Blazon Research Resources · Effect sizes

How to convert test statistics to effect sizes

Written by Thomas Solomon, PhD

Published: 27 Oct 2026

Convert a test statistic to an effect size →

Why study design matters

A t statistic summarizes an estimate relative to its standard error. It does not uniquely identify the standardizer required for every effect-size definition. The same numerical t value can arise from independent groups, paired measurements, one-sample tests or unequal-variance procedures, and these situations do not all imply the same standardized effect.

Convert t to an effect-size correlation r

The supported conversion is:

r = t / √(t² + df)

The sign of t is retained. This should be described as an effect-size correlation derived from t, not automatically as the raw Pearson correlation between two measured variables. A clinical-research primer gives the equivalent relationship based on t² and df. Lee 2015

Convert an independent-samples t statistic to Cohen’s d

For the conventional pooled-variance independent-samples t test:

d = t × √(1/n₁ + 1/n₂)

This requires the sample sizes in both groups and assumes the t statistic corresponds to the pooled-variance model.

Why Welch’s t is different

Welch’s test does not use the same pooled-variance structure. A pooled-SD Cohen’s d cannot be uniquely recovered from the Welch t statistic and group sample sizes alone. Statistical Converter therefore does not silently apply the pooled formula.

Convert paired or one-sample t to Cohen’s d_z

For a paired-samples or one-sample t statistic:

d_z = t / √n

For paired measurements, d_z standardizes the mean difference using the SD of the difference scores. It is therefore not interchangeable with every repeated-measures standardized effect. Literature on repeated-measures effect sizes emphasizes that the chosen standardizer matters for interpretation and comparability. Bakeman 2005 and Lakens 2017

Working with pre/post data? Use the Pre/Post & Change-Score Calculator →

Convert F to partial eta squared

For appropriate fixed-factor designs:

η_p² = (F × df_effect) / (F × df_effect + df_error)

Lakens gives this relationship and notes that its applicability depends on the design context; partial eta squared should not be treated as universally interchangeable with eta squared or generalized eta squared. Lakens 2013

Cohen’s d versus Hedges’ g

Cohen’s d based on sample SDs is biased upward in small samples. Hedges’ g applies a small-sample correction. Lakens notes this distinction and provides the familiar approximate correction, while the Statistical Converter uses the exact gamma-function correction implemented through log-gamma for numerical stability. Lakens 2013

Convert Cohen’s d and Hedges’ g →

Frequently asked questions

Can I calculate Cohen’s d from a t statistic?

Yes for some designs. For a pooled-variance independent-samples t test, t plus the two group sample sizes is sufficient. Paired or one-sample t statistics map to d_z, not the same pooled-SD effect.

Can I calculate Cohen’s d from a Welch t statistic?

Not uniquely from t and group sample sizes alone. Additional group summary statistics are needed, so the converter deliberately refuses that shortcut.

What does t → r produce?

It produces an effect-size correlation derived mathematically from t and df. It should not automatically be interpreted as an observed Pearson correlation between two variables. Lee 2015

Is partial eta squared the same as eta squared?

No. They coincide only in some simple designs. In factorial or more complex designs they use different denominators and should be labelled correctly. Lakens 2013

Which effect size should I use for repeated measures?

That depends on the inferential and comparability goal. d_z is directly related to the paired t statistic, but other standardized effects may be preferable for cross-study interpretation. Bakeman 2005 and Lakens 2017

References

1. Lakens D. Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Front Psychol. 2013;4:863. Lakens 2013
2. Lee DK. Statistical notes for clinical researchers: effect size. Lee 2015
3. Bakeman R. Recommended effect size statistics for repeated measures designs. Behav Res Methods. 2005. Bakeman 2005
4. Lakens D. Equivalence Tests: A Practical Primer for t Tests, Correlations, and Meta-Analyses. Lakens 2017

Convert test statistics to effect sizes →