Last updated: 2026-06-23

What an effect size measures

When two groups differ — a treatment group and a control group, students taught two different ways, machines running two settings — the natural first question is whether the difference is real. A significance test and its p-value answer that: they tell you whether the observed gap is larger than you would expect from random sampling alone. What a p-value does not tell you is how big the difference is. A p-value shrinks as the sample grows, so with enough data almost any difference, however trivial, becomes "statistically significant."

An effect size fills that gap. Cohen's d is the most widely used effect size for comparing two means. It expresses the distance between the group averages in units of standard deviation, which strips out the original measurement scale and lets you compare effects across studies, instruments, and fields.

The pooled standard deviation and the formula

To put the mean difference on a standardized scale, Cohen's d divides it by the pooled standard deviation — a single spread estimate that blends both groups, weighting each by its degrees of freedom:

Cohen's d is then the standardized mean difference:

The sign of d reflects direction — a positive value means Group 1 has the higher mean — and its magnitude reflects the size of the effect. Because the pooling assumes the two groups share a common population variance, d pairs naturally with the two-sample t-test, which makes the same assumption.

Small, medium, and large

Jacob Cohen suggested conventional benchmarks so that a bare number like 0.46 carries some intuition:

These are rules of thumb, not laws. Cohen himself warned against treating them mechanically. In a well-controlled lab task a d of 0.2 might be unremarkable, while in a large public-health intervention the same value could translate into thousands of affected people. The benchmarks are a starting point for interpretation, not a substitute for it — always read d against the context of the field and the practical stakes.

Worked example

Suppose Group 1 has a mean of 100 with a standard deviation of 15 from 30 observations, and Group 2 has a mean of 110 with the same standard deviation and sample size. First the pooled standard deviation, then d:

The magnitude, |d| ≈ 0.67, falls in the medium range, and the negative sign tells us Group 2 scored higher. Reported in plain terms: the two groups differ by about two-thirds of a standard deviation.

Relation to the t-test and small-sample correction

Cohen's d and the Two-Sample t-Test Calculator (Welch) describe the same comparison from two angles. The t-test asks whether the difference is distinguishable from zero; d asks how large that difference is. A thorough report gives both, just as a Z-Score Calculator is often reported alongside a percentile to convey both position and likelihood.

One refinement is worth noting. Cohen's d slightly overestimates the population effect in small samples. Hedges' g applies a simple correction factor — approximately 1 − 3 / (4(n₁ + n₂) − 9) — that shrinks d toward zero. The adjustment is negligible once the combined sample passes roughly 50, but for small studies reporting Hedges' g is the more honest choice.