Cohen’s d Calculator Paired t Test

This calculator computes Cohen’s d effect size for a paired t test. You enter the mean difference, the standard deviation of the differences, and the sample size; it returns Cohen’s d (with a small-sample correction option) and the 95% confidence interval for the true effect size.

Formula

Cohen’s d for paired samples is:

d = (mean difference) / (standard deviation of differences)

In symbols: d = M_diff / SD_diff

• M_diff – the average of the differences between paired observations (e.g., before minus after). Unit: same as the measurement.
• SD_diff – the standard deviation of those differences. Unit: same as the measurement.

This is the standard formula used in meta-analyses and textbooks. It expresses the effect size in standard deviation units: a d of 0.5 means the average difference is half a standard deviation of the differences.

Worked Example

Using the default values: mean difference = 2.5, SD of differences = 3.0, sample size = 30.

1. Divide the mean difference by the SD: 2.5 / 3.0 = 0.8333.
2. Cohen’s d = 0.833 (rounded to three decimals).
3. For the Hedges correction: multiply by (1 – 3/(4*30 – 1)) = (1 – 3/119) = 0.9748. So corrected d = 0.833 * 0.9748 = 0.812.
4. For the 95% CI: standard error = sqrt(1/30 + 0.833^2/(2*30)) = sqrt(0.03333 + 0.01157) = sqrt(0.04490) = 0.2119. Lower = 0.833 – 1.96*0.2119 = 0.833 – 0.415 = 0.418. Upper = 0.833 + 0.415 = 1.248.

The calculator shows d = 0.833, corrected d = 0.812, and 95% CI [0.418, 1.248].

When to Use Cohen’s d for Paired Data

Use this effect size when you have two measurements from the same subjects (before/after, left/right, matched pairs) and you want to quantify the magnitude of the change, not just whether it is statistically significant. Unlike the p-value, which depends on sample size, Cohen’s d is a standardized measure that allows comparison across studies.

Common applications:

• Pre-post intervention studies (e.g., test scores before and after a training program).
• Crossover trials where each participant receives both treatments.
• Reliability studies comparing two measurement methods on the same subjects.

Interpretation benchmarks (proposed by Cohen, 1988): d ≈ 0.2 is small, 0.5 is medium, 0.8 is large. These are rough guidelines, not absolute rules.

How to Choose Between Cohen’s d and Hedges’ g

Cohen’s d is a biased estimator for small samples; it tends to overestimate the population effect size. Hedges’ g applies a correction factor: g = d × (1 – 3/(4n – 1)), where n is the number of pairs. Use Hedges’ g when your sample size is less than about 50, or when you plan to compare your result with other studies that use g. For large samples, the difference is negligible.

Note: Some fields report a different version of Cohen’s d for paired tests that uses the standard deviation of the raw scores rather than the differences. That version is less common and not used here. Always check which formula a paper used when comparing effect sizes.

Common Mistakes and Misconceptions

One frequent error is using the standard deviation of the pre-test scores instead of the standard deviation of the differences. This inflates or deflates d unpredictably. Always compute the difference for each pair first, then take the mean and SD of those differences.

Another misconception: Cohen’s d is not a percentage. A d of 0.5 does not mean a 50% improvement. It means the average change is half a standard deviation of the change scores.

Rounding: Keep at least three decimal places in intermediate steps to avoid rounding errors in the confidence interval.

Limitations of the Formula

The formula assumes the differences are approximately normally distributed, especially for small samples. If the differences are heavily skewed, the effect size may still be meaningful, but the confidence interval may be inaccurate. The confidence interval formula used here is an approximation that works well for moderate sample sizes (n > 20) and moderate effect sizes. For more precise intervals, especially with small samples, use a bootstrap or the noncentral t distribution method.

Also, Cohen’s d is a point estimate. The confidence interval gives a range of plausible values for the true effect size. If the interval crosses zero, the effect is not statistically significant at the 0.05 level.

Frequently Asked Questions

Q: What is the difference between Cohen’s d and the t statistic?
The t statistic is the mean difference divided by the standard error (SD/√n). It measures the strength of evidence against the null hypothesis and depends on sample size. Cohen’s d divides by the SD alone, so it is independent of sample size and measures the magnitude of the effect.

Q: Can I use this calculator if I only have the t value and sample size?
Yes. You can compute d = t / √n. For example, if t = 2.5 and n = 30, d = 2.5 / √30 = 0.456. But you will need the mean difference and SD to get the confidence interval.

Q: Should I use Cohen’s d or Pearson’s r for paired data?
Both are valid effect sizes. Cohen’s d is more intuitive for mean differences. Pearson’s r (or the point-biserial correlation) is sometimes used for paired data, but it is less common. Choose the one that is standard in your field.

Q: How do I report Cohen’s d in a paper?
Report the value with a confidence interval: “Cohen’s d = 0.83, 95% CI [0.42, 1.25].” Also state the sample size and whether you used the correction for small samples.

Q: What if my mean difference is negative?
The sign of d indicates the direction of the effect. For example, if scores decrease after an intervention, the mean difference is negative, and d will be negative. When interpreting, focus on the absolute value; the sign is arbitrary depending on how you define the difference.

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in md | Mean difference | - | 2.5 | 0
in sd | SD of differences | - | 3.0 | 0
in n | Sample size (pairs) | - | 30 | 2
sel corr | Correction | None=0, Hedges g=1 | 0
out d | Cohen's d | - | md / sd | 3
out d_corr | Corrected d (Hedges g) | - | (md / sd) * (1 - 3 / (4 * n - 1)) | 3
out ci_low | 95% CI lower | - | (md / sd) - 1.96 * sqrt(1/n + (md/sd)^2 / (2*n)) | 3
out ci_high | 95% CI upper | - | (md / sd) + 1.96 * sqrt(1/n + (md/sd)^2 / (2*n)) | 3