Why P-Values Are Not Enough
A result that is "statistically significant (p < .05)" tells you that an observed effect is unlikely due to chance. What it does not tell you is how large or meaningful that effect actually is.
Consider this: a study with 10,000 participants finds a 0.3-point difference between groups and reports p < .001. Meanwhile, a study with 30 participants finds a 15-point difference but reports p = .08. The first result is significant while the second is not, yet the second may be far more meaningful. This happens because p values are heavily influenced by sample size.
This is why effect sizes matter. An effect size quantifies the magnitude of a result independently of sample size. APA 7th edition guidelines require effect size reporting alongside significance tests, and most journals now treat it as essential.
This guide covers the most commonly used effect size measures, their interpretation benchmarks, and how to report each one in APA format.
Cohen's d — Effect Size for Mean Differences
When to Use It
Cohen's d measures the difference between two group means in standard deviation units. It is the standard effect size for independent samples t-tests and paired samples t-tests.
Interpretation Benchmarks
Cohen (1988) proposed the following general guidelines:
| Cohen's d | Interpretation | |-------------|----------------| | 0.20 | Small effect | | 0.50 | Medium effect | | 0.80 | Large effect |
A d of 0.50 means the two group distributions overlap by about 67%. A d of 0.80 means the overlap drops to about 53%, a difference most people would readily notice.
APA Reporting Examples
Independent samples t-test:
An independent samples t-test showed that the experimental group (M = 82.40, SD = 10.25) scored significantly higher than the control group (M = 74.60, SD = 11.30) on the post-test, t(58) = 2.89, p = .005, d = 0.75.
Paired samples t-test:
A paired samples t-test revealed that depression scores were significantly lower after the intervention (M = 18.30, SD = 5.40) compared to before (M = 24.10, SD = 6.20), t(34) = 4.52, p < .001, d = 0.76.
Note that Cohen's d can exceed 1.0, so you include a leading zero (e.g., d = 0.75, not d = .75).
Hedges' g — Correcting Cohen's d for Small Samples
The Small-Sample Bias Problem
Cohen's d is the most widely reported effect size for group comparisons, but it has a known limitation: it systematically overestimates the true population effect size when sample sizes are small (roughly n < 20 per group). This bias occurs because the sample standard deviation in small studies tends to underestimate the population standard deviation, inflating the resulting d value.
How Hedges' g Corrects This
Hedges' g applies a correction factor to Cohen's d that adjusts for this small-sample bias. The correction shrinks the effect size estimate slightly, producing a less biased estimate of the population effect size. The smaller the sample, the larger the correction. As sample sizes increase, the correction becomes negligible.
When to Use Hedges' g
- Any study with small group sizes (fewer than 20 participants per group)
- Meta-analyses, where combining effect sizes from studies of varying sample sizes requires unbiased estimates
- Pilot studies, where sample sizes are inherently small
Rule of Thumb
For samples of n > 30 per group, Cohen's d and Hedges' g are nearly identical (typically differing by less than 1%). In these cases, reporting either is acceptable. For samples of n < 20, Hedges' g is the more appropriate choice.
APA Reporting Example
An independent samples t-test showed that participants in the mindfulness group (M = 4.20, SD = 1.15) reported significantly lower stress than the control group (M = 5.10, SD = 1.30), t(18) = 2.45, p = .025, g = 0.72.
Notice that the format is identical to Cohen's d — simply replace d with g. Many journals accept either measure, but if your study has fewer than 20 participants per group, using Hedges' g demonstrates methodological rigor.
Eta Squared and Partial Eta Squared — Effect Size for ANOVA
When to Use Them
Eta squared (η²) and partial eta squared (partial η²) are the standard effect size measures for analysis of variance (ANOVA). They express the proportion of total variance in the dependent variable that is accounted for by an independent variable.
The Difference Between η² and Partial η²
Confusing these two is one of the most common reporting errors in published research.
- η² (eta squared): Proportion of total variance explained by a factor. All factors' η² values sum to at most 1.
- Partial η²: Proportion of variance explained after removing other factors' effects. Values across factors can sum to more than 1.
