About this chapter
Correlation and regression is usually the most familiar question on the Statistics 1 paper, and the calculations are well answered. The marks that slip away are in the words: an interpretation with no context, a prediction made in the wrong units, or a regression line used backwards to predict x from y.
This chapter covers the summary statistics Sxx, Syy and Sxy, the product moment correlation coefficient and what it does and does not show, the regression line of y on x, explanatory and response variables, interpreting the gradient and intercept, reliable and unreliable predictions, and the effect of coding. The later questions include a unit conversion, coded data over time, and a data pair that has to be corrected.
The ten questions
- Q1 (5 marks): Sxx, Syy and Sxy from summary totals, then the PMCC.
- Q2 (5 marks): a regression line from Sxx and Sxy, a prediction, and why another prediction is unreliable.
- Q3 (5 marks): interpret a PMCC of 0.902 for revision and test scores, the explanatory variable, and a claim about cause.
- Q4 (5 marks): coded data: the PMCC for the original variables, then the regression line in x and y.
- Q5 (5 marks): a stationery spending model: interpret the gradient, predict in dollars, and explain why the intercept means nothing here.
- Q6 (13 marks): temperature and ice cream sales: summary statistics, the PMCC, a regression line and its use.
- Q7 (11 marks): fish length and mass: the PMCC, show a given regression line, and a prediction for a length given in metres.
- Q8 (13 marks): prices coded by year and value: the PMCC, why coding leaves it unchanged, and a regression line decoded back.
- Q9 (10 marks): age and blood pressure: the PMCC, the regression line, and the increase in age linked to a rise in blood pressure.
- Q10 (12 marks): a data pair recorded wrongly: correct Σy, Σy² and Σxy, then the new PMCC and line.
Key skills tested
Scatter diagrams. Describe the direction and strength of the correlation, naming both variables in context.
Summary statistics. Sxx = Σx² − (Σx)²/n and Sxy = Σxy − (Σx × Σy)/n.
The PMCC. r = Sxy ÷ √(Sxx × Syy), which always lies between −1 and 1. Give it to at least 3 significant figures.
Interpreting r. It measures linear correlation only, and correlation does not show that one variable causes the other.
The regression line. y = a + bx with b = Sxy ÷ Sxx and a = mean of y minus b times the mean of x. Write out the final equation.
Explanatory variable. The explanatory variable x is set or controlled, and the response y depends on it. Only predict y from x.
Interpreting a and b. b is the change in y for each increase of 1 in x, in context and with units. a may be meaningless if x = 0 is impossible.
Predictions. Predictions inside the range of the data (interpolation) are reliable; outside it (extrapolation) they are not. Check the prediction is even possible.
Coding. Coding does not change r. Substitute the codes into the coded line to get the line in the original variables.
Worked example
Question 1 from this chapter. For 10 pairs, Σx = 55, Σy = 320, Σx² = 385, Σy² = 10 650 and Σxy = 1920. (a) Find Sxx, Syy and Sxy. (b) Calculate the product moment correlation coefficient.
(a) Sxx = 385 − 55² ÷ 10 = 385 − 302.5 = 82.5. Syy = 10 650 − 320² ÷ 10 = 10 650 − 10 240 = 410. Sxy = 1920 − 55 × 320 ÷ 10 = 1920 − 1760 = 160.
(b) r = 160 ÷ √(82.5 × 410) = 160 ÷ 183.9 = 0.870.
Keep the full calculator value of the square root, and write r to 3 significant figures only at the end.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q4: summary statistics, the PMCC, showing a given regression line, interpreting the gradient, and a prediction needing metres converted to centimetres, as in Q6 and Q7.
- June 2022, Q2: the PMCC in context, a regression line, a prediction with units to watch, and the change in x linked to a given change in y, as in Q5 and Q9.
Where marks are lost
- No context. In June 2022 a large number just wrote "positive correlation" without naming the two variables.
- Not enough accuracy in a show that. In June 2024 a gradient given as 0.722 was not accurate enough to show the printed line. Give one more decimal place than the value you are shown.
- Units. Not converting 2.5 m into centimetres in June 2024, and substituting 7 000 000 instead of 7 in June 2022, both led to wrong predictions.
- The whole line instead of the gradient. In June 2022, to find the change in x for a given change in y, the rate of change was needed, but most substituted into the full equation.
- Predicting the wrong way. In June 2024 the hardest part went wrong when students used the regression of y on x to predict x.
Common questions
Can I write the regression line with fractions?
No. The June 2022 examiners did not accept fractions in a final regression equation, because a and b are estimates. Use decimals.
Does coding change the PMCC?
No. Adding a constant or multiplying by a positive constant leaves r unchanged.
When is a prediction unreliable?
When it uses a value outside the range of the data, or when the result is impossible, such as a negative length.
Where this chapter leads
- Statistics 1 Chapter 2: Measures of Location and Spread: the same summary totals and coding, used for means and variances.
- Statistics 1 Chapter 3: Representations of Data: spotting and removing anomalies before analysing data.