Cohen’s d CalculatorEffect Size for t-Tests, with Hedges’ g and Exact Confidence Intervals

Find Cohen’s d from means and standard deviations, raw data, paired pre/post scores or a t-statistic. You get Hedges’ g, Glass’s Δ, an exact 95% confidence interval, plain-language interpretation and the sample size you need for your next study.

4 study designs Exact noncentral-t CI d_z · d_av · d_rm for paired data U3, overlap and NNT Power and sample size

By Manzoor Ahmad •Reviewed and updated October 21, 2026

Formulas follow Cohen (1988), Hedges (1981) and Lakens (2013). Confidence intervals invert the noncentral t distribution as described by Cumming and Finch (2001). Results are cross-checked against SciPy, and all math runs in your browser.

What is Cohen’s d?

Cohen’s d is a standardized effect size that expresses the difference between two means in standard-deviation units: d = (M₁ − M₂) / SD_pooled. By Cohen’s benchmarks, d ≈ 0.2 is small, 0.5 is medium and 0.8 is large. Unlike a p-value, d does not grow with sample size, so it tells you how big a difference is, not just whether it exists.

Mean (M₁)

SD (s₁)

n₁

Mean (M₂)

SD (s₂)

n₂

d is positive when Group 1 scores higher. Treat Group 2 as the control if you want Glass’s Δ.

Confidence level

α for power

Results update as you type.

Effect size

d =—

Enter your data

Your interpretation appears here.

−2.5−1.7−0.800.81.72.5

Group 2 higherThe white bar shows the CIGroup 1 higher

How much do the two distributions overlap?

Both groups are drawn as normal curves with a common standard deviation, so the distance between their peaks equals d.

Group 1Group 2Overlap

What this effect means in plain words

Common-language translations of d (assuming normal data with equal SDs).

Control event rate for NNT:%

Power and sample size

Power of your current design to detect this d, and the sample needed to detect it next time.

Report it (APA 7th edition style)

—

Notes on your result

Step-by-step solution

How to Use This Cohen’s d Calculator

Pick your study design. Choose Two independent groups for separate people in each condition, Paired / pre–post when the same people are measured twice, One sample to compare against a fixed value, or From a t-statistic when a paper reports t and n only.
Enter your numbers. Type means, SDs and sample sizes, or switch to Raw data and paste the scores. The calculator then finds the means and SDs for you. If you only have raw scores and want to check them first, the Descriptive Statistics Calculator summarizes them in one pass.
Read d, Hedges’ g and the confidence interval. The ruler shows the direction, size and uncertainty at a glance. If the white CI bar crosses zero, the data are compatible with no effect.
Translate it. The plain-language cards show what share of one group beats the other group’s average, how much the distributions overlap, and the number needed to treat.
Plan ahead. The power card shows how many participants you need to detect this effect reliably in a follow-up study.

Cohen’s d Formula (and Its Variants)

Every version divides a mean difference by a standard deviation. The versions differ only in which standard deviation, and that choice changes the number considerably for paired designs.

COHEN’S d · INDEPENDENT GROUPS

d = (M₁ − M₂) / s_p
s_p = √[((n₁−1)s₁² + (n₂−1)s₂²) / (n₁+n₂−2)]

The pooled SD weights each group by its degrees of freedom, so it works with unequal sample sizes. Compute it on its own with the Pooled Standard Deviation Calculator.

HEDGES’ g · SMALL-SAMPLE CORRECTION

g = d × J
J = Γ(df/2) / [√(df/2) · Γ((df−1)/2)]

df = n₁ + n₂ − 2. The exact J removes d’s upward bias, which matters below about 20 per group. The familiar shortcut is J ≈ 1 − 3 / (4df − 1).

GLASS’S Δ · CONTROL-GROUP SD

Δ = (M₁ − M₂) / s_control

Use this when the treatment changes variability, so pooling the SDs would blur the comparison. This calculator treats Group 2 as the control.