For one-way ANOVA they are identical. For factorial designs they differ. Most software, including SPSS, reports partial η² by default.
Interpretation Benchmarks
| η² / partial η² | Interpretation | |-----------------|----------------| | .01 | Small effect | | .06 | Medium effect | | .14 | Large effect |
A partial η² of .10 means the independent variable accounts for 10% of the variance in the dependent variable after controlling for other factors.
APA Reporting Examples
One-way ANOVA:
A one-way ANOVA revealed a statistically significant effect of teaching method on achievement scores, F(2, 87) = 5.34, p = .007, η² = .11.
Factorial ANOVA (interaction effect):
The interaction between teaching method and gender was statistically significant, F(2, 84) = 3.92, p = .024, partial η² = .09.
Because η² and partial η² are proportions bounded between 0 and 1, the leading zero is omitted in APA format (e.g., .11 rather than 0.11).
Omega Squared — A Less Biased Alternative to Eta Squared
Why Eta Squared Overestimates
Eta squared (η²) is a descriptive statistic that describes your sample data, but it systematically overestimates the effect size in the population. This overestimation is particularly pronounced with small sample sizes or when there are many groups. The reason is that η² includes both systematic variance (the true effect) and some error variance in its numerator, inflating the estimate.
How Omega Squared Corrects This
Omega squared (ω²) provides a less biased estimate of the proportion of variance explained in the population. It applies a correction that accounts for the number of groups and sample size, resulting in a more conservative and accurate estimate. In practice, ω² values are always smaller than the corresponding η² values from the same data.
Interpretation Benchmarks
Omega squared uses the same general benchmarks as eta squared:
| ω² | Interpretation | |----|----------------| | .01 | Small effect | | .06 | Medium effect | | .14 | Large effect |
APA Reporting Example
A one-way ANOVA revealed a statistically significant effect of treatment condition on anxiety scores, F(3, 76) = 4.82, p = .004, ω² = .12.
When to Use Omega Squared
Many methodologists and an increasing number of journals now recommend ω² over η², particularly for studies with smaller sample sizes. If your field or target journal does not specify a preference, reporting partial η² remains acceptable since it is still the most commonly used ANOVA effect size. However, reporting ω² alongside partial η² demonstrates awareness of the bias issue and strengthens your methods section.
r and R² — Effect Size for Correlation and Regression
When to Use Them
The Pearson correlation coefficient r measures the strength and direction of a linear relationship between two continuous variables. It serves as its own effect size. In regression analysis, the coefficient of determination R² indicates the proportion of variance in the outcome explained by the predictors.
Interpretation Benchmarks
| r (absolute value) | Interpretation | |---------------------|----------------| | .10 | Small effect | | .30 | Medium effect | | .50 | Large effect |
Since R² is the square of r, the corresponding benchmarks are:
| R² | Interpretation | |------|----------------| | .01 | Small effect | | .09 | Medium effect | | .25 | Large effect |
APA Reporting Examples
Correlation:
There was a statistically significant positive correlation between study hours and exam scores, r(48) = .42, p = .003.
Regression:
The regression model was statistically significant, F(2, 97) = 18.45, p < .001, R² = .28, adjusted R² = .26, indicating that study hours and attendance explained 27.5% of the variance in exam scores.
Both r and R² are bounded by 1, so the leading zero is omitted.
Cramér's V — Effect Size for Chi-Square Tests
When to Use It
Cramér's V quantifies the strength of association between two categorical variables in a chi-square test of independence. For 2x2 tables it equals the phi coefficient (φ), but Cramér's V generalizes to larger tables.
Interpretation Benchmarks
For df* = 1 (a 2x2 table):
| Cramér's V | Interpretation | |--------------|----------------| | .10 | Small effect | | .30 | Medium effect | | .50 | Large effect |
Here df* is the smaller of (rows - 1) and (columns - 1). As df* increases, the benchmark thresholds decrease, so always consider the table dimensions when interpreting V.
