Cohen’s d from t Statistic Calculator
Convert a reported t statistic into the effect-size form appropriate to an independent-groups, paired-samples, or one-sample design. The workbench keeps the sign, exposes the sample-size assumption, adds a small-sample Hedges correction, and separates magnitude from statistical significance.
The estimated group difference is about 0.53 pooled standard deviations in the positive direction.
Reconstruct the study design
Test family
Use only when the t statistic came from the conventional equal-variance independent-samples test.
Reported statistic and sample sizes
For paired or one-sample mode, the first sample-size box means pairs or observations; the second is disabled.
Effect-size report
This quick interval is an approximation, not a noncentral-t confidence interval. Use original data or specialist software for publication-grade inference.
The positive sign follows the original t contrast. Interpret 0.53 standard deviations against outcomes, prior studies, uncertainty, and practical consequences—not a universal magnitude label.
What this conversion can recover
A t statistic combines a standardized difference with its sampling uncertainty. When the test design and sample sizes are known, algebra can recover a member of the Cohen’s d family. For two independent groups analyzed with a pooled-variance t test, the conversion is d = t multiplied by the square root of 1/n₁ + 1/n₂. For a paired or one-sample t test, the matching standardized difference-score measure is dz = t divided by the square root of n.
The calculator preserves the sign of t. A positive value means the contrast was positive in the direction defined by the original analysis, such as group 1 minus group 2. A negative value reverses that direction. The sign has no meaning without knowing how the researcher coded the contrast, so record the group order beside the result.
Independent-groups formula
df = n₁ + n₂ − 2
The default independent-groups example has t = 2.50, n₁ = 40, and n₂ = 50. The sample-size term is √(1/40 + 1/50) = √0.045, about 0.212132. Multiplying by 2.50 gives d = 0.530330. That value expresses the estimated mean difference in pooled within-group standard-deviation units under the conventional equal-variance test.
This shortcut is tied to the pooled-variance t statistic. A Welch t test uses a different standard error and fractional degrees of freedom; t and group sizes alone generally do not identify the exact pooled-standard-deviation d. If the article says Welch, Satterthwaite, unequal variances, robust standard errors, clustering, survey weights, or regression adjustment, obtain the means and standard deviations or use a conversion developed for that analysis.
Paired and one-sample formula
df = n − 1
For paired observations, the t test is applied to within-pair difference scores. Dividing t by √n produces dz, the mean difference divided by the standard deviation of those difference scores. For a one-sample test, it is the sample mean minus the reference value, divided by the sample standard deviation. The same algebra does not make the two study designs substantively interchangeable.
Repeated-measures research has several standardized mean differences. A value standardized by the difference-score standard deviation can differ from one standardized by the average condition standard deviation, especially when within-person correlation is high. Report the d variant explicitly. This calculator does not reconstruct dav or drm because t and n alone do not supply both condition standard deviations and their correlation.
Why Hedges’ g is also shown
Sample standardized mean differences can be upward-biased estimates of a population effect, most noticeably with small degrees of freedom. The result panel multiplies d by the common approximation J = 1 − 3/(4df − 1). For the default df of 88, J is about 0.9915 and g is about 0.5258.
The correction is modest in a large sample and stronger in a small sample. It does not repair a mismatched study design, dependence between supposedly independent observations, selective reporting, measurement error, or a biased sample.
Approximate uncertainty interval
For independent groups, the calculator approximates the variance of d as (n₁ + n₂)/(n₁n₂) + d²/(2df). For paired and one-sample designs, it uses 1/n + d²/(2df). It takes the square root to get an approximate standard error and applies the selected normal critical value around d. The default standard error is about 0.2159, giving a rough 95 percent interval from 0.1072 to 0.9534.
This symmetrical interval is useful for a quick audit but is not the preferred final interval for every design. Standardized effects have asymmetric sampling behavior in small samples. A noncentral-t method, bootstrap suited to the sampling structure, or a meta-analysis package can give a more defensible interval. If the original paper reports a confidence interval based on its raw data and model, prefer it to a reverse-engineered approximation.
r and r-squared translations
The workbench calculates r = t/√(t² + df), retaining the sign. This is an effect-size translation associated with the test and its degrees of freedom. Squaring r removes the direction and yields a variance-style proportion. In the default case, r is approximately 0.2575 and r² is 6.63 percent.
