Choosing & Using Statistical Tests

Knowing that a result must be tested for significance is one thing; knowing which test to run is another, and it is one of the most reliably examined skills in Edexcel Paper 3. The good news is that test selection is not a matter of memory or intuition — it is a mechanical decision fixed by just three features of the study: whether the hypothesis concerns a difference or a correlation, whether the design is independent (unrelated) or related, and the level of measurement of the data. Answer those three questions in order and the correct test drops out. Edexcel 9PS0 examines a specific set of five non-parametric tests — the sign test, Mann-Whitney U, Wilcoxon signed-rank, Spearman's rho and chi-square — and this lesson teaches you to select the right one from any described study, to state the criteria that justify each, and to compare a calculated value against a critical value to reach a conclusion. The formulae for chi-square and Spearman's rho are covered, and the decision logic is built into a table and a flowchart you can reproduce under exam conditions.

Key Definition: Statistical test selection is the process of choosing the correct inferential test for a study from three criteria — the type of hypothesis (difference vs correlation), the experimental design (independent vs related), and the level of measurement (nominal, ordinal or interval) — so that the analysis matches the data.

By the end of this lesson you will be able to:

• Select the correct Edexcel test from any described study using the three-question decision procedure.
• State the specific criteria (hypothesis type, design, level of data) that justify each of the five tests.
• Explain conceptually how the sign test, Mann-Whitney U, Wilcoxon, Spearman's rho and chi-square work, and apply the chi-square and Spearman formulae.
• Compare a calculated value against a critical value in the correct direction for each test, and state the conclusion in context.

---

Spec Mapping

Edexcel 9PS0 — Paper 3: Psychological Skills (Research Methods). This lesson develops the test-selection and test-application content assessed in Section A of Paper 3, where candidates are given a described study and asked to name and justify the appropriate statistical test, or to interpret a calculated value against a critical value. Our sequence teaches the three-question decision procedure first, then each test in turn, so the ordering reflects our teaching rationale rather than the specification's own.

Our lesson covers	Edexcel 9PS0 research-methods area
The three test-choice criteria (hypothesis, design, level of data)	Factors affecting the choice of statistical test
Sign test; Mann-Whitney U; Wilcoxon signed-rank	Tests of difference
Spearman's rho	Test of correlation
Chi-square ($\chi^2$χ2)	Test of association / difference in frequencies
Comparing calculated and critical values	Use of critical values in interpreting significance

Assessment Objectives. These items are dominated by AO2 (applying the three criteria to a novel study to select and justify a test; reading a critical-values table; interpreting a calculated value) and AO3 (evaluating whether a chosen test is appropriate and what its result implies), on an AO1 base of accurate knowledge of each test and its conditions. This is application-heavy: a scenario-free "list of tests" answer scores poorly, and the justification — naming the design and level of data — is where most marks sit.

Connects to…

• Inferential statistics (this course). The null/alternative and one-/two-tailed distinctions, the calculated-versus-critical logic, and Type I/II errors from the previous lesson are the machinery every test in this lesson relies upon.
• Quantitative data (this course). The level of measurement established in the descriptive-statistics lesson is one of the three criteria; misclassifying nominal, ordinal or interval data leads straight to the wrong test.
• Experimental methods & design (this course). Whether a design is independent groups or repeated measures / matched pairs — decided when the study was planned — is exactly the "related vs unrelated" criterion that selects between, say, Mann-Whitney and Wilcoxon.

---

The Three Questions That Fix the Test

Every test-selection decision reduces to three questions, asked in this order.

1. Difference or correlation? Does the hypothesis predict a difference between conditions or an association/relationship between two co-variables? This chooses which block of the table you are in.
2. Independent or related design? Independent (unrelated) = different participants in each condition (independent groups); related = the same participants in both conditions (repeated measures) or matched participants (matched pairs). For a correlation, the two scores are paired by definition, so this question does not apply.
3. What level of measurement? Nominal (categories/frequencies), ordinal (ranks), or interval (equal measured units). This chooses the column.

The cell where your answers intersect names the test. Because all five Edexcel tests are non-parametric, interval data are treated using the ordinal tests (Mann-Whitney, Wilcoxon, Spearman) unless a question specifies otherwise — the rank-based tests are perfectly valid on interval data, they simply convert the scores to ranks first.

