Correlation and regression

Bivariate data and scatter diagrams; describing correlation by its direction (positive/negative) and strength; why correlation does not imply causation; and using a given regression line y = a + bx to make and interpret predictions — including why prediction inside the data range (interpolation) is safe but outside it (extrapolation) is not. Interpreting the PMCC is flagged as A-level.

Statistics · Data presentation

Correlation and regression

Why it works

So far each value stood alone. Bivariate data comes in pairs

(x, y)

— a temperature and the ice-cream sales that day — and the question becomes whether the two move together. Plot each pair as a point on a scatter diagram and the shape of the cloud tells you.

Correlation describes a linear tendency in that cloud, in two parts:

Direction — positive correlation if $y$ tends to rise as $x$ rises (points

trend up to the right); negative if

y

tends to fall as

x

rises.

Strength — strong if the points lie close to a straight line, weak if

they are loosely scattered. If there is no linear pattern, there is no correlation.

Here is a scatter of sales against temperature — a strong positive correlation, with the regression line drawn through it:Correlation is not causation. A correlation between two variables does not mean one causes the other. Ice-cream sales and drownings rise together, but neither causes the other — a third variable, hot weather, drives both. Always ask whether a lurking variable could explain the link.

The regression line. When the points show linear correlation we summarise them with the regression line of $y$ on $x$ , written

y = a + bx,

where

x

is the explanatory (independent) variable — the one you control or that comes first — and

y

is the response (dependent) variable. The line is the "line of best fit" chosen to minimise the total squared vertical distance from the points (least squares); in AS you are given the equation or read it from a calculator, not asked to derive it.

Reading the equation in context. The two numbers carry meaning:

the gradient $b$ is the change in $y$ for each $1$ -unit increase in $x$ (e.g.

3

more sales per

1\,^\circ

C");

the intercept $a$ is the predicted $y$ when $x = 0$ — meaningful only if $x = 0$

is realistic and within the data.

Predicting — interpolation vs extrapolation. To predict, substitute the

x

-value:

Interpolation — predicting for an $x$ inside the range of the data. Reliable,

because the line is supported by nearby points.

Extrapolation — predicting for an $x$ outside the data range. Unreliable: you

have no evidence the linear trend continues, so the estimate can be far off.

A-level only — the product moment correlation coefficient (PMCC). The PMCC,

r

, puts a number on linear correlation:

-1 \le r \le 1.

Its sign gives the direction and its size gives the strength:

r

near

+1

is strong positive, near

-1

strong negative, and near

0

means no linear correlation (there could still be a non-linear relationship).

r

has no units. Calculating

r

is a Year-2 / A-level skill; at AS you describe correlation in words.