Probability and Significance

Probability, Significance, and the Use of Statistical Tables

Every inferential statistical test produces a test statistic — a number calculated from the data. To decide whether a result is statistically significant, the calculated test statistic must be compared against a critical value obtained from a statistical table. Understanding how to use statistical tables correctly is an essential A-Level skill.

The Logic of Significance Testing (recap)

Under the null hypothesis (H₀), any observed difference or correlation is due to chance. A statistical test calculates the probability — the p-value — of obtaining the observed test statistic (or a more extreme one) if H₀ is true. If p falls below the pre-specified significance level (α), H₀ is rejected and the result is declared significant. In psychology, α = 0.05 (5%) is the conventional threshold, though 0.01 and 0.001 are used when stricter evidence is required.

Critical Values and Statistical Tables

Rather than computing exact p-values (which requires specialist software), psychologists use statistical tables that list critical values — the threshold value of the test statistic that must be met or exceeded (or not exceeded, depending on the test) for the result to be significant at a given α.

To use a statistical table correctly, three pieces of information are needed:

1. Sample size (N) or degrees of freedom (df) — larger samples produce larger critical values for most tests.
2. Significance level (α) — 0.05, 0.01, or 0.001.
3. Number of tails — one-tailed (directional hypothesis) or two-tailed (non-directional hypothesis).

The calculated test statistic is then compared to the critical value. The direction of the comparison varies by test:

• For most non-parametric tests (sign test, Wilcoxon, Mann-Whitney): the result is significant if the calculated value is equal to or less than the critical value (smaller = more extreme).
• For correlation coefficients and t-tests: the result is significant if the calculated value is equal to or greater than the critical value in absolute terms (larger = more extreme).

One-Tailed and Two-Tailed Tests

A one-tailed (directional) test is appropriate when the research hypothesis specifies the direction of the expected effect — e.g. 'Group A will score higher than Group B'. It tests whether the effect falls in the predicted direction only. A two-tailed (non-directional) test is appropriate when the hypothesis predicts a difference but not its direction — e.g. 'There will be a difference between groups'. Two-tailed tests split α between both tails, making them more conservative (harder to reach significance) but appropriate when direction was not predicted in advance. The direction must be specified before data collection — choosing a one-tailed test after seeing the data inflates the Type I error rate.

Worked Example: Using a Wilcoxon Table

A researcher obtains a Wilcoxon T statistic of 4 with N = 10 (after excluding ties), using a one-tailed test at α = 0.05. The critical value from the Wilcoxon table at N=10, one-tailed, α = 0.05 is T ≤ 10. Since T = 4 ≤ 10, the result is significant. Conclusion: the difference between conditions is statistically significant (T = 4, N = 10, p < .05, one-tailed).