Statistics interpretation guide
Effect Size Chart: Cohen’s d, r, Eta-Squared and Cramer’s V
Effect size describes magnitude, but the scale changes with the research question and statistic. This guide connects each common measure to the contrast it was designed to summarize.
Answer first
Choose the effect size from the design, not from a universal chart
Use Cohen’s d or a related standardized mean difference for a contrast between means, r for strength and direction of association, eta-squared for variance attributed to an ANOVA effect in the sample, and Cramer’s V for association in a contingency table.
Threshold labels such as small, medium, and large are rough conventions, not universal facts. Interpret the estimate with confidence intervals, units, baseline risk, prior research, design, and consequences.
A method you can reuse
Match the measure to the comparison
Readers sometimes search for Hattie’s effect size chart 2024 or a single benchmark table. A dated chart cannot decide whether an effect matters in a new subject, population, outcome scale, or decision. The measure and context must come first.
Report both the raw effect when it is meaningful and the standardized effect when comparison across scales is useful. A five-point mean difference may be easier for a practitioner to understand than d = 0.50.
Name the exact contrast
Is the question about two means, a continuous association, explained variation, or categorical association?
Choose the measure and denominator
Standardization changes interpretation. State whether the denominator is a pooled SD, residual variance, total variation, or table size.
Calculate direction and magnitude
Keep the sign for measures where direction matters. Check group coding and reference categories.
Add uncertainty
A point estimate without a confidence interval can look more certain than the sample supports.
Interpret in the field
Compare with meaningful thresholds, prior studies, costs, risks, and the outcome’s real scale, not only generic labels.
Worked from start to finish
Four effect-size calculations from simple summaries
These separate examples show why one formula cannot cover every analysis.
M1 = 82, M2 = 76, pooled SD = 10
d = (82 – 76) / 10 = 0.60
Correlation:
r = 0.40, so r² = 0.16 in a simple bivariate interpretation
ANOVA share of variation:
SS between = 14, SS total = 20
eta² = 14 / 20 = 0.70
2 × 2 categorical association:
chi-square = 10, N = 100, min(r – 1, c – 1) = 1
Cramer’s V = √(10 / 100) = 0.316
The d value preserves direction only if the group order is stated. The correlation can be positive or negative. Eta-squared and Cramer’s V are nonnegative and require context from the design and variables.
The eta-squared example is deliberately small and unstable. Effect-size estimates can be biased upward in small samples, so report uncertainty and consider less biased alternatives where appropriate.
Effect size chart by research question
This chart is a selection guide, not a universal importance scale.
| Question | Common measure | Report with it |
|---|---|---|
| How far apart are two means? | Cohen’s d, Hedges’ g, raw mean difference | Group means, SDs, n, confidence interval |
| How strongly do two continuous variables move together? | Pearson r or Spearman rho | Scatterplot, n, interval, design |
| How much sample variation is attributed to an ANOVA effect? | Eta-squared, partial eta-squared, omega-squared | ANOVA table, design, interval when available |
| How strong is association in a contingency table? | Phi or Cramer’s V | Counts, percentages, chi-square, df |
| How much does an intervention change risk? | Risk difference, risk ratio, odds ratio | Baseline risk and confidence interval |
Statistical significance and effect size answer different questions. With a very large sample, a tiny effect can have a small p-value. With a small sample, a practically important effect can remain imprecise. Report both without letting either replace study quality.
Standardized measures help compare outcomes measured on different scales, but they can hide what a change means. Whenever possible, translate back to the original outcome, predicted probabilities, or an absolute risk difference.
What students report
Real student experiences, with context
These public comments are personal experiences, not universal outcomes. They are included because they show where students commonly get stuck and how the method above helps.
“My understanding of effect size is kinda weak”
Student discussion in r/statistics
Effect size is not one universal number. The correct measure depends on whether the contrast concerns means, association, explained variance, or categorical association.
“Getting it isn’t the problem; over-interpreting it is.”
Student discussion in r/statistics
Software can calculate a p-value instantly. The harder work is checking the design, assumptions, multiplicity, effect size, uncertainty, and practical importance.
Failure-mode review
Common problems and how to repair them
Using one benchmark in every discipline
Use field evidence and meaningful differences. A generic label is only a starting reference.
Mixing eta-squared with partial eta-squared
Name the exact statistic. Their denominators differ and values cannot be compared casually across designs.
Omitting group direction for d
State which mean was subtracted from which so the sign has meaning.
Reporting magnitude without uncertainty
Add a confidence interval or clearly explain why one is unavailable. Small samples can produce unstable estimates.
Before you submit or move on
A practical final check
- The effect size matches the variable types and design.
- The formula and denominator are named.
- Coding and group direction are explicit.
- Raw effects are reported where interpretable.
- Confidence intervals accompany point estimates when possible.
- Generic benchmarks are not treated as universal rules.
- Practical importance is separated from statistical significance.
Questions students ask
Frequently asked questions
What is the best effect size to use?
It depends on the research question and design. Use a standardized mean difference for mean contrasts, r for continuous association, eta measures for ANOVA effects, and Cramer’s V for categorical association.
Is Cohen’s d of 0.5 always medium?
No. That label is a rough convention. Importance depends on the outcome, field, baseline, uncertainty, costs, and consequences.
What is the difference between eta-squared and partial eta-squared?
They use different denominators. Partial eta-squared excludes variation associated with other effects from its denominator, so label the statistic clearly.
Should I report a p-value and effect size?
Usually yes, along with confidence intervals and descriptive statistics. They answer different questions and neither replaces design quality.
Can effect sizes be negative?
Directional measures such as d and r can be negative depending on group order or association direction. Magnitude-only measures such as eta-squared and Cramer’s V are nonnegative.
Sources and further reading
- American Statistical Association Statement on P-Values. Official principles for responsible p-value interpretation.
- NIST Engineering Statistics Handbook: One-Way ANOVA. Government reference for the one-way fixed-effects ANOVA model.
Forum quotations are short excerpts from public discussions. They describe individual experiences and have not been independently verified. Factual guidance in this article is grounded in the primary and institutional sources listed above.