# data[].attributes.headline
Probability distributions summary & crib sheet

# data[].attributes.subject
Probability distributions summary & crib sheet

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.description
You are viewing this site in an unsupported browser. See our browser support page for more information.

# data[].attributes.attachment_link.entities[].name
Confidence Interval for Population Means

# data[].attributes.attachment_link.entities[].name
Variance Formula and Calculating Statistical Variance

# data[].attributes.attachment_link.title
Population standard deviation | Descriptive statistics | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.entities[].name
Variance Formula and Calculating Statistical Variance

# data[].attributes.attachment_link.entities[].name
Sample Variance and Sample Standard Deviation

# data[].attributes.attachment_link.title
Variance of a population | Descriptive statistics | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.title
Range, variance and standard deviation as measures of dispersion | Khan Academy

# data[].attributes.attachment_link.title
Constructing a box and whisker plot | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.title
Box and whisker plot | Descriptive statistics | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.title
Means and medians of different distributions | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.title
Comparing means of distributions | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.title
Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy

# data[].attributes.attachment_link.title
Probability Venn Diagram Example : ExamSolutions

# data[].attributes.attachment_link.readable_text
What Are Mutually Exclusive Events?

As we grow up, we hear all sorts of sayings and idioms. Some of them seem to make a little sense, and then there are a couple that are so brilliantly simple. Have you ever heard the saying, 'You can't have your cake and eat it, too?' This saying is a perfect way to explain mutually exclusive events. The saying refers to the fact that you cannot both eat your cake and still have it front of you at the same time. Statistically speaking, having your cake, and eating your cake, are mutually exclusive.

Two events are defined to be mutually exclusive if they cannot happen at the same time. In other words, if one event happens, the other event cannot happen. Mutually exclusive events are sometimes referred to as disjointed events.

Examples and Formula

To help understand this definition, it is best to go through a few real-life examples. For simplicity's sake, let's call two events A and B, and we will see a few examples of events that are mutually exclusive and then a few that are not:

A: Today is Saturday.
B: Today is Tuesday.

A: A coin lands on heads.
B: The same coin lands on tails.

A: You have passed your statistics class.
B: You have failed your statistics class.

Notice that all of the pairs of events cannot happen at the same time. It would be impossible. If today is Saturday, today can't possibly also be Tuesday. If a coin lands on heads, that means it did not land on tails. It is also impossible to both pass and fail the same class.

What about this set, though?

A: Today is Saturday.
B: It is raining outside.

A: A coin lands on heads.
B: I rolled a die and it landed on four.

A: You passed your statistics class.
B: You passed your accounting class.

These events might sound similar to the previous ones, but they're not. These can happen at the same time. For instance, it is very possible to have a rainy Saturday, to toss a coin to get heads while rolling a die and getting a four, and to pass both your statistics and accounting classes.

In formulaic form, two events, A and B, can be expressed as mutually exclusive as:

P (A and B) = 0

That is, there is no probability of any overlap between the two events. Another way to think of mutually exclusive event is with this picture:

Mutually Exclusive Events vs. Independent Events

Many people confuse mutually exclusive events with independent events. While independent events have many interesting properties and should be discussed in their own lesson, remember that independent events are two events whose outcomes have no effect on one another. Think about the second set of examples we just talked about. These are all independent events. Knowing that today is Saturday does not affect, whatsoever, the chances of it raining.

# data[].attributes.attachment_link.keywords[]
mutually exclusive in statistics: definition

# data[].attributes.attachment_link.keywords[]
formula & examples - video & lesson transcript | study.com

# data[].attributes.attachment_link.title
Mutually Exclusive in Statistics: Definition, Formula & Examples - Video & Lesson Transcript | Study.com

# data[].attributes.attachment_link.description
As we grow up, we hear all sorts of sayings and idioms. Some of them seem to make a little sense, and then there are a couple that are so brilliantly simple. Have you ever heard the saying, 'You can't have your cake and eat it, too?'

