Effect Sizes: How to Measure and Report the Size of a Result

An effect size is a number that answers the question a p-value can't: not whether a result exists, but how big it is. Two studies can both report significance at p < .001 and mean completely different things. One found a difference that changes clinical practice. The other found one too small to notice. The effect size is what separates them, and it's why journals across psychology, medicine, education, and economics now expect one alongside every significance test.

This guide covers why significance alone is not enough and how the two families of effect sizes differ. It then walks through calculating Cohen's d by hand and matching the right measure to your analysis. The last two sections cover the small-medium-large benchmarks and how to report the result.

Which Effect Size Goes With Which Analysis

Almost every test has a matching effect size, and picking the wrong one is a common reviewer catch. Find the analysis you ran, and read across.

Why a P-Value Is Not Enough

A significance test asks one narrow question: assuming no effect exists, how surprising is this result? It returns a probability, not a magnitude. That's a useful question. It's just not the question most readers actually have, and the gap between the two is where a lot of overstated research lives.

The problem is that p-values respond to sample size. Collect enough data and almost any difference, however tiny, will cross the significance threshold. A study of 50,000 people can find a two-point difference on a 100-point scale at p < .001. The result is real in the narrow statistical sense and irrelevant in every other sense. Reporting only the p-value hides that.

It cuts the other way too. A genuinely large effect in a small sample can miss significance entirely. A write-up reporting only "no significant difference" then buries a finding worth following up. The effect size makes the magnitude visible either way, which is why reporting both is now standard rather than optional. For the underlying logic of significance itself, see the guide to hypothesis testing.

The Two Families

Effect sizes divide into two broad groups, and knowing which one you're looking at tells you how to read the number.

Difference-based measures

These express the size of a gap between groups in standard deviation units. Cohen's d is the best known, with Hedges' g and Glass's delta as close relatives. They answer "how far apart are these groups, relative to how much the data varies?" Because the units cancel out, a d of 0.5 means the same thing whether you measured blood pressure or reading scores.

Association-based measures

These express how strongly two variables relate, or what share of the variance one accounts for in the other. Pearson's r, r squared, eta-squared, and Cramer's V all belong here. The variance-explained members run from 0 to 1 and are read as proportions. The mechanics of r, the most common member of this family, are covered in the guide to what the correlation coefficient measures.

Cohen's d, Worked by Hand

Seeing d calculated once makes it stop being abstract. Suppose you compared exam scores for 10 students who received tutoring against 10 who didn't.

Step 1: Pool the standard deviations

Cohen's d divides the difference between means by a single standard deviation representing both groups. The pooled standard deviation is a weighted combination of the two, with each group weighted by its degrees of freedom.

Step 2: Standardize the difference

Now divide the raw gap between the means by that pooled value. The units of measurement disappear, leaving a pure number.

Step 3: Read the result

A d of 0.91 means the two group means sit about nine tenths of a standard deviation apart. By Cohen's conventions that's a large effect, and unlike the 6.3-point raw difference, it's directly comparable to results from studies that used a different exam.

A note on Hedges' g

Cohen's d runs slightly high in small samples, overstating the effect. Hedges' g applies a correction factor that shrinks the estimate, and with 20 students here it brings d of 0.91 down to g of 0.87. The correction is negligible once samples pass about 20 per group. With small samples, g is the better-behaved choice, and many journals now prefer it by default.

Effect Sizes for ANOVA

With three or more groups, the effect size question changes shape. It's no longer "how far apart are these two means?" but "how much of the variation is tied to group membership?" The answer comes as a proportion.

Eta-squared divides the variance attributed to the grouping factor by the total variance. Say the between-groups sum of squares is 245.6 and the total is 1,180.4. Eta-squared is then 0.21, so about 21 percent of the variance in the outcome is associated with group membership.

Partial eta-squared is what most software reports for factorial designs. It divides the effect's variance by that effect's variance plus the error variance, ignoring other factors in the model. This matters because partial eta-squared values are larger than eta-squared values in the same analysis, and they don't sum to 1 across effects. Reporting one while labeling it the other inflates your apparent effect, which is a common and easily caught error.

Omega-squared is a less biased alternative to eta-squared that adjusts for sample size, and some fields now prefer it. Whichever you use, name it explicitly rather than writing "effect size = 0.21," so the reader knows which quantity they're reading. The tests themselves are covered in the guide to ANOVA.

Effect Sizes for Categorical and Binary Outcomes

When the outcome isn't continuous, the d and eta families don't apply, and different measures take over.