Decision flowchart

graph TD
    A[What does the hypothesis test?] -->|Difference| B[What is the design?]
    A -->|Correlation / association| C[Level of measurement?]
    B -->|Independent groups<br/>UNRELATED| D[Level of measurement?]
    B -->|Repeated measures / matched pairs<br/>RELATED| E[Level of measurement?]
    D -->|Nominal| D1[Chi-square]
    D -->|Ordinal / interval| D2[Mann-Whitney U]
    E -->|Nominal| E1[Sign test]
    E -->|Ordinal / interval| E2[Wilcoxon signed-rank]
    C -->|Nominal| C0[Chi-square<br/>test of association]
    C -->|Ordinal / interval| C1["Spearman's rho"]

Decision table

	Nominal	Ordinal / Interval
Difference — independent (unrelated)	Chi-square ($\chi^2$χ2)	Mann-Whitney U
Difference — related (repeated measures / matched pairs)	Sign test	Wilcoxon signed-rank
Correlation / association	Chi-square ($\chi^2$χ2)	Spearman's rho ($r_s$rs​)

Working through the three questions in order makes the choice all but automatic. First, decide difference or correlation — this picks the row block. Second, for a difference, decide independent or related — this picks the row. Third, identify the level of measurement — this picks the column. Chi-square appears twice because it serves both as a test of difference between independent categories and as a test of association between two categorical variables; either way it requires nominal (frequency) data.

Exam Tip: This table is the single most-tested piece of methods knowledge on Paper 3. When you name a test, always justify it by stating all three criteria: "a Mann-Whitney U test, because the hypothesis predicts a difference, the design is independent groups (unrelated), and the data are ordinal." Naming the test alone rarely earns full marks — the justification is where the credit lies.

Worked scenarios — choosing the right test and ruling out the wrong ones

The surest way to master selection is to reason not only about the correct test but about why the others are wrong, because eliminating the near-misses forces you to check each criterion. Three short scenarios show the discipline.

Scenario 1. A researcher measures reaction time (in milliseconds) for the same 15 participants in a quiet room and again in a noisy room, predicting slower times in noise. Working the three questions: the hypothesis is a difference, the design is related (the same people in both conditions — repeated measures), and reaction time in ms is treated as ordinal for a non-parametric test. The correct test is therefore Wilcoxon signed-rank. Why not the others? Mann-Whitney is ruled out because the design is not independent (it is the same participants, not two separate groups); the sign test is a defensible but weaker choice that would throw away the magnitude of each change; Spearman is wrong because this is a difference, not a correlation; chi-square is wrong because the data are measured scores, not frequency counts in categories.

Scenario 2. A researcher records whether 80 people (40 dog-owners, 40 cat-owners) describe themselves as "introvert" or "extrovert", testing for an association between pet ownership and personality type. Here both variables are categorical (nominal — each person falls into a box), the groups are independent, and the prediction is an association, so the correct test is chi-square. Why not the others? Mann-Whitney, Wilcoxon and Spearman all require at least ordinal data and cannot be run on pure categories; the sign test needs a related design and a direction of change, neither of which is present. The give-away that points to chi-square is that the data are counts of people in categories rather than scores or ranks.

Scenario 3. A researcher ranks 12 students on both hours of sleep and concentration score, predicting that more sleep goes with better concentration. The hypothesis is a correlation and the data are ordinal (ranks), so the correct test is Spearman's rho. Why not the others? All three difference tests (sign, Mann-Whitney, Wilcoxon) are ruled out immediately because there is no manipulation and no comparison of conditions — there is a single group with two paired measures; chi-square is wrong because the data are ranks, not frequencies. Practising this "choose and eliminate" routine is what converts the decision table from something memorised into something understood.

---

The Five Tests in Detail

The sign test

Use the sign test when the hypothesis predicts a difference, the design is related, and the data are nominal — or can be reduced to the direction of change (better/worse, up/down) for each participant.

• How it works: find the sign ($+$+, $-$− or $0$0) of each participant's change, discard the zeros, let $N$N be the number remaining, and let the calculated value $S$S be the count of the less frequent sign.
• Criteria: difference · related · nominal.
• Decision rule: the result is significant if the calculated value is equal to or less than the critical value.

The sign test deliberately throws away information — it uses only the direction of each change, not its size, so a change of $-5$−5 counts the same as $-1$−1. This makes it easy to calculate but relatively insensitive, which is why the Wilcoxon test (below) is preferred when the magnitude of each change is available.

Mann-Whitney U

Use Mann-Whitney U when the hypothesis predicts a difference, the design is independent (unrelated), and the data are ordinal (or interval treated as ordinal).

