Effect Size Calculator for Paired t Test

This calculator computes Cohen’s d_z for a paired t test from the mean difference, the standard deviation of the differences, and the sample size. You enter those three values and get the effect size, which tells you how large the difference is in standard deviation units.

Formula

Cohen’s d_z for a paired t test is:

d_z = (mean difference) / (standard deviation of the differences)

Where:

• mean difference = average of (post-test score − pre-test score) across all pairs
• standard deviation of the differences = sample standard deviation (n−1 denominator) of those difference scores

This is the most common effect size reported for paired t tests. It is also exactly the t statistic divided by the square root of the sample size: d_z = t / √n. That relationship is useful when you only have the t value and n.

Worked Example

Using the default values: mean difference = 5, SD of differences = 10, n = 30.

d_z = 5 / 10 = 0.50

So the effect size is 0.50, which is conventionally a medium effect. The t statistic would be 0.50 × √30 ≈ 2.74, which is significant at the 0.05 level (two-tailed) for df = 29.

When to Use This Effect Size

Use Cohen’s d_z when you have paired data: the same subjects measured twice (e.g., before/after), or matched pairs. It answers “how large is the change relative to the variability of the changes?â€� It is appropriate for:

• Pre-test/post-test designs
• Crossover trials
• Matched case-control studies
• Repeated measurements on the same subjects

It is not appropriate for independent groups (use Cohen’s d or Hedges’ g instead).

How to Read the Result

Cohen’s d_z is in standard deviation units of the difference scores. A value of 0.2 is small, 0.5 medium, 0.8 large (Cohen’s benchmarks). But these are rough guidelines; what matters is your field and context. For example, in some medical research, even a small effect can be clinically meaningful.

Because d_z uses the SD of the differences, it can be larger than d for independent groups even when the raw means are similar, because paired designs remove between-subject variability.

Common Mistakes and Misconceptions

• Using the SD of the pre-test or post-test scores instead of the SD of the differences. That gives a different, usually smaller, effect size.
• Forgetting that d_z is not the same as Cohen’s d for independent groups. They are not interchangeable.
• Rounding too early: compute the SD of differences with full precision, then divide.
• Using the population SD (divide by n) instead of sample SD (divide by n−1). Most software uses sample SD.

A common misconception is that d_z is a “biased� measure. It is not biased, but it is sensitive to the correlation between the two measurements. High correlation leads to larger d_z for the same mean difference.

Alternatives and Related Measures

There are other effect sizes for paired designs:

• Cohen’s d_av: uses the average of the two standard deviations (pre and post) as the denominator. It is more conservative and comparable to independent-groups d.
• Hedges’ g_z: a bias-corrected version of d_z for small samples.
• Glass’s Δ: uses the SD of the control or pre-test group.
• Common language effect size (CLES): probability that a randomly selected difference is positive.

For meta-analysis, you may need to convert d_z to other metrics, but that requires additional information.

Limitations and Caveats

This calculator assumes the differences are approximately normally distributed, which is the same assumption as the paired t test itself. If the distribution is heavily skewed, consider a nonparametric test and report a different effect size (e.g., rank-biserial correlation).

Also, d_z does not account for the reliability of the measurement. If your measure is noisy, the effect size will be underestimated. And it does not tell you about practical significance—only statistical magnitude.

Frequently Asked Questions

How do I compute d_z from a t statistic?

d_z = t / √n. For example, if t = 2.74 and n = 30, then d_z = 2.74 / 5.477 ≈ 0.50.

What is the difference between d_z and d_av?

d_z uses the SD of the difference scores; d_av uses the average of the two SDs. d_z is usually larger because paired differences have less variability. d_av is more comparable to independent-groups d.

Can I use this calculator if I only have the means and standard deviations of each group?

No, you need the standard deviation of the differences, which depends on the correlation between the two measurements. If you only have pre and post SDs, you cannot compute d_z without the correlation.

Is Cohen’s d_z the same as Cohen’s d for paired samples?

Yes, in most textbooks, “Cohen’s d for paired samplesâ€� refers to d_z. But some authors use d_av. Always check which one is reported.

What if my differences are not normally distributed?

The paired t test may be invalid. Consider using the Wilcoxon signed-rank test and report the rank-biserial correlation as the effect size.

How many decimal places should I report?

Usually two decimals are enough, but in meta-analysis, report at least three to avoid rounding errors.

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in md | Mean difference | - | 5 | 0
in sd | SD of differences | - | 10 | 0
in n | Sample size (pairs) | - | 30 | 2
out dz | Cohen's dz | - | md / sd | 2
out d_avg | Cohen's dav (approx) | - | md / sd | 2