APA Reporting Example
A chi-square test of independence indicated a significant association between gender and major choice, χ²(2, N = 200) = 12.56, p = .002, V = .25.
Understanding Effect Size Through Real Research Examples
Abstract benchmarks become far more meaningful when grounded in actual research scenarios. The following examples illustrate how the same effect size metric can carry different practical implications depending on the context.
Example 1: Educational Intervention
A school district implements a peer tutoring program and measures its impact on standardized math scores. The result: d = 0.40. What does this mean in practice? A student performing at the 50th percentile without tutoring would be expected to perform at approximately the 66th percentile with tutoring. In a class of 30 students, this translates to roughly 5 additional students scoring above the class median. For an educational intervention that costs relatively little to implement, this is a meaningful improvement.
Example 2: Clinical Psychology
A randomized controlled trial examines cognitive behavioral therapy (CBT) for generalized anxiety disorder. The result: d = 0.75. This means the average patient who received CBT improved more than approximately 77% of patients in the waitlist control group. In clinical terms, this often represents the difference between meeting and not meeting diagnostic criteria for anxiety — a genuinely life-changing outcome for patients.
Example 3: Public Health
A large-scale vaccination study reports an odds ratio of 0.30 for infection risk. This means vaccinated individuals had 70% lower odds of infection compared to unvaccinated individuals. Even though this translates to a relatively modest effect size in standardized terms, when applied to millions of people, it can prevent hundreds of thousands of infections.
The Key Lesson
The same d value can carry vastly different practical significance depending on the domain, the cost of the intervention, the severity of the outcome, and the population size affected. A d of 0.20 in education may matter less than a d of 0.20 in life-saving medical treatment. Always interpret effect sizes within the specific context of your research question.
Confidence Intervals for Effect Sizes
Why Point Estimates Are Not Enough
APA 7th edition explicitly recommends reporting confidence intervals (CIs) for effect sizes, not just the point estimate. A point estimate like d = 0.75 tells you the best single guess for the population effect size, but it says nothing about the precision of that estimate.
What a Confidence Interval for d Means
A 95% CI for Cohen's d provides a range of plausible values for the true population effect size. For example, d = 0.75, 95% CI [0.32, 1.18] means you can be reasonably confident that the true effect size falls somewhere between 0.32 (a small-to-medium effect) and 1.18 (a large effect).
Interpreting the Width
- Wide CI (e.g., [0.10, 1.40]): Low precision. The true effect could be trivially small or very large. This typically occurs with small sample sizes.
- Narrow CI (e.g., [0.60, 0.90]): High precision. You have a good estimate of the true effect size. This typically occurs with large sample sizes.
- CI crossing zero (e.g., [-0.15, 0.85]): The true effect might be zero or even in the opposite direction. This is consistent with a non-significant result.
APA Reporting Example
The experimental group scored significantly higher than the control group, t(58) = 2.89, p = .005, d = 0.75, 95% CI [0.22, 1.27].
The confidence interval provides critical context that the point estimate alone cannot. In this case, while d = 0.75 suggests a medium-to-large effect, the CI indicates the true effect could be as small as 0.22 (small) or as large as 1.27 (very large). This level of transparency helps readers assess the robustness of your findings.
Visualizing Effect Sizes: Distribution Overlap
Making Abstract Numbers Intuitive
One of the most effective ways to understand what an effect size means is to visualize how much two group distributions overlap. When two groups have identical means (d = 0.0), their distributions overlap completely. As d increases, the distributions pull apart.
Overlap at Different Effect Sizes
| Cohen's d | Distribution Overlap | Practical Meaning | |-------------|---------------------|-------------------| | 0.0 | 100% | Identical distributions | | 0.2 | ~85% | Almost indistinguishable; differences visible only in aggregate | | 0.5 | ~67% | Noticeable difference; most individuals still overlap | | 0.8 | ~53% | Obvious difference; about half the distributions still overlap | | 1.0 | ~45% | Very clear difference; less than half overlap | | 1.5 | ~30% | Dramatic difference; minimal overlap | | 2.0 | ~19% | Extreme difference; distributions barely overlap |
Cohen's U3 Statistic
Another way to interpret effect sizes is through Cohen's U3, which indicates what percentage of the lower-scoring group the average person in the higher-scoring group exceeds.
