You run an independent-samples t-test on two groups of 4,000 students. The p-value comes back at .001. Significant. But when you check the means, Group A scored 72.3 and Group B scored 72.9. That is a 0.6-point difference on a 100-point scale. No teacher would restructure a course over half a point. The test detected a real difference, yes, but a difference too small to care about. This is the problem that effect sizes solve.
Why P-Values Are Not Enough
A p-value tells you the probability of seeing your data (or something more extreme) if the null hypothesis were true. It answers one question: is there likely an effect? It does not answer the follow-up question that actually matters to your reader: how big is it?
The reason is arithmetic. The test statistic in most frequentist tests is a ratio: signal divided by noise. The signal is the observed difference or relationship. The noise is the standard error, which shrinks as sample size grows. With n = 50 per group, a 2-point mean difference might produce t = 1.41 and p = .16. Run the same study with n = 5,000 per group and that same 2-point gap could yield t = 14.1 and p < .001. The effect did not change. The sample did.
Effect size strips sample size out of the equation. Cohen's d, for instance, divides the mean difference by the pooled standard deviation, not by the standard error. Whether you tested 30 people or 30,000, a d of 0.50 means the same thing: the groups differ by half a standard deviation. That is what makes effect sizes comparable across studies, and it is why power analysis relies on them to determine how many participants you need before you collect any data.
Effect Size Measures by Test Type
Different statistical tests call for different effect size measures. The choice depends on what kind of variables you are testing and what kind of comparison you are making. Here is a breakdown of the most common ones.
Cohen's d: T-Tests
Cohen's d quantifies the standardized distance between two group means. The formula is straightforward: take the difference between the two means and divide it by the pooled standard deviation. If Group A has M = 78.2 (SD = 10.4) and Group B has M = 72.6 (SD = 11.1), the pooled SD is approximately 10.75. The difference is 5.6, so d = 5.6 / 10.75 = 0.52. That is a medium effect by Cohen's (1988) conventions: 0.20 is small, 0.50 is medium, 0.80 is large. It works for both independent-samples and paired-samples t-tests, though the paired version uses the standard deviation of difference scores in the denominator.
Eta-Squared and Partial Eta-Squared: ANOVA
When you compare three or more groups, ANOVA replaces the t-test and eta-squared (η²) replaces Cohen's d. Eta-squared is the proportion of total variance explained by the factor: η² = SSeffect / SStotal. If your ANOVA table shows SSbetween = 420 and SStotal = 3,000, then η² = 420 / 3,000 = .14, a large effect. Cohen's benchmarks for η² are .01 (small), .06 (medium), and .14 (large).
Partial eta-squared (ηp²) is what SPSS reports by default. In a one-way ANOVA the two measures are identical, but in factorial designs they diverge. Partial η² uses SSeffect / (SSeffect + SSerror), removing the variance accounted for by other factors. This makes it larger than η² when multiple predictors are in the model. Most thesis committees accept partial η² as the standard for ANOVA results, and the same .01 / .06 / .14 benchmarks are commonly applied.
Pearson r: Correlation
For bivariate correlation, the correlation coefficient itself is the effect size. No extra calculation needed. Cohen's benchmarks: r = .10 (small), .30 (medium), .50 (large). An r of .35 between study hours and GPA means study hours account for about 12% of the variance in GPA (r² = .12). That is a medium effect with practical meaning in an educational context.
Cramér's V: Chi-Square
Chi-square tests compare frequencies across categories, so a standardized mean difference makes no sense here. Cramér's V fills the gap. For a 2×2 table, the benchmarks mirror Pearson's r: .10 (small), .30 (medium), .50 (large). For larger tables, the thresholds depend on the degrees of freedom. A 3×2 table with V = .25 represents a medium effect, while the same V in a 5×4 table would be considered large.
R²: Regression
R² tells you the proportion of variance in the dependent variable explained by all predictors in the model. In a multiple regression predicting thesis grade from hours studied, supervisor meetings, and prior GPA, an R² of .41 means the model accounts for 41% of the variation in grades. Cohen's benchmarks: .02 (small), .13 (medium), .26 (large). For individual predictors, report the standardized beta coefficient (β) or the semi-partial correlation squared (sr²).
Odds Ratios: Logistic Regression
When the outcome is binary (pass/fail, admitted/rejected), logistic regression produces odds ratios. An OR of 2.5 means the odds of the outcome are 2.5 times higher for a one-unit increase in the predictor. An OR of 1.0 means no effect. Values below 1.0 indicate a protective or negative association. There are no universal small/medium/large benchmarks for odds ratios because interpretation depends on the base rate and the context, but an OR between 1.5 and 2.5 is generally considered meaningful in social science research.