PAIRED DESIGNS · LAKENS (2013)

d_z = M_diff / s_diff
d_av = M_diff / ((s₁ + s₂) / 2)
d_rm = d_z × √(2(1 − r))

d_z drives the paired t-test and power. d_av and d_rm are comparable with between-subjects d.

Converting a t-statistic to Cohen’s d

For an independent-samples t-test, d = t × √(1/n₁ + 1/n₂). For a paired or one-sample test, d_z = t / √n. This is how meta-analysts recover effect sizes from papers that report only t, and it is the fourth tab of the calculator. To move between d and a correlation, use r = d / √(d² + a) with a = (n₁ + n₂)² / (n₁n₂). The Correlation Coefficient Calculator then tests and interprets that r.

Which Effect Size Should You Report?

Design	Effect size	Standardizer	Best used when
Two independent groups	Cohen’s d	Pooled SD	Groups have similar SDs and n ≥ 20 each
Two small groups	Hedges’ g	Pooled SD × J	Any n below about 20, or for meta-analysis
Treatment vs control	Glass’s Δ	Control SD	The intervention changes the spread of scores
Pre–post, same people	d_av or d_rm	Average SD of the two times	Comparing with between-subjects studies
Pre–post, for power	d_z	SD of differences	Planning sample size for a paired t-test
One sample vs norm	d	Sample SD	Testing against a known population value

For categorical outcomes such as yes/no or category counts, Cohen’s d is the wrong tool. Use Cramér’s V from the Chi Square Test Calculator instead.

Worked Example: Cohen’s d for Two Independent Groups

This is the calculator’s default, an illustrative dataset. Forty-two students revise with active recall (M = 78.4, SD = 9.6) and forty revise by re-reading (M = 72.1, SD = 11.2). How big is the difference?

Pooled SD: s_p = √[(41 × 9.6² + 39 × 11.2²) / 80] = √[(3,778.56 + 4,892.16) / 80] = 10.41.
Cohen’s d: (78.4 − 72.1) / 10.41 = 0.61, a medium effect.
Hedges’ g: with df = 80, J = 0.9906, so g = 0.60. The correction barely matters at this sample size.
Exact 95% CI: t = 0.605 × √(42 × 40 / 82) = 2.74, and inverting the noncentral t distribution gives d between 0.16 and 1.05.
Interpretation: 72.7% of active-recall students score above the re-reading average (U3), the distributions overlap by 76%, and a random active-recall student beats a random re-reader 66.6% of the time.
Power: this design had 77% power to detect d = 0.61. A replication needs 44 per group for 80% power, or 59 per group for 90%.

APA write-up: Students who used active recall (M = 78.40, SD = 9.60) scored higher than students who re-read (M = 72.10, SD = 11.20), t(80) = 2.74, p = .008, d = 0.61, 95% CI [0.16, 1.05].

Worked example: why paired d can look enormous

Twelve trainees are scored before and after a course. Scores rise by 5.17 points on average, but because every person improves by a similar amount, the SD of the differences is only 1.53. That gives d_z = 3.38. The scores themselves have SDs near 8, so the difference relative to how much people vary is d_av = 0.65. Both numbers are correct, but they answer different questions. Report d_av (or d_rm = 0.62) when you compare with between-subjects research, and use d_z for paired power analysis. The calculator shows all three side by side.

Cohen’s d Interpretation: Small, Medium and Large

Cohen proposed 0.2, 0.5 and 0.8 as rough benchmarks for when nothing better is available. Sawilowsky (2009) extended the scale to 1.2 (very large) and 2.0 (huge). These labels are conventions, not laws. In education, d = 0.25 can be practically important, while in a lab manipulation 0.5 may be unremarkable. Whenever you can, compare with effects from your own field.