| Cohen's d | U3 (percentile of higher group) | |-------------|--------------------------------| | 0.2 | 58% | | 0.5 | 69% | | 0.8 | 79% | | 1.0 | 84% | | 1.5 | 93% |
At d = 0.8, the average person in the higher-scoring group performs better than 79% of people in the lower-scoring group. This translation from standard deviation units to percentiles makes effect sizes immediately understandable to non-statistical audiences, such as clinicians, educators, and policymakers.
Effect Size Benchmarks Vary by Field
Cohen's Benchmarks Are Defaults, Not Universal Rules
Cohen (1988) himself described his small/medium/large conventions as guidelines for when researchers have no better frame of reference. They were never intended to be applied mechanically across all disciplines. In practice, what counts as a meaningful effect size varies dramatically across fields.
Education
Hattie (2009), in his synthesis of over 800 meta-analyses of educational interventions, identified d = 0.40 as the "hinge point." Effects above this threshold represent interventions that produce meaningful improvements beyond what students would gain through normal development. By this standard, many interventions considered to have "small" effects by Cohen's criteria are actually producing educationally significant results.
Clinical Psychology
In clinical psychology, even d = 0.20 can be clinically meaningful when the condition is severe. A small reduction in symptoms of psychosis, suicidality, or chronic pain can substantially improve quality of life. The clinical significance of an effect depends on the severity of the disorder and the availability of alternative treatments.
Social Psychology
Meta-analyses in social psychology show that the typical effect size is between d = 0.20 and d = 0.40. What would be considered "small" by Cohen's standards is actually quite typical in this field. Expecting large effects from subtle social manipulations is unrealistic, and researchers in this area should calibrate their expectations accordingly.
Medical Research and Public Health
In medical research, even tiny effects measured by odds ratios close to 1.0 (e.g., OR = 0.95) can save thousands of lives when applied at population scale. A medication that reduces heart attack risk by 5% may seem trivial in terms of effect size, but across millions of patients it prevents tens of thousands of heart attacks.
Recommendation
Rather than relying solely on Cohen's conventions, compare your effect sizes to the distribution of effects reported in prior studies within your specific research area. Many fields now publish meta-analytic benchmarks that provide discipline-specific reference points. This approach yields more meaningful interpretation than applying one-size-fits-all labels.
Effect Size Summary Table
The following table provides a quick reference for all major effect size measures and their interpretation benchmarks.
| Statistical Test | Effect Size Measure | Small | Medium | Large | |-----------------|-------------------|-------|--------|-------| | t-test | Cohen's d | 0.20 | 0.50 | 0.80 | | ANOVA | η² / partial η² | .01 | .06 | .14 | | Correlation | r | .10 | .30 | .50 | | Regression | R² | .01 | .09 | .25 | | Chi-square | Cramér's V | .10 | .30 | .50 |
Important: These are general guidelines, not rigid rules. Cohen himself called them conventions for when no better basis is available. In some fields, a "small" effect can have substantial real-world impact. Always interpret effect sizes within your research context.
Common Mistakes to Avoid
Confusing η² with Partial η²
SPSS labels its output "Partial Eta Squared," but many researchers report the value as plain η². In factorial designs the two differ, so always specify which you are reporting using partial η² or ηp².
Reporting Only Significance Without Effect Size
Stating "p < .05" without an effect size does not meet APA 7th edition standards. Report an effect size for every inferential test, whether significant or not. Non-significant effect sizes are valuable for power analyses and meta-analyses.
Mechanically Applying Cohen's Benchmarks
Labeling every d = 0.45 as "medium" without considering context is an oversimplification. Compare your effect sizes to prior studies in your field for more meaningful interpretation.
Getting the Leading Zero Wrong
Values that cannot exceed 1 (p, r, η², R², V) omit the leading zero (e.g., .42). Values that can exceed 1 (Cohen's d, M, SD) include it (e.g., 0.75). Mixing up this rule is a frequent formatting error.
