Testing for zero correlation with the PMCC

Hypotheses and table

H0: ρ = 0, H1: ρ > 0, ρ < 0 or ρ ≠ 0, where ρ is the population PMCC.

Critical values come from the product moment column of the correlation table (one-tailed levels 10%, 5%, 2.5%, 1%, 0.5%; double these for two-tailed tests).

PMCC critical values, one-tailed (Pearson tables)

Example

n = 10, Sxx = 48.4, Syy = 126.9, Sxy = 55.2. r = 55.2/sqrt(48.4 × 126.9) = 0.7043.

H0: ρ = 0, H1: ρ > 0 at 5%: critical value 0.5494. 0.7043 > 0.5494, so reject H0: there is evidence of positive correlation.

Assumptions and choice of test

The PMCC test assumes the data come from a bivariate normal distribution (each variable roughly normal, scatter roughly elliptical). If this is doubtful, or the data are ranks, use Spearman's test instead.

The two tests can give different results: the PMCC detects linear association in the values, Spearman's detects monotonic agreement in the ranks.

Worked example

A sample of 12 pairs gives r = -0.62. Test at the 5% level whether there is correlation (two-tailed).

1. H0: ρ = 0, H1: ρ ≠ 0.
2. Two-tailed 5% uses the 2.5% column: 0.5760.
3. |-0.62| > 0.5760.

Answer: Reject H0: there is evidence of correlation (negative) between the variables.

Check yourself

1. n = 10, Sxx = 48.4, Syy = 126.9 and Sxy = 55.2. Calculate r. Give your answer to 3 significant figures.
   Show answer

   0.704. 55.2/sqrt(48.4 × 126.9) = 0.7043.

2. Find the critical value of r for a one-tailed test at the 5% level with n = 12. Give your answer to 4 decimal places.
   Show answer

   0.4973. From the table: 0.4973.

3. Find the critical value of r for a two-tailed test at the 1% level with n = 20. Give your answer to 4 decimal places.
   Show answer

   0.5614. Two-tailed 1% uses the 0.5% column: 0.5614.

4. n = 10, r = 0.704, H1: ρ > 0 at 5% (critical value 0.5494). What is the conclusion?
   • Reject H0: there is evidence of positive correlation
   • Do not reject H0: there is no evidence of correlation
   • Reject H0: there is evidence of negative correlation
   • The test cannot be done with n = 10
   Show answer

   Reject H0: there is evidence of positive correlation. 0.704 > 0.5494.

5. n = 15, r = -0.47. Test H0: ρ = 0 against H1: ρ < 0 at the 2.5% level. What is the outcome?
   • Reject H0, because 0.47 > 0.4409
   • Reject H0, because r is negative
   • Do not reject H0, because 0.47 < 0.5140
   • Do not reject H0, because r is negative
   Show answer

   Do not reject H0, because 0.47 < 0.5140. The 2.5% one-tailed critical value is 0.5140; -0.47 is not beyond -0.5140.

6. What assumption is needed for the PMCC hypothesis test?
   • The data come from a bivariate normal distribution
   • The data are ranks
   • The sample size is at least 30
   • The correlation is positive
   Show answer

   The data come from a bivariate normal distribution. The PMCC critical values are derived assuming bivariate normality.

7. Exam marks in two subjects look roughly normal and the scatter diagram is elliptical. Which test is appropriate?
   • Spearman's test, because the data are marks
   • A χ² test, because there are two variables
   • The PMCC test, because the data appear bivariate normal
   • No test, because marks cannot be correlated
   Show answer

   The PMCC test, because the data appear bivariate normal. Bivariate normal data suit the PMCC test, which uses the actual values.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S3
• Pearson Edexcel Mathematical Formulae and Statistical Tables

Last updated 8 November 2026. Spotted a mistake? Tell us.