The effect size calculator above converts a raw difference between two group means into a number that no longer depends on the units you measured in. Feed it two means, two standard deviations and two sample sizes and it returns Cohen's d, the small-sample bias correction known as Hedges' g, Glass's delta, eta squared, and a confidence interval around d. A difference of five points is meaningless on its own. Five points when the typical spread within each group is fourteen points is a small shift; five points when the spread is one point is enormous. Effect size is the arithmetic that tells those two situations apart.
Arb Digital publishes this page because the number that decides whether a result matters is almost never the p-value. A conversion test on a hundred thousand sessions will return a significant p-value for a difference nobody could act on, and a careful study of thirty people will return a non-significant p-value for a difference that would be worth chasing with a bigger sample. The p-value answers a question about the data being unusual under a null hypothesis; the effect size answers the question about how big the thing is. They are different questions and you need both.
What This Effect Size Calculator Does
It handles the two-independent-groups case, which is the design effect sizes were invented for. You supply summary statistics rather than raw data, because summary statistics are what appear in a paper, a report or a statistics package's output, and they are sufficient for every quantity on this page. The tool pools the two standard deviations, divides the mean difference by the result, corrects for the upward bias that appears in small samples, converts the same information into the variance-explained metrics, and puts an interval around the estimate.
The pooling itself is the part most often done wrong by hand, and it is why this page exists alongside our pooled standard deviation calculator. That tool gives you the pooled SD as an end product for any number of groups; this one takes the same pooled quantity as an intermediate step and carries it through to a standardised difference. If you only need the yardstick, use the pooled SD page. If you need the ratio, stay here.
How to Use It
- Enter the two means. Put the treatment or variant group first. The sign of d follows the order you choose, so a positive d means group 1 scored higher.
- Enter the two standard deviations. Use the sample SD with the n minus 1 denominator, which is what almost every software package reports by default. Our standard deviation calculator will produce it from raw numbers if you only have a list of values.
- Enter the two sample sizes. These control both the pooling weights and the size of the small-sample correction, so approximate values will give you an approximate answer.
- Set the confidence level. Ninety-five percent is conventional; the interval widens as you raise it.
- Read d, then g, then the interval. If the sample sizes are under about fifty per group, quote g rather than d.
The Formulas and How They Are Calculated
Cohen's d is the mean difference divided by the pooled standard deviation. The pooled SD is not the average of the two SDs. It is the square root of a weighted average of the two variances, with each variance weighted by its own degrees of freedom: sp = √[((n1−1)s12 + (n2−1)s22) / (n1+n2−2)]. Averaging the standard deviations directly gives a different, wrong answer whenever the two SDs differ, and the error grows with the difference.
Hedges' g multiplies d by a correction factor J = 1 − 3/(4df − 1), where df is n1+n2−2. This is the standard approximation to the exact gamma-function form and it is accurate to well under a thousandth for any df above about ten. Glass's delta divides the mean difference by the control group's SD alone rather than the pooled value, which is the right choice when the treatment is expected to change the spread as well as the centre.
Eta squared is the proportion of total variance attributable to group membership. For two groups it is obtained from the t statistic as η2 = t2/(t2+df), where t itself is d multiplied by √(n1n2/(n1+n2)). The confidence interval uses the large-sample standard error SEd = √[(n1+n2)/(n1n2) + d2/(2(n1+n2))]. The NIST/SEMATECH e-Handbook's section on sample sizes required uses the same standardised shift, writing the difference to be detected in units of sigma for exactly this reason.
Small, Medium and Large Are Conventions, Not Facts
Cohen proposed 0.2, 0.5 and 0.8 as rough labels for small, medium and large values of d, and he was explicit that they were a fallback for researchers with no better information in their own field. They are not thresholds, they were not derived from data, and they carry no meaning about whether an effect is worth acting on. The calculator prints the conventional label because readers expect it, but treat it as a rough sense of scale rather than a verdict.
The reason to distrust the labels is that the same d means completely different things depending on what is being measured and what it costs to obtain. A d of 0.1 on a mortality outcome across a national population is a very large effect in every sense that matters. A d of 1.2 on a self-report questionnaire administered immediately after an intervention, by the person who delivered it, may be almost entirely demand characteristics. Neither of those readings comes from the number. They come from the design, the outcome and the field, and the labels cannot substitute for knowing them.
Why the Bias Correction Matters More Than People Expect
Cohen's d systematically overestimates the population effect size in small samples. The bias comes from the pooled standard deviation, which is a downward-biased estimator of the population sigma; a denominator that is too small makes the ratio too big. The correction factor J is always slightly below one and always shrinks the estimate toward zero.
The size of the shrinkage is easy to underestimate. With ten per group, df is 18 and J is about 0.958, so a d of 1.00 becomes a g of 0.958. With five per group, df is 8 and J is about 0.903, a ten percent reduction. Above about fifty per group the correction falls below one percent. Report g whenever either group is small: it is never worse than d, and a meta-analysis built from many small studies reporting uncorrected d inherits an upward bias in every input.
Effect Size and Significance Are Not the Same Question
The t statistic this page computes internally is d multiplied by a function of sample size alone. That single line of algebra explains the whole relationship between the two concepts. Hold the effect size fixed and grow the sample and t rises without limit; the p-value from our p-value calculator falls toward zero while the effect size does not move at all. Significance is effect size multiplied by evidence, and evidence is bought with sample size.