• Odds ratio. The standard effect size for a binary outcome, and the quantity logistic regression produces directly. An odds ratio of 2.08 means each one-unit increase in the predictor roughly doubles the odds of the outcome. Values above 1 increase the odds, below 1 decrease them, and 1 means no effect.
• Risk ratio. Common in medicine and epidemiology, and easier to interpret than the odds ratio because it compares probabilities rather than odds. The two converge only when the outcome is rare, so they aren't interchangeable.
• Cramer's V. The effect size for a chi-square test of association, running from 0 to 1. For a 2x2 table it reduces to phi. Chi-square results are frequently reported with the test statistic and p-value alone, which leaves out the magnitude entirely.
• Rank-biserial correlation and epsilon-squared. The counterparts for Mann-Whitney and Kruskal-Wallis respectively, covered alongside the tests in the guide to non-parametric tests.

What the Benchmarks Actually Mean

Cohen's small, medium, and large labels are the most cited numbers in this whole area, and the most misused. Here they are, alongside the equivalents for other measures.

Cohen proposed these as a fallback for researchers with nothing better to go on, and he said so at the time. They were not derived from data about any particular field. Since then, meta-analyses in several disciplines have found that typical effects run well below Cohen's medium. Judging your result against his conventions can therefore make an ordinary finding look disappointing, or a modest one look impressive.

The better move is to compare your effect against the published range in your own literature. If interventions in your area typically produce a d around 0.25, your d of 0.35 is a strong result. The conventional label for it is beside the point. When no field-specific comparison exists, use Cohen's benchmarks and say explicitly that you're doing so.

One more caution: statistical magnitude and practical importance are different things. A small standardized effect on a widely prescribed medication can matter enormously at population scale. A large effect on a measure nobody cares about matters not at all. The number sets up the argument about importance; it doesn't settle it.

Confidence Intervals Around Effect Sizes

An effect size is an estimate from a sample, so it carries uncertainty, and journals increasingly expect that uncertainty reported. Consider a d of 0.91 with an interval running from 0.02 to 1.80. It's a very different finding from a d of 0.91 with an interval from 0.75 to 1.07, though the point estimates match.

Wide intervals are the norm in small samples, and reporting them honestly is what stops a single underpowered study from being read as settled. Whether an interval includes zero also tells you what the significance test would have said, which makes the interval the more informative of the two.

Effect Size, Power, and Sample Size

Effect size does its most useful work before you collect any data. A power analysis asks how many participants you need to reliably detect an effect of a given size. The effect size you assume is the main input. Smaller assumed effects require substantially larger samples.

That assumed value should come from prior published research or pilot data where possible, falling back on Cohen's conventions only when neither exists. Optimistic assumptions are the most common route to an underpowered study. Assume a d of 0.8, find a d of 0.3, and the study was never large enough to detect what was actually there. The full procedure, including how to translate an assumed effect into a target n, is covered in the guide to calculating sample size.

Reporting Effect Sizes in APA Format

The effect size goes in the same sentence as the significance test, not in a separate paragraph and not only in a table. A complete report of the tutoring example reads:

A few conventions hold across measures. Name the specific measure rather than writing "effect size," since d, g, and partial eta-squared are not interchangeable. Italicize the symbol and leave the number upright. Report two decimal places for most measures. Include the confidence interval where your software provides one. And interpret the number in the surrounding text, because a reader shouldn't have to know Cohen's conventions by heart to understand what you found.

Common Mistakes

• Reporting significance without magnitude. A p-value alone tells the reader an effect probably isn't zero. It says nothing about whether it's worth caring about.
• Treating Cohen's benchmarks as authoritative. They're a fallback for when field-specific comparisons don't exist. Compare against your own literature first.
• Confusing eta-squared with partial eta-squared. Partial values are larger in the same analysis. Label which one you're reporting.
• Calling a large effect important without argument. Statistical magnitude and practical significance are separate claims. Make the second one explicitly, with reference to the outcome's real-world stakes.
• Reporting an effect size that doesn't match the test. Cohen's d belongs to mean comparisons, not chi-square tables. Use the pairing your analysis calls for.
• Omitting the confidence interval. A point estimate from a small sample can be wildly imprecise, and the interval is what shows it.
• Computing an effect size only after a significant result. Nonsignificant findings need effect sizes too. A large effect that missed significance is a signal the study was underpowered, and that's worth reporting rather than hiding.

Where to Go Next

Effect sizes sit alongside the tests that produce them. For the comparisons that generate a Cohen's d, see the guides to t-tests and ANOVA. For the association measures, see what the correlation coefficient measures and using correlation coefficients in research papers, which covers reporting conventions in detail. For the standard deviation that sits underneath every difference-based measure, see the guide to standard deviation and variance.

When the analysis is written up, the results section is where a subject-matter editor earns their place. A reviewer who spots a mislabeled effect size, or a magnitude claim the number doesn't support, will question everything around it. Editor World's academic editing, dissertation editing, and journal article editing services include review of statistical reporting. You choose your own editor by field, so the person reading your results knows your discipline's conventions.

Frequently Asked Questions

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