# data[].attributes.headline
Summary , includes explaination of Seasonal Variation

# data[].attributes.subject
Summary , includes explaination of Seasonal Variation

# data[].attributes.attachment_link.description
A secondary school revision resource for GCSE Maths about higher level data handling and probability

# data[].attributes.attachment_link.title
Moving Averages, Trend Line and Seasonal Variation

# data[].attributes.attachment_link.description
A GCSE Statistics help video to go through the main ideas on calculating moving averages for time series data and how to then plot and draw a trend line to then calculate the mean seasonal variation to predict future values.

# data[].attributes.attachment_link.description
This video will allow you to see how the normal distribution can be used to look at the spread of a distribution around the sample mean. It is seen that a lot of natural systems can be described through the normal distribution and many real life applications use it to help describe patterns within their businesses e.g.

# data[].attributes.attachment_link.description
An example of a typical GCSE Statistics Question on using the binomial distribution to calculate the probability of a given event.

# data[].attributes.attachment_link.description
A secondary school revision resource for GCSE Maths about foundation level data representation, line graphs, frequency polygons and stem and leaf diagrams

# data[].attributes.body
Using and comparing population pyramids

# data[].attributes.attachment_link.readable_text
Population structure means the 'make up' or composition of a population. Looking at the population structure of a place shows how the population is divided up between males and females of different age groups.

Population structure is usually shown using a population pyramid. A population pyramid can be drawn up for any area, from a whole continent or country to an individual town, city or village.

The following graphs show the population pyramids of an MEDC (the UK) and an LEDC (Mozambique), for 2000 and in 2025 using projected figures. The left side of each pyramid shows the number of men in each age group, the right side shows the number of women in each age group.

Notice how in the UK 2000 pyramid there is a bulge in the area of the 30-34 and 35-39 age groups, with the numbers thereafter reducing fairly steadily as the ages increase. This matches stage 4 of the demographic transition model.

Compare this to the 2025 pyramid, which would be stage 5 in the model. Here the bulge extends much further, covering the age groups 30-64, with the numbers beginning to reduce significantly only after 64.

Now compare the UK population pyramids with those for Mozambique:

In this graph, notice that in 2000 the 0-4 age group contained the largest number of people, with the numbers thereafter declining steadily as the ages increase. The graph matches stage 1 in the model.

In the second graph, the largest group in Mozambique in 2025 is still the 0-4 age group, but there are nearly as many people in the 5-29 age groups. Now the population pyramid matches stage 2.

# data[].attributes.attachment_link.title
Population structure and population pyramids

# data[].attributes.attachment_link.description
Population structure means the 'make up' or composition of a population. Looking at the population structure of a place shows how the population is divided up between males and females of different age groups. Population structure is usually shown using a population pyramid.

# data[].attributes.attachment_link.title
Compare histograms using center and spread

# data[].attributes.attachment_link.description
In this lesson you will learn how to analyze two histograms by comparing the center and spread of the data sets.

# data[].attributes.body
-Drawing a histogram
-Using a histogram to complete the table
-Using the table to complete the histogram

# data[].attributes.attachment_link.title
9-1 GCSE Maths - Histograms - Unequal Class Intervals - Frequency Density -Higher

# data[].attributes.body
FYI:
Skewed right = Positive Skew
Skewed left = Negative Skew

# data[].attributes.attachment_link.title
Shapes of Distributions - MathBitsNotebook(A1 - CCSS Math)

# data[].attributes.attachment_link.description
MathBitsNotebook Algebra 1 CCSS Lessons and Practice is free site for students (and teachers) studying a first year of high school algebra.

# data[].attributes.attachment_link.title
Histograms GCSE Example 4 Finding the median from a Histogram

# data[].attributes.attachment_link.description
This channel is managed by up and coming UK maths teachers. Videos designed for the site by Steve Blades, retired Youtuber and owner of m4ths.com to assist learning in UK classrooms. Designed for the Edexcel spec but applicable to AQA, OCR,MEI and WJEC.

