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Teachers: log in to access the following: Slides in PPTX (with click-to-reveal answers) Slides in PDF (one slide per page, suitable for importing into IWB software) Worksheet (with space for student work) Handout (slides with exercises only; 4 per page for reduced printing) Links to past exam questions Teachers: log in to access these.

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S6a - Scatter graphs, correlation and causation

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One fascinating—and disturbing—observation from the 20th century was the link between childhood exposure to lead and crime rates. You’ve probably heard of unleaded petrol but for much of the 20th century, cars ran on petrol with lead additives. Emissions from cars and from other sources caused a rise in lead being ingested by children, and around twenty years later, higher crime rates were observed. As leaded petrol was phased out, crime rates fell. What was particularly interesting is that different parts of the world banned leaded petrol at different times, and crime only around 20 years after the peak of lead emissions did violent crime rates start to fall. Does this strong correlation imply a causation? Here is a BBC article, and below is an Australian documentary on the subject:

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This topic is covered in lessons S4a, S4b, S4c, S4d, S4e and S4f.

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Big data is a buzz phrase across many industries with implications for accountants, engineers of driverless cars, health professionals, producers of online content and advertisers – as raised in this quick overview:

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Visualising lower and upper quartiles

You have come across the median before. The idea of the median is to partition a data set into two halves, but of course this isn't strictly possible if the data set contains an odd number of values. Nevertheless, whether the data set has an odd or even number of values, if listed in order, there will be an equal number of values to the left of the median and to the right. Note that some values to the left and/or right might be equal to the median; but the number of values to the left and right will be equal. You can see this in the applet below. Drag the slider to change the number of values in the data set.

The idea of quartiles is to partition a data set into four quarters. Again, this isn't strictly possible unless the number of values in the set is a multiple of 4. We can however do a pretty good job as follows:

1. Make sure the values are in order, and find the median.

2. Look at the values to the left of the median, and find the median of these. This gives you the lower quartile.

3. Look at the values to the right of the original median, and find the median of these. This gives you the upper quartile.

By doing this, the following will all be equal:

the number of values to the left of the lower quartile,

the number of values between the lower quartile and the median,

the number of values between the median and the upper quartile, and

the number of values to the right of the upper quartile.

Visual representation of boxplots

Suppose the points on the number line below represent the scores (out of 100) achieved by 23 students on a test. Move the points around to see how the boxplot is affected:

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You have come across the median before. The idea of the median is to partition a data set into two halves, but of course this isn't strictly possible if the data set contains an odd number of values.

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Chelsea’s positions in the Premier League from 2012 to 2016 were: 6, 3, 3, 1, 10.
Arsenal’s positions in the Premier League from 2012 to 2016 were: 3, 4, 4, 3, 2.

Which team enjoyed the greater success? Which team suffered the lower dips in performance? What was the range in finishing positions for each of the two teams? Which team would you say was more consistent over the five year period?

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Teachers: log in to access the following:

Slides in PPTX (with click-to-reveal answers)

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Worksheet (with space for student work)

Handout (slides with exercises only; 4 per page for reduced printing)

Links to past exam and UKMT questions

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Teachers: log in to access the following: Slides in PPTX (with click-to-reveal answers) Slides in PDF (one slide per page, suitable for importing into IWB software) Worksheet (with space for student work) Handout (slides with exercises only; 4 per page for reduced printing) Links to past exam and UKMT questions Teachers: log in to access these.

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While you can find the mode of a set of numbers (just like you can find the median and mean), the mode is useful because it makes sense for “nominal data” too. For example, you could find the modal colour of car, or the modal dessert in a restaurant – but you wouldn’t be able to find the median or mean in these cases. Sometimes we don’t need to know all of the information about a dataset. A restaurant may not be interested in the exact numbers of each dessert sold, but might be interested in what is most popular to make sure it doesn’t run out of ingredients for that dessert.