Situations for Non-parametric tests

The key idea. Earlier tests — the $z$-test and $t$-test — assume the population is normally distributed. If that assumption is false (the data are skewed, or you simply have no reason to believe normality), those tests are not valid. A non-parametric test (also called distribution-free) makes no assumption about the shape of the population.

What do they test instead of the mean? Because we no longer have a tidy normal model, we stop talking about the mean $\mu$ and instead test the median $m$ — the middle value, which always exists and is meaningful for any distribution.

The trade-off. Non-parametric tests are more robust (they work in more situations) but, when the data really are normal, they are slightly less powerful than the matching parametric test — they are a little more likely to miss a real effect. You use them precisely when normality cannot be assumed.

The three tests you must know (and choose between):

• Sign test — uses only the sign of each difference. Weakest assumptions, lowest power.
• Wilcoxon signed-rank test — uses signs and ranks of the sizes of the differences. Needs a symmetric distribution.
• Wilcoxon rank-sum (Mann–Whitney) test — for two independent samples; pools and ranks everything.

Cambridge tip. If a question says "it cannot be assumed that the population is normally distributed", that is your cue to reach for a non-parametric test — and to state the hypotheses in terms of the median $m$.

Examiners reward the correct choice of test — get it wrong and the whole question collapses. Work through three questions in order.

Question 1 — How many samples?

• One sample (or a single set of paired differences) → sign test or Wilcoxon signed-rank test.
• Two independent samples → Wilcoxon rank-sum (Mann–Whitney) test.

Question 2 — Is it paired? A paired design has each observation in sample A naturally matched to one in sample B (before/after the same person, twins, the same plot measured twice). You then work with the differences, which reduces the problem to a single sample.

Question 3 — Can we assume symmetry? The Wilcoxon signed-rank test requires the population (of values, or of differences) to be symmetric about its median. If symmetry cannot be assumed, fall back to the sign test, which assumes nothing about shape.

Why two tests for one sample? The signed-rank test uses more information (the sizes of the differences as well as their signs), so it is more powerful — but only valid when the distribution is symmetric. The sign test throws away the sizes and keeps only the signs, so it works for any shape but is weaker.

Cambridge tip. When a question gives you a paired data set and says "assume the differences are symmetrically distributed", that phrase is a deliberate green light for the signed-rank test. No symmetry statement → use the sign test.

Every non-parametric test in 9231 is built around a hypothesis about the median.

Single-sample / paired tests (sign, signed-rank). Let $m$ be the population median (for a paired test, $m$ is the median of the differences). State: $H_0: m = m_0 \qquad H_1: m \ne m_0 \;(\text{two-tailed}) \;\;\text{or}\;\; m > m_0,\; m < m_0 \;(\text{one-tailed})$

For a paired test about "no change", the hypothesised difference is usually $0$: $H_0: m_d = 0 \qquad H_1: m_d \ne 0$

Two-sample test (rank-sum / Mann–Whitney). Here the null is about the identity of the two populations: $H_0: \text{the two populations are identical} \qquad H_1: \text{one is shifted (located higher / lower / differently)}$

Often written in terms of a difference in medians, $H_0: m_A = m_B$.

Decide the tail FIRST. Read the question: "has the median increased?" is one-tailed; "has the median changed?" is two-tailed. The tail decides which critical value to read and whether to halve the probability.

Cambridge tip. Always define your symbol — "let $m$ be the population median" — before writing $H_0$. Examiners want the parameter named, and a vague "$H_0$: no difference" can lose the mark.