Omitting Effect Size from Chi-Square Results
Many researchers report χ² and p without Cramér's V. Effect sizes should accompany all statistical tests, including those with categorical data.
Frequently Asked Questions
Can Cohen's d be greater than 1?
Yes. Cohen's d is unbounded and can take any positive value. A d of 1.0 means the two group means differ by exactly one standard deviation. A d of 1.5 means they differ by one and a half standard deviations. Values above 1.0 are uncommon but occur regularly in research with strong manipulations or highly distinct populations (for example, comparing expert musicians with non-musicians on auditory tasks).
What does a negative effect size mean?
A negative effect size reflects the direction of the difference, not its magnitude. It simply means the group you designated as "Group 1" scored lower than "Group 2." If you reverse the group labels, the sign flips. When interpreting magnitude, use the absolute value. For example, d = -0.60 and d = 0.60 represent the same size of effect in opposite directions.
Which effect size should I report for my analysis?
The appropriate effect size depends on the statistical test you are using. For t-tests, report Cohen's d (or Hedges' g for small samples). For ANOVA, report partial η² (or ω²). For correlations, r itself is the effect size. For regression, report R². For chi-square tests, report Cramér's V. Refer to the summary table above for a quick reference.
Does a large effect size prove causation?
No. Effect size quantifies the magnitude of a relationship or difference, but it does not establish causation. A large d in an observational study may reflect confounding variables rather than a causal mechanism. Causal claims require appropriate research designs (such as randomized controlled trials), not simply large effect sizes.
What is the effect size for non-parametric tests?
For the Mann-Whitney U test, the rank-biserial correlation (r) is the standard effect size. For the Wilcoxon signed-rank test, r = Z / sqrt(N) is commonly used. For the Kruskal-Wallis test, epsilon squared (ε²) or eta squared based on ranks can be reported. For the Friedman test, Kendall's W serves as the effect size. These measures use the same small/medium/large interpretation framework as their parametric counterparts.
How does SPSS output effect sizes?
SPSS reports partial η² by default for ANOVA procedures (found in the "Tests of Between-Subjects Effects" table when you check "Estimates of effect size"). However, SPSS does not automatically calculate Cohen's d for t-tests — you must compute it manually or use a dedicated tool. For regression, SPSS provides R² in the Model Summary table. For chi-square, you need to request Cramér's V through the Crosstabs procedure (under Statistics > Phi and Cramér's V).
What is the relationship between sample size and effect size?
Effect size and sample size are theoretically independent. A large sample does not produce a larger effect size, and a small sample does not produce a smaller one. However, small samples produce less precise estimates of effect size (wider confidence intervals), which means the observed d in a small study may differ substantially from the true population d. This is one reason why Hedges' g correction is recommended for small samples.
Should I report effect size for non-significant results?
Yes. APA 7th edition requires effect size reporting for all inferential tests, regardless of whether the result is statistically significant. Non-significant results with effect size estimates are valuable for several reasons: they inform power analyses for future studies, they contribute to meta-analyses, and they prevent publication bias by providing a complete picture of the evidence. A non-significant result with d = 0.45 tells a very different story than one with d = 0.02.
Using StatMate to Calculate Effect Sizes Automatically
StatMate's statistical calculators automatically compute effect sizes alongside every test result.
- T-test calculator: Outputs Cohen's d with 95% confidence intervals
- ANOVA calculator: Provides both η² and partial η²
- Correlation calculator: Reports r and R² together
- Chi-square calculator: Computes Cramér's V automatically
All results follow APA 7th edition conventions, so you can paste them directly into your manuscript. This eliminates manual calculation errors and saves significant writing time.
Wrapping Up
Effect sizes transform statistical results from a simple "significant or not" verdict into a meaningful statement about magnitude. While p values indicate whether an effect likely exists, effect sizes tell you whether it matters in practice. Mastering Cohen's d, η²/partial η², r/R², and Cramér's V ensures your research communicates both statistical rigor and real-world relevance.