards, the analysis of which we will leave for another time.
Our Final Answer
In case you want to try these strategies out yourself, we've created a series of charts which will give you the ideal strategy given the minimum amount you want to win for any of the three rule sets. While we can't point to a single strategy and say it's the best strategy to follow, looking at these charts and gauging how risky you would want to be would take you a long way to maximizing your potential earnings.

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Play Your Cards Right Archives - Game Show Theory

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The game of Card Sharks is one of the best remembered shows to come out the Goodson-Todman stable. It ran for three years from 1978 - 1981, had a successful revival run from 1986 to 1989, and spawned several international versions, most famous of which is the British version (titled "Play Your Cards Right"), which ran for a total of 13 years.

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Marbleslides are great

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Desmos offers best-in-class calculators, digital math activities, and curriculum to help every student love math and love learning math.

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Higher Risk of Job Automation in Lower Paying Jobs

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"The Secret Rules of Modern Living Algorithms" - Documentary.

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for probability and risk

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Numeracy

"Numeracy is a proficiency which is developed mainly in Mathematics but also in other subjects. It is more than an ability to do basic arithmetic. It involves developing confidence and competence with numbers and measures. It requires understanding of the number system, a repertoire of mathematical techniques, and an inclination and ability to solve quantitative or spatial problems in a range of contexts. Numeracy also demands understanding of the ways in which data are gathered by counting and measuring, and presented in graphs, diagrams, charts and tables."

Secondary National Strategy, Mathematics at key stage 3

Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths main page links to more activities designed for students in upper Secondary/High school.

If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.

It may be worth remembering that if Transum.org should go offline for whatever reason, there are mirror site at Transum.info that contains most of the resources that are available here on Transum.org.

When planning to use technology in your lesson always have a plan B!

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This game is about predicting whether the next playing card to be turned over will have a higher or lower value than the previous card.