You are viewing a free preview of this lesson.
Subscribe to unlock all 10 lessons in this course and every other course on LearningBro.
OCR names five non-parametric inferential tests, and you are expected to know, for each one, the exact combination of circumstances that makes it the correct choice — and to be able to compare a calculated value against a critical value to reach a conclusion. This is the single most reliably examined piece of Component 01 procedural knowledge: "which test should the researcher use, and justify your answer" appears in almost every paper.
This lesson takes each of the five tests — Binomial sign test, Wilcoxon signed-ranks, Mann-Whitney U, Spearman's rho and Chi-square (χ²) — and fixes it against the three-question logic from the last lesson (difference or association; related or independent design; level of data). We build a single decision table, note each test's critical-value rule, and work a complete example of choosing a test and interpreting the result. Master this table and you have secured a block of marks that many candidates leave on the table.
| This lesson covers | OCR H567 Component 01 sub-area | AO focus |
|---|---|---|
| Binomial sign test (difference, related, nominal) | 1.3 — the five named non-parametric tests | AO1; AO2 |
| Wilcoxon signed-ranks (difference, related, ordinal+) | 1.3 — the five named tests | AO1; AO2 |
| Mann-Whitney U (difference, independent, ordinal+) | 1.3 — the five named tests | AO1; AO2 |
| Spearman's rho (association, ordinal+ paired) | 1.3 — the five named tests | AO1; AO2 |
| Chi-square χ² (association, independent, nominal) | 1.3 — the five named tests | AO1; AO2 |
| Comparing calculated vs critical value | 1.3 — statistical tables of critical values | AO2 |
Referenced descriptively; see the official OCR H567 specification document for exact wording. This lesson develops AO1 (the criteria defining each test) and AO2 (selecting the correct test for a scenario and interpreting a calculated value against a critical value).
Every test choice is fixed by three answers, and it is worth over-learning them because they recur in every worked example that follows:
These three answers select one of the five tests uniquely. The table below is the object to memorise. Mastering it is disproportionately valuable: test-selection questions appear on essentially every Component 01 paper, they are quick to answer once the logic is secure, and — unlike some evaluation questions — they have a definitively correct answer, so they are among the most reliable marks in the whole component to bank.
A little care with each question pays off. For the first question, remember that an experiment (with a manipulated IV) is looking for a difference between conditions, whereas a correlational study measures two variables to detect an association — the word "relationship" or "link" in a question usually signals an association, while "compare", "affect" or "difference between" signals a difference. For the second question, "related" covers both repeated-measures (the same people twice) and matched-pairs (different but matched people) designs, because in both cases each score in one condition is naturally paired with a score in the other; "independent" means genuinely separate, unconnected groups. For the third question, recall from Lesson 5 that nominal data are counts in categories, whereas ordinal data can be ranked — a distinction that decides between the sign test and Wilcoxon (for related differences) and between chi-square and Mann-Whitney (for independent differences). Working through the questions deliberately, rather than pattern-matching to a half-remembered example, is what makes test selection reliable.