Cohen’s d	U3 (share above the other mean)	Overlap	Probability of superiority	NNT (CER 20%)	r (equal n)
0.2 · Small	57.9%	92.0%	55.6%	16.5	0.100
0.5 · Medium	69.1%	80.3%	63.8%	6.0	0.243
0.8 · Large	78.8%	68.9%	71.4%	3.5	0.371
1.2 · Very large	88.5%	54.9%	80.2%	2.3	0.514
2.0 · Huge	97.7%	31.7%	92.1%	1.5	0.707

Values come from the calculator’s own engine: U3 = Φ(d), overlap = 2Φ(−|d|/2), probability of superiority = Φ(d/√2), and NNT follows Furukawa and Leucht (2011).

The confidence interval matters as much as d

A d of 0.6 from 20 people and a d of 0.6 from 2,000 people are very different findings. The interval shows the range of population effects compatible with your data. This calculator uses the exact noncentral-t method rather than the common ±1.96 × SE shortcut, which is noticeably off for small samples and large effects. For intervals around each group’s mean itself, use the Confidence Interval Calculator.

Cohen’s d and Sample Size: How Many Participants Do You Need?

Effect size is the key input for power analysis. Smaller effects need much larger samples, because halving d roughly quadruples n. These sample sizes come from exact noncentral-t power for a two-sided test at α = 0.05:

Cohen’s d	80% power, per group	90% power, per group	80% power, paired (d_z)	90% power, paired (d_z)
0.2	394	527	199	265
0.3	176	235	90	119
0.4	100	133	52	68
0.5	64	86	34	44
0.6	45	60	24	32
0.8	26	34	15	19
1.0	17	23	10	13
1.2	12	16	8	10

Planning a survey or trial? Use your d from a pilot or prior study as the input, ideally the lower end of its confidence interval rather than the point estimate. Then confirm the margin of error with the Sample Size Calculator.

Common Cohen’s d Mistakes

Using d_z when comparing with between-subjects studies. As the worked example shows, it can be five times larger than d_av for the same data.
Mixing up SD and SE. Papers often report the standard error. Convert it first with SD = SE × √n, or check it with the Standard Deviation Calculator.
Reporting d without its interval. A point estimate from a small study is a weak guide to the true effect.
Treating the 0.2 / 0.5 / 0.8 labels as fixed. Judge practical importance in context.
Pooling very different SDs. If one SD is more than about double the other, report Glass’s Δ alongside d.
Using d on heavily skewed data or data with outliers. Extreme values inflate the SD and shrink d. Check each group with the Interquartile Range Calculator, and consider reporting medians and IQRs.
Ignoring direction. State which group is higher. The sign of d depends on the order you entered the groups.

Frequently Asked Questions

How do you calculate Cohen’s d?

Subtract one group mean from the other and divide by the pooled standard deviation. The pooled SD is the square root of [(n1 − 1)s1² + (n2 − 1)s2²] / (n1 + n2 − 2). For example, means of 78.4 and 72.1 with a pooled SD of 10.41 give d = 6.3 / 10.41 = 0.61.

What is a good Cohen’s d value?

By Cohen’s conventions, 0.2 is small, 0.5 is medium and 0.8 is large, and Sawilowsky added 1.2 as very large and 2.0 as huge. Whether a value is good depends on the field. In education and health interventions, effects of 0.2 to 0.4 are often practically meaningful.

Can Cohen’s d be negative or greater than 1?

Yes to both. A negative d means the second group has the higher mean, so the sign shows only direction. d is measured in standard-deviation units and has no upper limit. Values above 1 mean the means differ by more than one standard deviation.

What is the difference between Cohen’s d and Hedges’ g?

Both standardize a mean difference by the pooled SD. Cohen’s d slightly overestimates the population effect in small samples. Hedges’ g multiplies d by a correction factor J that removes this bias. With more than about 20 people per group, the two are nearly identical.

How do I calculate Cohen’s d for a paired t-test?

There are several versions. d_z divides the mean difference by the SD of the difference scores and equals t divided by the square root of n. d_av divides the mean difference by the average of the two SDs and is comparable with between-groups studies. This calculator reports d_z, d_av and d_rm together.

How do you get Cohen’s d from a t value?

