Bivariate data, scatter diagrams and types of correlation

35 min

Does revising more pay?

A teacher asks eight S5 pupils how many hours they revised in the week before a test, and records each pupil's mark out of 50 (illustrative data). Each pupil gives a pair of numbers. One list of marks, as in S3 statistics, describes one variable. Pairs let us ask how two variables move together, and whether one can be used to predict the other. That is bivariate statistics, used to relate rainfall and harvests, fertiliser and yield, or advertising and sales.

Definition

Bivariate data are pairs $(x_{i}, y_{i})$ , $i = 1, \dots, n$ , of values of two variables $X$ and $Y$ measured on the same individuals. Usually $X$ is the explanatory (independent) variable and $Y$ the response (dependent) variable.

A scatter diagram (scatter plot) shows each pair as the point $(x_{i}, y_{i})$ on a pair of axes. The mean point is $(\bar{x}, \bar{y})$ , where $\bar{x} = \frac{1}{n}\sum x_i$ and $\bar{y} = \frac{1}{n}\sum y_i$ .

The revision data

Revision time $x$ (hours)	2	3	5	6	8	9	11	12
Test mark $y$ (out of 50)	17	21	24	30	31	33	38	46

Here $\bar{x} = \frac{56}{8} = 7$ and $\bar{y} = \frac{240}{8} = 30$ .

Definition

Correlation is the tendency of two variables to vary together. From a scatter diagram:

positive correlation: $y$ tends to increase as $x$ increases (points rise to the right);
negative correlation: $y$ tends to decrease as $x$ increases (points fall to the right);
no correlation: no trend;
linear correlation when the points lie near a straight line, non-linear when they follow a curve;
strong when the points lie close to the line or curve, weak when they are widely spread, and perfect when they lie exactly on a line.

Worked example 1: describing a scatter diagram

Describe the correlation in the revision data and interpret it.

Solution. The points rise from left to right and lie close to a straight line: a strong positive linear correlation. Pupils who revised longer tended to score higher. The point $(9, 33)$ is a little below the trend and $(12, 46)$ a little above, but no point is far from the others.

Worked example 2: predicting the type of correlation

State the type of correlation you expect, and why, between: (a) the altitude of a place and its average air temperature; (b) a pupil's height and arm span; (c) a pupil's shoe size and mark in a French test.

Solution. (a) Negative: air temperature falls as you climb, which is why Rwanda's high volcano slopes are cooler than the eastern lowlands. (b) Positive and strong: longer bodies tend to have longer arms. (c) None expected: shoe size has nothing to do with French.

Worked example 3: a line of best fit by eye

Draw a line of best fit through the mean point of the revision data and use it to estimate the mark of a pupil who revised 10 hours.

Solution. A good line passes through $(7, 30)$ with about as many points above it as below, for example through $(2, 17)$ and $(12, 43)$ , of gradient $\frac{43 - 17}{12 - 2} = 2.6$ . At $x = 10$ : $y \approx 30 + 2.6 \times 3 \approx 38$ marks. Lesson 4 replaces "by eye" with the regression line, which everyone computes the same way.

Warning

Correlation is not causation. Sales of cold drinks and of umbrellas may both rise in some months without one causing the other; a third variable (the weather, the season) may drive both.
Do not predict far outside the data. The line says nothing about 30 hours of revision: the mark cannot exceed 50.
Choose the axes deliberately: the explanatory variable goes on the horizontal axis.

Tip

Draw the cross at $(\bar{x}, \bar{y})$ first. Every sensible line of best fit passes through it, and the regression lines of lesson 4 always do.

Corbettmaths: Scatter Graphs Correlation (about 2 min). A quick summary of positive, negative and no correlation, as in this lesson's small diagrams.

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Practice

Questions 1 to 3 are routine; 4 to 6 need more care; 7 asks for a judgement.

Name the explanatory and the response variable in: (a) the amount of fertiliser and the maize harvest of a plot; (b) the age of a car and its value.
State the type of correlation you expect between the age of a car and its resale value.
For the pairs $(1, 9), (2, 8), (4, 7), (5, 5), (7, 4), (8, 2)$ , find the mean point and describe the correlation.
Draw a scatter diagram of $(10, 52), (15, 60), (20, 61), (25, 70), (30, 72), (35, 81)$ (illustrative: $x$ = kg of fertiliser per plot, $y$ = kg of beans). Describe the correlation.
For the data of question 4, draw a line of best fit through the mean point and estimate the yield for 28 kg of fertiliser.
The points of a scatter diagram lie close to a U-shaped curve. Is the correlation "none"? Explain.
A study finds that towns with more hospitals have more deaths each year. Does building hospitals cause deaths? Give a better explanation.

Show answersHide answers

(a) Explanatory: fertiliser; response: harvest. (b) Explanatory: age; response: value.
Negative: older cars tend to be worth less.
$\bar{x} = \frac{27}{6} = 4.5$ , $\bar{y} = \frac{35}{6} \approx 5.83$ . Strong negative linear correlation.
The points rise steadily, close to a line: strong positive linear correlation.
Mean point $(22.5, 66)$ . A line through it with gradient about 1.1 gives $y \approx 66 + 1.1 \times 5.5 \approx 72$ kg (answers near 71 to 73 are fine).
No: there is a clear relationship, but it is non-linear. A linear measure of correlation would miss it.
No. Bigger towns have both more hospitals and more people, and so more deaths: population is the hidden third variable.

Sources and credits

Written by Ganzaa: the explanations, the scatter diagrams (plotted from the data), the worked examples, the practice items and the quiz questions. The revision and fertiliser data are illustrative, not real measurements.
Video: Scatter Graphs Correlation, Corbettmaths, embedded from YouTube.

1Does revising more pay?

2The revision data

Does revising more pay?

The revision data