• How it works: pool the scores from both groups, rank them all from lowest to highest, then compare the rank totals of the two groups. If the two groups were really the same, high and low ranks would be evenly mixed; if one group's scores cluster systematically higher, its rank total will stand out. The test statistic is U.
• Criteria: difference · independent · ordinal.
• Decision rule: the result is significant if the calculated value of $U$U is equal to or less than the critical value.

Wilcoxon signed-rank

Use Wilcoxon when the hypothesis predicts a difference, the design is related, and the data are ordinal (or interval treated as ordinal).

• How it works: find the difference between each participant's two scores, rank the sizes of those differences (ignoring sign), then sum the ranks of the positive and of the negative differences separately. Under the null the two rank sums balance; if one direction dominates, the smaller sum (the test statistic T) becomes small.
• Criteria: difference · related · ordinal.
• Decision rule: the result is significant if the calculated value of $T$T is equal to or less than the critical value.

Wilcoxon is the more powerful relative of the sign test: both suit related designs, but Wilcoxon uses the magnitude of each change (via the ranks) rather than only its direction, so it extracts more from the same data.

Spearman's rho

Use Spearman's rho when the hypothesis predicts a correlation and the data are ordinal (or interval treated as ordinal).

• How it works: rank each of the two co-variables separately, find the difference $d$d between the two ranks for each pair, and combine these into a single coefficient between $-1$−1 and $+1$+1.
• Criteria: correlation · ordinal.
• Decision rule: the result is significant if the calculated value of $r_s$rs​ is equal to or greater than the critical value.

The formula is

$r_s = 1 - \frac{6 \sum d^2}{n(n^2 - 1)}$rs​=1−n(n2−1)6∑d2​

where $d$d is the difference between the two ranks for each pair and $n$n is the number of pairs. The result always lies in the range $-1 \leq r_s \leq +1$−1≤rs​≤+1: a value near $+1$+1 indicates a strong positive relationship, near $-1$−1 a strong negative one, and near $0$0 little or no monotonic relationship.

Chi-square

Use chi-square when the data are nominal (frequency counts in independent categories) and the hypothesis predicts either a difference between those categories or an association between two categorical variables.

• How it works: compare the observed frequencies with the frequencies expected if the null were true, cell by cell, in a contingency table.
• Criteria: difference or association · independent · nominal.
• Decision rule: the result is significant if the calculated value of $\chi^2$χ2 is equal to or greater than the critical value.

The formula is

$\chi^2 = \sum \frac{(O - E)^2}{E}$χ2=∑E(O−E)2​

where $O$O is each observed frequency and $E$E the corresponding expected frequency. The larger the gap between what is observed and what chance would predict, the larger $\chi^2$χ2 becomes. Chi-square is entered into the critical-values table using degrees of freedom, not $N$N: for a contingency table, $df = (\text{rows} - 1) \times (\text{columns} - 1)$df=(rows−1)×(columns−1), so a $2 \times 2$2×2 table has $df = 1$df=1.

Exam Tip: The direction of the calculated-versus-critical comparison differs by test, and stating it wrongly loses the conclusion mark. For the sign test, Mann-Whitney (U) and Wilcoxon (T), the calculated value must be equal to or less than the critical value. For Spearman's rho and chi-square, it must be equal to or greater than the critical value. A handy memory aid: the correlation and association tests (rho, $\chi^2$χ2) need to be bigger.

---

Two Worked Applications

Worked example — chi-square

A researcher asks 100 people whether they prefer therapy A or B, and records their sex, predicting an association between sex and preference.

	Prefer A	Prefer B	Row total
Male	30	20	50
Female	20	30	50
Column total	50	50	100

First identify the test: the data are frequencies in categories (nominal), the groups are independent, and the prediction is an association — so chi-square is correct. The expected frequency for each cell, if there were no association, is $E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}$E=grand totalrow total×column total​. Every cell here expects $\frac{50 \times 50}{100} = 25$10050×50​=25. Applying the formula, each of the four cells contributes $\frac{(30-25)^2}{25} = 1$25(30−25)2​=1 or $\frac{(20-25)^2}{25} = 1$25(20−25)2​=1, so $\chi^2 = 1 + 1 + 1 + 1 = 4.0$χ2=1+1+1+1=4.0. With $df = (2-1)(2-1) = 1$df=(2−1)(2−1)=1, the critical value at $p \leq 0.05$p≤0.05 (two-tailed) is $3.84$3.84. As the calculated value ($4.0$4.0) exceeds the critical value ($3.84$3.84), the result is significant: there is a significant association between sex and therapy preference.

Worked example — Spearman's rho