| Test | Looking for | Design | Level of data | Mnemonic hook |
|---|---|---|---|---|
| Binomial sign test | Difference | Related | Nominal | "Sign" = simple +/− differences |
| Wilcoxon signed-ranks | Difference | Related | Ordinal (or above) | Related ranks → Wilcoxon |
| Mann-Whitney U | Difference | Independent | Ordinal (or above) | Independent ranks → Mann-Whitney |
| Spearman's rho | Association (correlation) | — (paired scores) | Ordinal (or above) | Correlation of ranks → Spearman |
| Chi-square (χ²) | Association | Independent | Nominal (frequencies) | Categories in a contingency table → chi-square |
graph TD
A["What is the study testing?"] --> B{"Difference or Association?"}
B -->|"Association / correlation"| C{"Level of data?"}
C -->|"Nominal (frequencies)"| D["Chi-square (χ²)"]
C -->|"Ordinal or above (paired)"| E["Spearman's rho"]
B -->|"Difference"| F{"Design: related or independent?"}
F -->|"Related (repeated / matched)"| G{"Level of data?"}
G -->|Nominal| H["Binomial sign test"]
G -->|"Ordinal or above"| I["Wilcoxon signed-ranks"]
F -->|"Independent (different people)"| J{"Level of data?"}
J -->|Nominal| D
J -->|"Ordinal or above"| K["Mann-Whitney U"]
style D fill:#8e44ad,color:#fff
style E fill:#8e44ad,color:#fff
style H fill:#2980b9,color:#fff
style I fill:#2980b9,color:#fff
style K fill:#27ae60,color:#fff
Notice that chi-square appears twice — it is used for an association between two categorical (nominal) variables, and this can equally be framed as a test of difference in frequencies between independent groups. In both framings the data are nominal frequencies arranged in a contingency table and the groups are independent, so chi-square is the test.
Before working through the tests individually, it helps to see the structure of the table, because the five tests are not an arbitrary list but a systematic grid. Reading down the "difference" branch: for a related design, the level of data splits the choice between the sign test (nominal) and Wilcoxon (ordinal); for an independent design, it splits between chi-square (nominal) and Mann-Whitney U (ordinal). Reading the "association" branch: the level of data splits between chi-square (nominal) and Spearman's rho (ordinal). Every test therefore occupies a unique cell defined by its answers to the three questions, and no two tests share the same combination. This is why the three-question method is guaranteed to identify the correct test: once you have fixed difference/association, related/independent, and level of data, exactly one cell — one test — remains. Memorising the grid as a grid, rather than as five disconnected facts, is far more robust under exam pressure.
Use when you are testing a difference, the design is related (same or matched participants, e.g. a before-and-after study), and the data are nominal — reducible to a direction of change (did each participant go "up" or "down"?). The calculated statistic is S, the number of the less frequent sign. Ties (no change) are discarded and N is reduced accordingly.
Critical-value rule: the calculated S must be less than or equal to the critical value to be significant.
Example use: a therapist rates each client's mood as "improved" or "not improved" after treatment (same clients, categorical outcome).
The sign test is the simplest of the five because it throws away everything except the direction of each participant's change. It asks only: did each person go up, down, or not change? Participants who changed are classified by the sign of their change (+ or −), ties are discarded, and the statistic S is simply the count of the less common sign. The logic is intuitive: if the treatment had no effect, ups and downs should be roughly equal, so a very lopsided split (almost everyone improving) is unlikely by chance and signals a real effect. Its simplicity is both a strength — it works even when the data are merely nominal (improved / not) — and a weakness, because by ignoring the size of each change it wastes information that a more sensitive test could use. When the data carry more than direction (a measurable magnitude of change), Wilcoxon is preferred precisely because it uses that extra information.
Use for a difference, a related design, and ordinal (or higher) data. Unlike the sign test, it uses the magnitude of each difference by ranking the differences. The calculated statistic is T (or W), the smaller of the sums of the positive and negative ranks.
Critical-value rule: the calculated T must be less than or equal to the critical value to be significant.
Example use: the same participants rate their anxiety on a 1–10 scale before and after a relaxation technique.
Wilcoxon is best understood as the sign test's more sophisticated sibling. It shares the sign test's situation — a difference in a related design — but instead of registering only the direction of each participant's change, it also takes account of the magnitude. The procedure calculates the difference for each participant, ranks those differences by size (ignoring their signs at first), then sums the ranks of the positive differences separately from the ranks of the negative differences; the smaller of the two rank-sums is the statistic T. The intuition is that if the treatment had no effect, the positive and negative changes should be similar in both number and size, so the two rank-sums should be roughly equal; a small T means the changes went overwhelmingly one way, both in direction and in magnitude, which is unlikely by chance. Because Wilcoxon uses the magnitude information that the sign test discards, it is more powerful — better able to detect a real effect — which is why it is preferred whenever the data are at least ordinal (a measurable amount of change), reserving the sign test for the cruder case where only direction is available.
