Sxx, Sxy and the product moment correlation coefficient

The S sums

• Sxx = Σx² - (Σx)² / n
• Syy = Σy² - (Σy)² / n
• Sxy = Σxy - (Σx)(Σy) / n

These are in the formula booklet. Show a numerical expression, not just the answer: examiners require it.

The PMCC

r = Sxy / sqrt(Sxx × Syy)

Example: n = 10, Σx = 55, Σy = 420, Σx² = 385, Σxy = 2510, Σy² = 18200.

Sxx = 385 - 55²/10 = 82.5; Sxy = 2510 - 55 × 420/10 = 200; Syy = 18200 - 420²/10 = 560.

r = 200 / sqrt(82.5 × 560) = 0.9305. A common slip is forgetting the square root.

Interpreting r

• r close to 1: strong positive linear correlation.
• r close to -1: strong negative linear correlation.
• r close to 0: little or no linear correlation (there may still be a non-linear relationship).

In an exam, interpret in context: 'there is strong positive correlation, so heavier rabbits tend to have longer ears'. Simply saying 'close to 1' is not enough.

Limitations and coding

r only measures linear correlation, can be distorted by outliers, and does not show causation. Linear coding of x or y (such as x = (p - 100)/5) does not change r, except that multiplying by a negative number changes its sign.

Worked example

For the data above, interpret r in context, where x is hours of sunshine and y is ice creams sold.

1. r = 0.9305, close to 1.
2. So there is strong positive linear correlation.

Answer: Days with more sunshine tend to have higher ice-cream sales.

Check yourself

1. n = 10, Σx = 55, Σx² = 385. Find Sxx.
   Show answer

   82.5. 385 - 3025/10 = 82.5.

2. n = 10, Σx = 55, Σy = 420, Σxy = 2510. Find Sxy.
   Show answer

   200. 2510 - 55 × 420/10 = 2510 - 2310 = 200.

3. n = 10, Σy = 420, Σy² = 18200. Find Syy.
   Show answer

   560. 18200 - 176400/10 = 560.

4. Sxx = 82.5, Syy = 560, Sxy = 200. Find the PMCC, r. Give your answer to 3 significant figures.
   Show answer

   0.930. r = 200/sqrt(82.5 × 560) = 0.930.

5. Sxx = 40, Syy = 90, Sxy = -48. Find the PMCC.
   Show answer

   -0.8. r = -48/sqrt(3600) = -48/60 = -0.8.

6. The PMCC between x and y is 0.84. The x values are coded by p = (x - 100)/5. What is the PMCC between p and y?
   • 0.168
   • 0.84
   • 4.2
   • -0.84
   Show answer

   0.84. Linear coding with a positive scale factor does not change r.

7. The PMCC for a data set is 0.03. What can you conclude?
   • There is no relationship of any kind
   • There is strong positive correlation
   • The variables are independent
   • There is almost no linear correlation, though there may be a non-linear relationship
   Show answer

   There is almost no linear correlation, though there may be a non-linear relationship. r only measures linear association.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1
• Pearson Edexcel IAL Statistics S1 (WST01) examiners' report, October 2024

Last updated 7 November 2026. Spotted a mistake? Tell us.