Reference Table: Tests, Measures, and Benchmarks
| Statistical Test | Effect Size Measure | Small | Medium | Large |
|---|---|---|---|---|
| Independent t-test | Cohen's d | 0.20 | 0.50 | 0.80 |
| Paired t-test | Cohen's d | 0.20 | 0.50 | 0.80 |
| One-way ANOVA | η² / ηp² | .01 | .06 | .14 |
| Factorial ANOVA | Partial ηp² | .01 | .06 | .14 |
| Pearson correlation | r | .10 | .30 | .50 |
| Chi-square (2×2) | Cramér's V | .10 | .30 | .50 |
| Chi-square (larger) | Cramér's V | Depends on df; see Cohen (1988), Table 7.1 | ||
| Multiple regression | R² | .02 | .13 | .26 |
| Logistic regression | Odds ratio (OR) | Context-dependent; OR of 1.5–2.5 often meaningful | ||
| Mann-Whitney U | r = Z / √N | .10 | .30 | .50 |
How to Calculate the Most Common Measures
Cohen's d by hand requires three numbers from your SPSS or R output: the two group means and the pooled standard deviation. The pooled SD formula is: √[((n1 − 1) × SD1² + (n2 − 1) × SD2²) / (n1 + n2 − 2)]. Say Group 1 (n = 45) has M = 65.3 and SD = 12.8, and Group 2 (n = 42) has M = 59.1 and SD = 13.5. The pooled SD works out to about 13.14. So d = (65.3 − 59.1) / 13.14 = 0.47, just under the medium threshold.
Eta-squared from ANOVA output is even simpler. Look at your ANOVA summary table and find two values: SSbetween (the sum of squares for your factor) and SStotal. Divide the first by the second. If SSbetween = 210.5 and SStotal = 3,842.0, then η² = 210.5 / 3,842.0 = .055. That sits just below the medium threshold of .06. In SPSS, check the "Estimates of effect size" box under Options in the One-Way ANOVA dialog to get partial η² automatically. Academic Stats Agent includes all of these effect size calculations in its output by default, formatted and ready for your results section.
Common Mistakes with Effect Sizes
The most frequent mistake is not reporting one at all. We still see theses that present only F-values and p-values for every ANOVA, with no mention of η². APA 7th edition (2020) is explicit: effect sizes should accompany every inferential test result. A results section without them looks outdated and will draw feedback from any methodologically aware reviewer.
The second mistake is using the wrong effect size for the test. Reporting Cohen's d after an ANOVA or η² after a t-test is technically possible with conversions, but it creates confusion. Match the measure to the test: d for t-tests, η² or partial η² for ANOVA, V for chi-square, r for correlation, R² for regression. The reference table above makes the mapping clear.
Third: treating Cohen's benchmarks as rigid rules. Cohen himself warned against this in his 1988 book. He described the .20 / .50 / .80 thresholds as provisional, intended for situations where no field-specific norms existed. In clinical psychology, a d of 0.30 for a brief therapy intervention might be impressive. In pharmacology, a d of 0.30 for a drug side effect might be alarming. In educational research, a d of 0.30 on a standardized test can represent months of learning gains. Context determines whether an effect is "small" or "large," not a one-size-fits-all table. Use Cohen's values as rough orientation, then interpret your effect in terms of what it means in your specific field.
A fourth error, and a subtle one, is confusing η² with partial η² and applying the same benchmarks without noting which one you are reporting. In a two-way ANOVA with an interaction term, partial η² for the main effect will always be larger than η² because the denominator excludes variance from the other factor. Clearly label which version you report, and your reader can interpret it correctly.
Reporting Effect Sizes in APA Format
APA 7th edition expects effect sizes inline with every test result. The format follows the same pattern as the test statistic: italicized symbol, equals sign, value. For a t-test: t(58) = 2.87, p = .006, d = 0.74. For ANOVA: F(3, 116) = 4.23, p = .007, ηp² = .10. For chi-square: χ²(1, N = 200) = 6.41, p = .011, V = .18. For correlation: r(73) = .42, p < .001.
A well-written results sentence integrates the effect size into the interpretation. Instead of stopping at "the difference was statistically significant," write something like: "Intervention participants scored significantly higher on post-test anxiety (M = 24.1, SD = 4.8) than control participants (M = 29.6, SD = 5.3), t(78) = 4.98, p < .001, d = 1.09, a large effect." That single sentence gives the reader means, spread, test statistic, significance, and practical magnitude. Nothing else is needed.
Key takeaway: A p-value tells you whether an effect probably exists. The effect size tells you whether anyone should care. Report both, match the measure to the test (Cohen's d for t-tests, η² for ANOVA, Cramér's V for chi-square, R² for regression), and interpret the number in the context of your field rather than relying on Cohen's benchmarks alone.