Correlation and regression

Maths: Applications and Interpretation · Statistics and probability. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths: Applications and Interpretation topics

The idea

A scatter diagram shows pairs of values (x, y) as points. If the points lie close to a straight line the two variables have a linear correlation: positive when y rises as x rises, negative when y falls as x rises.

Pearson's r measures how strong the linear relationship is. The regression line of y on x is the line of best fit found by technology, and you use it to predict y from a given x.

Spearman's r_s is Pearson's coefficient worked out on the ranks of the data instead of the values. It measures whether y always increases (or always decreases) as x increases, even along a curve, and it is affected less by outliers. Correlation does not prove that one variable causes the other.

Rules to remember

−1 ≤ r ≤ 1: r near 1 is strong positive, r near −1 is strong negative, r near 0 is little or no linear correlation.
A line of best fit drawn by eye passes through the mean point (x̄, ȳ); so does the regression line.
Regression line y = ax + b: a is the change in y for each increase of 1 in x; b is the value of y when x = 0.
Use the y on x line only to predict y from x, and only for x inside the range of the data (interpolation).
Extrapolation, predicting outside the range of the data, is unreliable because the pattern may not continue.
Spearman's r_s: rank each variable, then find r for the ranks; equal values share the average of their ranks.
r_s = 1 means y always increases when x increases; the points need not lie on a straight line.

Common mistake

Students say that a value of r near 0 means there is no relationship. It only means no linear relationship; the points may follow a clear curve, so look at the scatter diagram.

Worked example

Five students revise for x = 2, 4, 5, 7, 9 hours and score y = 40, 48, 55, 71, 66 marks. Find Spearman's rank correlation coefficient.

Rank x from smallest to largest: 1, 2, 3, 4, 5.
Rank y in the same way: 40 → 1, 48 → 2, 55 → 3, 71 → 5, 66 → 4, giving 1, 2, 3, 5, 4.
Enter the two lists of ranks in the calculator and find r: r_s = 0.9.
Check: the rank differences are 0, 0, 0, −1, 1, so Σd² = 2 and r_s = 1 − (6 × 2)/(5 × (5² − 1)) = 1 − 12/120 = 0.9.

Answer: r_s = 0.9, a strong positive rank correlation.

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Topic 4: Statistics and Probability

Keep going

← Measures of central tendency and dispersion Probability: Venn diagrams, tree diagrams and conditional probability →Take a Maths: Applications and Interpretation test