Use for a difference, an independent design (different participants in each condition), and ordinal (or higher) data. Scores from both groups are ranked together and the ranks compared. The calculated statistic is U (the smaller of the two U values).
Critical-value rule: the calculated U must be less than or equal to the critical value to be significant.
Example use: one group revises with music, a separate group revises in silence, and their recall is ranked.
Mann-Whitney U is the independent-design counterpart to Wilcoxon: both use ranks to test a difference on ordinal data, but Mann-Whitney compares two separate groups of people rather than one group measured twice. The procedure ranks all the scores from both groups together as a single combined set, then examines how the ranks are distributed between the two groups; the statistic U reflects the degree of overlap. The intuition is that if the two groups came from the same population (no effect), high and low ranks should be evenly mixed between them; if one group systematically holds the higher ranks, that separation is unlikely by chance and signals a real difference. Confusing Mann-Whitney with Wilcoxon is one of the most frequent errors, and the discriminator is always the design: same or matched participants → Wilcoxon (related); different, unconnected participants → Mann-Whitney (independent). Both require the data to be at least ordinal.
Use for an association (correlation) between two variables measured on ordinal (or higher) scales, where each participant contributes a pair of scores. It measures the strength and direction of the correlation of the ranks. The calculated statistic is rho (rs), ranging from −1 to +1.
Critical-value rule: the calculated rho must be greater than or equal to the critical value to be significant. (Correlation coefficients flip the rule — bigger is more significant.)
Example use: is there a relationship between hours revised and exam rank position?
Spearman's rho is the odd one out among the five because it measures an association rather than a difference, and this changes both its interpretation and its critical-value rule. Instead of asking "do two conditions differ?", it asks "as one variable rises, does the other rise (positive), fall (negative), or show no pattern (zero)?" — the same directional language as the correlation of Lesson 1, but now applied to ranked data. The statistic rs runs from −1 (perfect negative) through 0 (none) to +1 (perfect positive), so its magnitude tells you the strength of the relationship and its sign tells you the direction. Because a correlation is "stronger" the further the coefficient is from zero, the significance rule flips relative to the difference tests: the calculated rs must be greater than or equal to the critical value to be significant. Remembering that Spearman is a correlation, and that correlations therefore behave "the opposite way round" from the difference tests when compared to the table, prevents one of the most common errors on this topic.
Use for an association between two nominal (categorical) variables with an independent design, where data are frequencies (counts) arranged in a contingency table. It compares observed frequencies with those expected if there were no association. The calculated statistic is χ², and degrees of freedom are found from (rows − 1) × (columns − 1).
Critical-value rule: the calculated χ² must be greater than or equal to the critical value to be significant. (Like Spearman, bigger is more significant.)
Example use: is there an association between sex (male/female) and preferred learning style (visual/auditory/kinaesthetic)?
Chi-square is the test for frequency data — counts of how many cases fall into each combination of categories, arranged in a contingency table. Its logic is a direct comparison of observed frequencies (what actually occurred) against expected frequencies (what would occur if the two variables were completely unrelated). The statistic χ² grows larger as the observed counts depart further from the expected counts; a large χ² therefore signals that the pattern of frequencies is unlikely under the assumption of no association, so the null is rejected. Like Spearman, chi-square requires the calculated value to be greater than or equal to the critical value to be significant — another test where "bigger is more significant". The degrees of freedom, calculated as (rows − 1) × (columns − 1), determine which row of the table to read and reflect the size of the contingency table. Chi-square has specific requirements that examiners sometimes probe: the data must be frequencies (not percentages or means), each case must fall into only one cell (independence of observations), and expected frequencies should not be too small. When you see categorical variables and a table of counts from independent groups, chi-square is almost always the intended test.
Subscribe to continue reading
Get full access to this lesson and all 10 lessons in this course.