1.3 - Choosing tests and reading critical values

Parametric assumptions and measurement

A statistical test is chosen to fit the research question, measurement and design, rather than whichever gives a significant result.

Parametric tests make specified assumptions about population distributions and their parameters. H567 requires the criteria for using them, rather than the calculation of named parametric tests. The conventional criteria taught here are at least interval-level data, an approximately normal relevant population distribution, no extreme scores that seriously distort the analysis, and approximately equal population variances where the comparison requires that assumption. Equal variances means similar spread, not identical means or numerically identical sample SDs. Some statistical tests have different assumptions, so a supplied test's actual conditions take priority over a universal checklist.

A 1–5 rating is normally ordinal: its order is meaningful, but equal psychological steps are not established. Stopwatch times have equal measured units and satisfy H567's interval requirement, although their distribution may still be skewed. Interval data alone do not prove that parametric assumptions are met.

Non-parametric tests make fewer or different distributional assumptions and may use ranks or category frequencies. They still have conditions: the correct design, usable measurements and independent observations or independent pairs, as applicable. They are not assumption-free, and not every non-normal dataset calls for the same test.

Selecting the five non-parametric tests

First decide whether the investigation seeks a difference between conditions or a relationship between variables.

For Wilcoxon, the size of a paired difference must be meaningfully rankable; a scale that only supports ordering individual categories may be better analysed by signs. A sign test can also discard magnitude from numerical paired measurements and keep only their direction. Both have independent pairs, even though the two measurements within a pair are related. For chi-square, categories must not overlap and expected frequencies must be adequate for the approximation; sparse counts may require another justified analysis rather than blindly applying it.

For example, separate groups rate two study rooms on an ordered satisfaction scale: Mann–Whitney U fits the difference, ordinal scores and independent groups. If the same people supply usable paired scores in both rooms and the non-zero differences can be ranked, Wilcoxon fits. Recording only whether each person's second score increased or decreased permits the sign test. Asking whether revision-hours rank relates to recall-score rank instead calls for Spearman. Comparing yes/no preferences between two independent groups calls for chi-square.

Justify the selection using the actual scenario. “It is non-parametric” alone does not explain why one of these five tests fits rather than another.

Reading a critical-value table

An observed value is the calculated test statistic, and a critical value is its rejection boundary for a specified test and threshold.

Before reading a table, select the intended significance level and the appropriate one- or two-tailed test. A justified directional hypothesis predicts a particular direction and may call for a one-tailed test; a non-directional hypothesis asks about either direction and uses a two-tailed test. Do not change the tail after seeing the findings. A result in the opposite direction does not support a directional prediction merely because its magnitude is large.

Read the correct sample-size or degrees-of-freedom entry. Mann–Whitney uses the two separate group sizes. Wilcoxon and sign tables use the number of usable non-zero/no-change-excluded pairs. Spearman uses the number of complete paired observations. A contingency-table chi-square uses degrees of freedom, $(r-1)(c-1)$, where $r$ and $c$ count rows and columns of categories, excluding totals.

Always follow the supplied table's explicit convention. A negative Spearman coefficient can be significant: for a two-tailed critical magnitude 0.648, a coefficient −0.80 qualifies because its magnitude is 0.80. In a directional test its sign must also match the predicted direction.

Consider an original illustrative independent category investigation with observed chi-square 4.50. Its 2×2 table has $(2-1)(2-1)=1$ degree of freedom. A supplied upper-tail critical-value extract is:

At the preselected 0.05 level, $4.50>3.841$, so reject the null of no categorical association. It is not significant at 0.01 because $4.50<6.635$. The table brackets its p-value; it does not provide an exact p-value of 0.05. No statistical decision alone proves causation or absence of an effect.

Using statistical symbols accurately

Mathematical symbols state precise relationships and prevent ambiguous significance statements.

The source specification prints $\sim$ as a required symbol. Its meaning depends on context: it must not be silently treated as exact equality. For an approximate numerical answer, write “approximately” or use $\approx$, such as $\sqrt{8}\approx2.83$. Proportionality means a constant ratio under stated conditions; it is stronger than merely having a positive correlation.

Use $\leq$ or $\geq$ when a critical rule includes equality. “Observed U is lower” refers to a statistic comparison, whereas $p<0.05$ refers to the probability under the null model. They are different quantities. Report significance with a threshold, research context and null decision, then keep its causal and practical limits visible.