Do not call r² the percent of individual outcomes caused by treatment. Its exact interpretation depends on design and model. Converting between effect-size metrics can help synthesis, but a converted value does not gain more information than the statistic from which it was derived.
Probability of superiority and overlap
To make a standardized difference more tangible, the calculator shows two optional normal-model translations. Probability of superiority is Φ(d/√2), the modeled chance that a random observation from the positively shifted distribution exceeds one from the other distribution. With d = 0.5303, that is about 64.62 percent in the direction of the contrast. A negative d produces a value below 50 percent.
Distribution overlap is 2Φ(−|d|/2), approximately 79.08 percent for the default. Both translations assume two normal distributions with equal standard deviations. Real distributions may be skewed, bounded, multimodal, or unequal in spread, so these are communication aids rather than empirical probabilities calculated from raw observations.
Magnitude is not statistical significance
A p value asks how compatible the data are with a null model under stated assumptions; d estimates standardized magnitude. The same d can produce different t statistics when sample sizes change, and a very small effect can be statistically detectable in a huge sample. Conversely, a practically important estimate can have wide uncertainty in a small sample.
Common 0.2, 0.5, and 0.8 labels are rough historical conventions, not natural laws. A 0.1-standard-deviation shift can matter greatly for a low-cost population intervention, while a larger score change may be trivial for an unreliable surrogate outcome. Compare with domain-specific thresholds, prior studies using the same measure, cost, harms, baseline risk, and the full confidence interval.
Assumptions to recover from the methods section
| Question | Why it changes the conversion |
|---|---|
| Independent, paired, or one sample? | It determines whether sample sizes enter as √(1/n₁ + 1/n₂) or √n. |
| Pooled or Welch t? | The independent-groups shortcut matches the pooled-variance statistic. |
| What was the contrast direction? | It determines what a positive or negative sign means. |
| Were observations clustered or weighted? | Nominal sample counts may not reproduce the model standard error. |
| Was t adjusted by covariates? | The resulting effect may be partial or model-based, not a raw standardized mean difference. |
Reporting checklist
Name the exact effect-size variant, give its sign and group order, identify the t test, report t and degrees of freedom, include both sample sizes, and state whether g was bias-corrected. Add an interval and explain the outcome in its original units when available. If the conversion came from a secondary report, note that the result was reconstructed rather than computed from individual observations.
Keep more digits during calculation than in the final prose. Two decimal places may be sufficient for narrative reporting, but retain the unrounded value in analysis files. A reproducible record should include the source statistic, all entered values, the selected design, the formula version, and any discrepancy with the source’s reported effect.
Frequently asked questions
Can I use this for a Welch independent-samples t test?
Not as an exact pooled-d recovery. Welch’s standard error is not determined by t and sample sizes alone; use group means and standard deviations or an appropriate Welch conversion.
Why is paired-samples d often different from independent-groups d?
Paired dz uses the standard deviation of within-pair differences. Correlation between repeated observations affects that standardizer.
Does a negative Cohen’s d mean the effect is bad?
No. It means the contrast is negative in the coded direction. Whether that is favorable depends on the outcome and group ordering.
Should I report d or Hedges’ g?
Follow the field and synthesis plan. Hedges’ g adds a small-sample bias adjustment; state the correction and standardizer used.
Is the displayed interval exact?
No. It is a normal approximation. Publication work may require a noncentral-t, bootstrap, or model-specific interval.
Can I calculate d from only a p value?
Usually not uniquely. You also need the test family, degrees of freedom or sample sizes, tail convention, and enough detail to reconstruct t.
If raw group summaries are available, calculate the standardized difference independently with the Cohen’s d effect size calculator instead of relying only on the t-based conversion.
References
NIST Dataplot — Hedges’ g and Cohen’s d definitions
NIST/SEMATECH e-Handbook — Analysis of paired observations
National Library of Medicine — Calculating and reporting effect sizes for t tests