One and two sample non-parametric tests — Edexcel A-Level Statistics
Test yourself on One and two sample non-parametric tests with PEARSON EDEXCEL A-Level practice questions.
7 days Premium · Then free forever · No card, no charge
One and two sample non-parametric tests explained
Non-parametric tests are used when the population distribution is not assumed normal.
Read the full explanation
For a single sample, the sign test investigates whether the population median equals a hypothesised value by counting how many observations lie above and below that value; the Wilcoxon signed-rank test also uses the sizes of the differences from the hypothesised median, ranking their absolute values and attaching the original signs. For a paired model, you calculate the difference within each pair, then apply the sign test or Wilcoxon signed-rank test to those differences to investigate whether the population median difference is zero. The sign test is quicker and uses only the direction of each difference; the signed-rank test uses both direction and magnitude, so it is more powerful when the differences are roughly symmetric. Both require independent observations and a random sample.
Your focus
- Carry out a sign test to investigate a population median for a single sample or a paired model.
- Carry out a Wilcoxon signed-rank test to investigate a population median for a single sample or a paired model.
- Interpret the results of these tests in context and explain the difference between the sign test and the Wilcoxon signed-rank test.
One and two sample non-parametric tests exam tips
Marking Points
- State the null hypothesis as population median = hypothesised value (single sample) or population median difference = 0 (paired model), and the alternative as one- or two-sided as appropriate.
- For the sign test, count the number of observations (or paired differences) above the hypothesised median and compare with the binomial distribution with n = number of non-zero differences and p = 0.5.
- For the Wilcoxon signed-rank test, calculate differences from the hypothesised median (or paired differences), rank the absolute differences ignoring zeros, assign the original signs to the ranks, and sum the positive ranks (or the smaller of the positive and negative rank sums).
- Compare the test statistic with the critical value from the appropriate table at the chosen significance level, or calculate a p-value, and interpret the result in the context of the original question.
- Recognise that the sign test uses only the direction of differences, while the Wilcoxon signed-rank test uses both direction and magnitude, making it more powerful when the distribution of differences is symmetric.
- Handle ties correctly: discard zero differences in the sign test; in the signed-rank test, assign average ranks to tied absolute differences.
Examiner Tips
- 💡Always define the population median (or median difference) clearly in the context of the question before stating the hypotheses.
- 💡Show your working: for the sign test, state the number of positive and negative differences; for the signed-rank test, show the ranks and the sum of positive ranks.
- 💡Check whether the test should be one-tailed or two-tailed based on the wording of the alternative hypothesis, and use the correct critical value.
- 💡When interpreting the result, relate it back to the original context and avoid simply saying 'accept the null hypothesis'; say 'there is insufficient evidence to reject the null hypothesis'.
Common Mistakes
- Misunderstanding: the sign test and Wilcoxon signed-rank test are parametric tests for means. Correction: both are non-parametric tests for the population median, and they do not assume a normal distribution.
- Misunderstanding: in the Wilcoxon signed-rank test, you rank the original observations rather than the differences. Correction: you must first calculate the differences from the hypothesised median (or the paired differences), then rank the absolute values of those differences.
- Misunderstanding: zero differences are included in the sign test or signed-rank test. Correction: zero differences are discarded before carrying out either test.
- Misunderstanding: the sign test and signed-rank test always give the same conclusion. Correction: they may differ because the signed-rank test uses more information (magnitudes) and can be more powerful when its assumptions hold.