# data[].attributes.body
Calculating Frequency Density and drawing a histogram

# data[].attributes.attachment_link.readable_text
Histograms are best used for large sets of data, especially when the data has been grouped into classes. They look a little similar to bar charts or frequency diagrams.
In histograms, the frequency of the data is shown by the area of the bars and not just the height. Histograms are most commonly used for continuous data.
Histograms often have bars of varying width, i.e. unequal class intervals. This is not a problem as we are dealing with area, not just the height.

The vertical axis of a histogram is labelled frequency density and is calculated by the following formula:
You may be given a question like this:
The ages of people sunbathing on a beach somewhere on a Greek island were recorded and organised into the frequency table below. Draw a histogram of this data.

To draw the histogram we need to calculate the frequency densities for each class interval. By extending the table using 2 extra columns we can easily calculate the frequency densities required. The workings have been shown for the first 2.

All we now need to do is draw this onto graph paper and we have our histogram.

The ages will be on the x-axis (from 0 to 100 on a continuous scale).

Frequency density will be on the y-axis (from 0 to 3).

Cumulative frequency is kind oflike a running total. We add each frequency to the ones before to get an 'at least' total.

These cumulative frequencies ('at least' totals) are plotted against theupper class boundaries to give us a cumulative frequency curve.

The distances a javelin is thrown during a sports day is recorded. The data is organised into the frequency table shown below:
The cumulative frequency column is the column you will be expected to add for yourself.
To draw the cumulative frequency curve we simply plot the cumulative frequencies against the upper end of each class interval.
Remember to always place the cumulative frequencies on the y-axis.
One of the best reasons for drawing a cumulative frequency curve is that it is simple to read off estimates for the median and quartiles even when faced with a large amount of data.

# data[].attributes.attachment_link.description
HistogramsHistograms are best used for large sets of data, especially when the data has been grouped into classes. They look a little similar to bar charts or frequency diagrams. In histograms, the frequency of the data is shown by the area of the bars and not just the height.

# data[].attributes.attachment_link.description
A frequency polygon is another way to show the information in a frequency table. It looks a little bit like a line graph. To make a frequency polygon, you just need to plot a few points and then join the points by straight lines. So what points do you need to plot?

# data[].attributes.body
How to draw one!

# data[].attributes.body
How to draw a cumulative Frequency diagram

# data[].attributes.attachment_link.title
Drawing a Cumulative Frequency Diagram (GCSE Mathematics Handling Data)

# data[].attributes.body
Interquartile Range - In an exam, u may be asked to compare two sets of data using the IQR, & median etc.

# data[].attributes.body
Cumulative Frequency - This MyMaths lesson describes how to make a CF curve very nicely.

# data[].attributes.body
https://www.youtube.com/watch?v=R0teHrZh7T8

# data[].attributes.headline
Stem and Leaf Diagram basic Exam style Questions and Answers

# data[].attributes.subject
Stem and Leaf Diagram basic Exam style Questions and Answers

# data[].attributes.body
http://www.castlefordacademy.com/useful/Grade%20D/27_stem%20and%20leaf.pdf

http://www.mathsgenie.co.uk/resources/49_stem-and-leafans.pdf

# data[].attributes.body
http://www.bbc.co.uk/schools/gcsebitesize/maths/statistics/representingdata1rev2.shtml

# data[].attributes.headline
Finding the y-intercept if the graph doesn't go to zero.

# data[].attributes.subject
Finding the y-intercept if the graph doesn't go to zero.

# data[].attributes.body
You may need to scroll down the page to find the relevant section with an example.

# data[].attributes.attachment_link.description
What's Spearman's Rank? As with most maths, it requires a formula and some prior knowledge. They're likely to ask you about it in the exam, so revise, and learn! In a nutshell, it shows how strong correlation is, from 1 being strong positive, to 0 being no correlation, and -1 being strong negative.