For an independent-samples t-test, multiply t by the square root of (1/n1 + 1/n2). For a paired or one-sample t-test, divide t by the square root of n. The From a t-statistic tab does this and also adds the confidence interval and Hedges’ g.

Can I calculate Cohen’s d with unequal sample sizes?

Yes. The pooled standard deviation weights each group’s variance by its degrees of freedom, so it handles unequal group sizes correctly. Very unequal sizes reduce precision, which you will see as a wider confidence interval.

How is the confidence interval for Cohen’s d calculated?

The exact method converts d to a t value, finds the two noncentrality parameters of the noncentral t distribution that put your observed t at the 2.5th and 97.5th percentiles, and converts those back to the d scale. It is more accurate than the normal approximation, especially for small samples.

When should I use Glass’s delta instead of Cohen’s d?

Use Glass’s delta when the treatment is likely to change the spread of scores as well as the mean. Dividing by the control group SD keeps the yardstick unaffected by the intervention. It is common in clinical and educational trials with a clear control condition.

How do I convert Cohen’s d to a correlation r?

Use r = d divided by the square root of (d² + a), where a = (n1 + n2)² / (n1 × n2). With equal groups, a equals 4, so d = 0.5 corresponds to r ≈ 0.24. The calculator shows this conversion as a tile for your data.

What sample size do I need to detect a medium effect?

For d = 0.5 with a two-sided test at α = 0.05, you need 64 participants per group for 80 percent power and 86 per group for 90 percent power. A paired design needs 34 pairs for 80 percent power when d_z is 0.5.

How do I report Cohen’s d in APA format?

Report it after the test statistic, italicized and with two decimals, followed by its confidence interval. For example: t(80) = 2.74, p = .008, d = 0.61, 95% CI [0.16, 1.05]. Keep the leading zero for d because it can exceed 1, but drop it for p-values.

Methodology and Sources

How the numbers are computed. The exact confidence interval for d comes from inverting the noncentral t cumulative distribution with bisection. That distribution is evaluated by 800-panel Simpson integration over the chi-distributed denominator. Hedges’ J uses the exact gamma-function form. Power uses the noncentral t distribution with a two-sided critical t, and required sample sizes are found by doubling and then bisection. Glass’s Δ uses the large-sample standard error from Hedges and Olkin (1985). Before release, every function was cross-checked against SciPy 1.18 (scipy.stats.t, nct and norm) and agreed to at least eight decimal places. The required sample sizes also match published G*Power values: 64 per group for d = 0.5 and 26 per group for d = 0.8 at 80% power.

References

Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.
Hedges, L. V. (1981). Distribution theory for Glass’s estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128. doi:10.3102/10769986006002107
Glass, G. V. (1976). Primary, secondary, and meta-analysis of research. Educational Researcher, 5(10), 3–8. doi:10.3102/0013189X005010003
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863. doi:10.3389/fpsyg.2013.00863
Cumming, G., and Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61(4), 532–574. doi:10.1177/0013164401614002
Sawilowsky, S. S. (2009). New effect size rules of thumb. Journal of Modern Applied Statistical Methods, 8(2), 597–599. doi:10.22237/jmasm/1257035100
Furukawa, T. A., and Leucht, S. (2011). How to obtain NNT from Cohen’s d. PLoS ONE, 6(4), e19070. doi:10.1371/journal.pone.0019070
Hedges, L. V., and Olkin, I. (1985). Statistical Methods for Meta-Analysis. Academic Press.

Disclaimer: This calculator is an educational and analytical aid. Verify results used in published research, clinical decisions or regulatory submissions with validated statistical software, and consult a qualified statistician where needed.

Manzoor Ahmad

Founder of DexoCalc · 13+ years in web development and SEO

Manzoor builds and personally reviews every DexoCalc calculator. Years of reading clinical and supplement research, where “statistically significant” so often hides a trivial effect, are why this tool puts the confidence interval and plain-language meaning beside every d. Each formula is cross-checked against reference statistical software before release.

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