This has two practical consequences. First, a significant result in a very large sample tells you almost nothing about magnitude on its own, so always report the effect size next to the p-value. Second, a non-significant result in a small sample is not evidence of no effect. The confidence interval is what distinguishes the two cases: an interval from −0.05 to 0.08 says the effect is small; an interval from −0.6 to 0.9 says the study could not tell. Both are non-significant and they mean opposite things. Use our t-test calculator for the formal test and read the two outputs together.
Choosing Between d, Glass's Delta and Eta Squared
Use Cohen's d or Hedges' g when the two groups plausibly share a population variance and you want a difference expressed in standard deviation units. This is the default for a randomised two-group comparison, where randomisation gives you a reason to expect similar spread.
Use Glass's delta when the intervention is expected to change the variability, not just the mean. Pooling a treatment SD that has been inflated or compressed by the treatment itself contaminates the yardstick with the very thing you are measuring, and the control SD is the cleaner reference. The calculator uses group 2 as the control for this reason, so enter your baseline group second.
Use eta squared when you want a proportion-of-variance figure comparable to what an analysis of variance reports. It reads intuitively as the fraction of total variation explained by group membership, but it is bounded at zero, which biases it upward in small samples. Our ANOVA calculator handles the multi-group case where it comes into its own, and Penn State's STAT 500 Applied Statistics course notes cover the two-sample comparison these measures sit on top of.
A Worked Example You Can Check Yourself
The default values are a treatment group with a mean of 105 and an SD of 15 across 30 people, against a control group with a mean of 100 and an SD of 14 across 30 people. The pooled variance is (29×225 + 29×196)/58 = 12,209/58 = 210.5, so the pooled SD is 14.509. Cohen's d is 5/14.509 = 0.345.
Degrees of freedom are 58, so J = 1 − 3/231 = 0.987 and Hedges' g is 0.340. Glass's delta uses the control SD alone: 5/14 = 0.357. The t statistic is 0.345 × √(900/60) = 0.345 × 3.873 = 1.335, giving eta squared of 1.3352/(1.3352+58) = 0.030 — group membership accounts for about three percent of the total variation. The standard error of d is √(60/900 + 0.3452/120) = 0.260, so the 95 percent interval runs from about −0.17 to 0.85.
That interval crosses zero, which means the same data would produce a non-significant t-test. The point estimate looks like a small-to-moderate effect and the interval is consistent with everything from a small effect in the opposite direction to a large one in the stated direction. Sixty people cannot distinguish those. Our confidence interval calculator shows the same widening behaviour for means, and the honest conclusion here is that the study needs to be larger, not that the treatment failed.
Arb Digital's analysts size an effect before they call it a win, and size the sample before they call a flat result a failure.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Averaging the two standard deviations — pooling weights the variances by degrees of freedom, and the shortcut diverges from the correct value as soon as the two SDs differ.
- Entering standard errors instead of standard deviations — a standard error is the SD divided by the square root of n, so using it inflates every effect size on this page by roughly a factor of √n.
- Quoting uncorrected d from a tiny sample — with under about twenty per group the bias is large enough to change how the result reads, and g costs nothing to report.
- Treating 0.8 as a finish line — the conventional labels were offered as a last resort in the absence of field benchmarks, not as a scale of importance.
- Reporting a point estimate with no interval — effect sizes from small samples are extremely unstable, and the interval is the only part of the output that shows it.
Related Free Tools From Arb Digital
Get the spread from raw values with the standard deviation calculator, combine several groups' spreads with the pooled standard deviation calculator, run the formal comparison with the t-test calculator, convert a statistic into a tail probability with the p-value calculator, extend the comparison past two groups with the ANOVA calculator, or put an interval around an estimate with the confidence interval calculator. Every statistics tool we publish is listed on the free online tools hub.
Frequently Asked Questions
There is no universal answer. Cohen offered 0.2, 0.5 and 0.8 as rough labels for small, medium and large, but he presented them as a fallback for fields with no benchmarks of their own. Whether an effect matters depends on the outcome being measured and the decision it feeds.
They use the same formula, but Hedges' g multiplies the result by a correction factor slightly below one that removes the upward bias present in small samples. The two agree closely above roughly fifty observations per group and diverge as samples shrink.
Pooling averages the variances, not the standard deviations, and it weights each variance by its own degrees of freedom. Averaging the standard deviations directly gives a different number whenever the two groups differ in spread or in size.
When the treatment is likely to change the variability as well as the mean. Pooling would then contaminate the yardstick with the effect being measured, so Glass's delta uses the control group standard deviation on its own.
Yes. The sign simply records which group had the higher mean, and it flips if you swap the order of entry. Only the magnitude describes the size of the effect.
No. Significance depends on the effect size and the sample size together. A large effect in a very small sample can easily be non-significant, and a tiny effect in a very large sample will usually be significant.
Both describe the same difference in different units. Eta squared expresses it as the proportion of total variance explained by group membership, and for two groups it can be derived directly from the t statistic that d produces.
This page explains a statistical calculation for educational purposes only. It is not medical, legal or financial advice, and an effect size on its own cannot establish that one treatment, product or intervention is better than another.