# data[].attributes.attachment_link.title
Scatter Graphs - Correlations - Lines of Best fit - Modelling

# data[].attributes.attachment_link.description
A video to go through the kind of questions asked about scatter graphs, correlation, lines of best fit, finding the equation of the line of best fit and interpreting the line of best fit.

# data[].attributes.headline
Finding the Gradient of a line of best fit.

# data[].attributes.subject
Finding the Gradient of a line of best fit.

# data[].attributes.attachment_link.title
Gradient of a Best Fit Line (GCSE Mathematics Handling Data)

# data[].attributes.attachment_link.description
Uploaded by missjohnstonmaths on 2010-11-01.

# data[].attributes.body
Remember interpolation is always more reliable than extrapolation.

# data[].attributes.attachment_link.title
Interpolation and Extrapolation: Estimating Values from a Graph

# data[].attributes.body
You should always plot the mean point before drawing your line of best fit.

# data[].attributes.body
If a change in one variable causes a direct change in the other variable - we have a causal relationship.

# data[].attributes.body
If points lie approximately on a straight line then there is a linear relationship and we say the variables are associated.

# data[].attributes.attachment_link.title
Spearman's Rank Correlation Coefficient : ExamSolutions Maths Revision

# data[].attributes.attachment_link.description
Go to http://www.examsolutions.net/ for the index, playlists and more maths videos on Spearman's rank correlation coefficient and other maths topics.

# data[].attributes.attachment_link.readable_text
Double bar graphs, also called double bar charts, help us to compare or present more than one kind of information, situations, or events instead of just one by using bars.

What you see below is a double bar chart.Take a close look at it and study it, so you can become familiar with its features

After you have examined the double bar chart above, here are a few good observations you can make from the graph:

The scale is on the left of the graph and it is 10 units.

The title is "Scores on a Fractions Test with or without Preparation"

Preparation helped students to score higher

The lowest score without preparation is 45 and the highest score is 85

The lowest score with preparation is 55 and the highest score is 100.

The students who made the most improvement are Darline, Carla, and John. Their score improved by 15.

The student who made the least improvement is Peter. Although Jetser's score is still slow, he improved by 10 points while Peter's improved only by 5 points

How to construct a double bar graph:

1. Decide what title you will give the graph

2. Decide if you want horizontal or vertical bars

3. Choose a scale

4. Put label on the axes

5. Draw the bars

Example

Use the table below to contruct a double bar chart. We will follow all steps outlined above to construct this graph

1. The title can be clearly seen from the table.

2. We will choose vertical bars

3. Since the scores differ from one another mostly by 5, 10, 15, or 20, it make sense to chose a scale of 10

If the variation between scores were like 1, 2, 3, 4, or 5, it would have been better to choose a scale of 1 or 2

4. We put names on the x axis and scores on the y axis. If we had decided to make horizontal bars, we would have put names on the y axis and scores on the x axis

5. Finally, we draw bars. The double bar graph is shown below:

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

# data[].attributes.attachment_link.description
Double bar graphs, also called double bar charts, help us to compare or present more than one kind of information, situations, or events instead of just one by using bars.

# data[].attributes.attachment_link.title
Cumulative Frequency Step Polygon For Simple Discrete Data - GCSE Statistics Revision - XtremePapers | Advancing knowledge is in our DNA

# data[].attributes.attachment_link.description
We add an extra column to the frequency table called the Cumulative Frequency column. Into this column we start off with the first frequency, i.e. 6. We then add to this the number in the next frequency column. i.e. 6 + 15 = 21.

# data[].attributes.headline
Advantages and disadvantages of each type of diagram

# data[].attributes.subject
Advantages and disadvantages of each type of diagram

# data[].attributes.attachment_link.title
BBC - GCSE Bitesize: Drawing a pie chart

# data[].attributes.attachment_link.description
A secondary school revision resource for GCSE Maths about foundation level data representation, pie charts and frequency diagrams