Scatter diagrams, correlation and the line of best fit by semi-averages

Last time

Before we start: last time, in Price relatives and weighted aggregate index numbers, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Two variables together

Bivariate data pair two measurements for each item, such as the height and the mass of each person. A scatter diagram shows whether they are related. This lesson covers describing correlation, and drawing a line of best fit by the method of semi-averages.

Correlation

Positive correlation: as one variable increases, the other tends to increase. Negative correlation: as one variable increases, the other tends to decrease. No correlation: there is no clear pattern. Strong correlation means the points lie close to a straight line; weak means they are widely scattered about it.

Correlation is not causation

Two variables can be correlated without one causing the other. Ice cream sales and drownings both rise in hot weather: a third variable, temperature, links them. So describe the relationship, and claim a cause only if there is a sensible reason.

The method of semi-averages

Put the data in order of x, and split them into a lower half and an upper half. Find the mean of x and the mean of y for each half: these are the two semi-average points. Plot both points and join them with a straight line: the line of best fit. It also passes through the overall mean point.

Six points, worked

Six points in order of x, split into two halves.

The line of best fit

The six points, with the semi-average points (2, 4) and (5, 8) in blue, joined by the line of best fit. The line passes through the overall mean point, (3.5, 6): a useful check.

The equation of the line

The gradient, b, is the change in y over the change in x between the semi-average points: 8 minus 4, over 5 minus 2, which is 4 over 3, about 1.33. Substitute one point to find a: 4 equals a plus 4 over 3 times 2, so a is 4 over 3, also about 1.33. So the line is y equals 1.33 plus 1.33 x.

Using the line

Read off, or calculate, y for a given x to make an estimate. Estimates within the range of the data, interpolation, are fairly reliable. Estimates outside the range, extrapolation, are unreliable, as the pattern may not continue. Interpret the gradient in context: for example, each extra week adds about 1.33 centimetres.

Mistakes that lose marks

Mistake one: not putting the data in order of x before splitting them into halves. Mistake two: finding the gradient from two of the original points instead of the semi-averages. Mistake three: trusting an extrapolated estimate. The fix: order, halves, means, and a check through the overall mean.

The full-marks tip

The full-marks tip: plot the semi-average points clearly and label them, then check the line passes through the overall mean. Comment on reliability: say whether the estimate is interpolation or extrapolation.

On one card

So, on one card: bivariate data. Keep it for revision.