Distributions and Correlation

1 · The shape of a set of data

Imagine a class of students each timing how long it takes them to solve the same puzzle, to the nearest minute. You get a long list of numbers like 3, 5, 4, 2, 6, 4, … On its own the list tells you very little. So we count how many students took 1 minute, how many took 2 minutes, and so on. That count is called the frequency. (Making a frequency table is covered on the frequency tables page, and drawing the charts on the statistical charts page.)

Draw the frequencies as bars and the data suddenly has a shape. The word for the whole pattern (which values happen, and how often) is the distribution of the data. When you describe a distribution, you talk about three things:

• Centre: roughly where the middle of the data is. We measure it with an average: the mean, the median or the mode. (How to calculate each one is on the averages and range page.)
• Spread: how spread out the values are. At KS3 we measure it with the range = biggest value − smallest value.
• Shape: is it symmetrical (the left half is roughly a mirror image of the right half) or skewed (lopsided: most values bunched up at one end, with a long thin "tail" stretching out to the other end)?

Drag the tops of the bars up and down to make your own distribution, or try the ready-made shapes. Watch the mean (orange) and the median (plum) markers, and read the description underneath.

Things to notice as you play:

• For a symmetrical distribution, the mean and the median sit together in the middle.
• For a skewed distribution, the long tail pulls the mean towards it. The median is much less affected. If the tail stretches to the right (towards big values) we say it is skewed to the right, also called positive skew. If the tail stretches to the left, it is skewed to the left or negative skew.
• The direction of the skew is the direction of the tail, not the direction of the big pile. That catches lots of people out.

2 · Outliers: the value that doesn't fit

An outlier is a value that is much bigger or much smaller than the rest of the data. It sits on its own, far away from everything else. An outlier might be a mistake (someone typed 60 instead of 6) or it might be real but unusual (one friend gets a lot of pocket money).

Here is the weekly pocket money of nine friends. Eight values are fixed. Drag the hollow dot (or tap anywhere along the line) to change the ninth friend's amount, and watch which measures react.

3 · Comparing two distributions

A very common question gives you two sets of data and says "compare them". A good comparison always has two parts:

1. Compare an average (usually the mean or the median): which group is higher or lower on average?
2. Compare the spread (the range): which group is more consistent? Consistent means the values are all close together, so the smaller range is more consistent.

And each part must be in context: say what the numbers mean in real life ("scored higher", "were faster"), not just "the mean is bigger". Pick a pair of data sets, look at the dot plots, then build your two sentences and press Check.

4 · Scatter graphs and correlation

So far each piece of data was one number per person. Sometimes we measure two things about each person or object, like the hours each student revised and their test score. Data in pairs like this is called bivariate data ("bi" means two, "variate" means something that varies).

To see if the two things are linked, plot each pair as one point on a scatter graph: one measurement across, the other up. (How to plot the points is on the statistical charts page.) Then step back and look at the cloud of points. The word for a link between the two is correlation:

• Positive correlation: as one goes up, the other tends to go up too. The points slope uphill from left to right.
• Negative correlation: as one goes up, the other tends to go down. The points slope downhill.
• No correlation: no pattern; the points are scattered all over the place.

Correlation can also be strong (points close to a straight line) or weak (there's a trend, but the points are spread widely around it). Use the buttons and the slider.

Try it: name that correlation

Look at the cloud of points and describe it in two words: strength and direction.

5 · Lines of best fit and estimating

When there is correlation, you can draw a line of best fit: a single straight line, drawn with a ruler, that follows the trend of the points as closely as possible. "By eye" means you judge it by looking; there is no formula to use at KS3. A good line of best fit:

• follows the direction of the trend (uphill for positive, downhill for negative);
• has roughly the same number of points on each side of it, and passes close to as many as possible;
• does not have to go through (0, 0), and does not have to go through any particular point (not even the first or last);
• ignores outliers.

Below is a shop's record of the temperature each day and the number of ice creams it sold. Drag the two orange handles to place your line. The feedback checks your line against the rules above.

To read an estimate: start at the value you know on one axis, go straight to the line, then straight across (or down) to the other axis and read the value. Using the line inside the range of the data you collected is called interpolation, and it is fairly reliable. Using it outside that range is called extrapolation, and it is not reliable: the pattern may not carry on. This shop only measured days between 14 °C and 30 °C. Who knows what happens at 0 °C (the line says a negative number of ice creams!) or at 45 °C?

6 · Correlation does not mean causation

On hot days, ice cream sales go up. On hot days, the number of people with sunburn goes up too. So ice cream sales and sunburn have a strong positive correlation. Does eating ice cream give you sunburn? Of course not! Both are caused by a hidden third factor: hot, sunny weather.

Causation means one thing actually makes the other happen. Correlation only tells you that two things move together. They might move together because:

• one really does cause the other (more time in the sun → more sunburn);
• a third factor causes both (hot weather → ice creams and sunburn);
• it's a coincidence (two things that both happened to rise over the same years).

Sort each card. Does the first thing directly cause the second, or is it correlation only (a hidden third factor or coincidence)?

7 · Common mistakes with scatter graphs

These are the mistakes examiners see again and again. Pick one to see it drawn (in red), then press Show the fix to see it done properly (in green).

Worked examples

Quick check

What you've learned

• The distribution of a set of data is its whole pattern. Describe it by its centre (an average: mean, median or mode), its spread (the range = biggest − smallest) and its shape.
• Symmetrical: both halves roughly mirror each other, and the mean and median are close. Skewed: bunched at one end with a long tail; the skew is named after the direction of the tail, and the tail pulls the mean towards it.
• An outlier is a value far away from the rest. It changes the range and the mean a lot, the median hardly at all.
• To compare two distributions, write one sentence about an average and one about the spread, both in context: "On average … because the mean is higher/lower; … is more consistent because the range is smaller."
• A scatter graph plots bivariate data as points. Positive correlation: both go up together. Negative: one goes up as the other goes down. None: no pattern. Strong means close to a line; weak means a loose trend. Always describe it in context.
• A line of best fit is one straight ruled line that follows the trend with roughly equal points on each side. It doesn't have to go through (0, 0), and you never join the dots.
• Estimates from the line inside the data (interpolation) are fairly reliable; outside the data (extrapolation) they are not.
• Correlation does not mean causation: two things can rise together because of a hidden third factor or by coincidence.