Sample size for a paired t-test

When to use it

The same people measured twice, before and after, or under two conditions.
Matched pairs, such as twins, two eyes of one person, or matched case and control.

Not for

Two independent groups. Use the two-sample t-test.
More than two measurements per person. Use repeated-measures ANOVA.

Notation

$n$: Number of complete pairs
$\Delta$: Mean within-pair difference
$\sigma_d$: Standard deviation of the differences
$d$: Standardised effect, Δ / σ_d
$\lambda$: Noncentrality of the t distribution
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$\tau$: Number of tails: 2 for a two-sided test, 1 for one-sided
$r_b$: Correlation between the outcome and a baseline covariate
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Average number of participants per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters per group
$\ell$: Expected loss of whole clusters
$q$: Expected dropout

Formulas

The smallest n with enough power

\Pr\!\left(T'_{n-1,\lambda} > t_{n-1,\,1-\alpha/\tau}\right) \ge 1-\beta, \qquad \lambda = d\sqrt{n}, \quad d = \frac{\Delta}{\sigma_d}

A paired t-test is a one-sample t-test on the within-pair differences, so it uses the same noncentral t calculation with n − 1 degrees of freedom [1]. What sets it apart is σ_d: the more strongly the two measurements are correlated, the smaller it is, and the fewer pairs are needed.

SD of the differences from the two measurements

\sigma_d = \sqrt{\sigma_1^{2} + \sigma_2^{2} - 2\,\rho_{12}\,\sigma_1\sigma_2}

When only each measurement’s SD and the correlation between them are known.

Adjusting for a baseline covariate

n' = \left\lceil n\,(1 - r_b^{2}) \right\rceil + 1

Analysing with a baseline measurement as a covariate (ANCOVA) removes the share r_b² of the outcome’s variance the test would otherwise have to overcome. One participant per group is added back for the degree of freedom the covariate uses [3].

Clustering

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho

When participants are recruited or randomised in clusters, observations within a cluster repeat information, and the sample is multiplied by the design effect. With equal clusters (CV = 0) this is 1 + (m − 1)ρ [4, 5].

Few clusters

F_t = \left(\frac{t_{2(C-1),\,1-\alpha/\tau} + t_{2(C-1),\,1-\beta}}{z_{1-\alpha/\tau} + z_{1-\beta}}\right)^{2}

Optional. A cluster trial is analysed on cluster-level summaries, so its critical values come from t on 2(C − 1) degrees of freedom rather than the normal distribution. With few clusters this costs participants; because C depends on the sample, it is iterated until it settles [6].

All adjustments together

n_{\text{adj}} = \left\lceil n (1 - r_b^{2})\,\mathrm{DE}\;F_t \right\rceil + 1

The adjustments multiply, in this order, and the result is rounded up once. Any adjustment not used contributes a factor of 1.

Dropout

n_{\text{recruit}} = \left\lceil \frac{n_{\text{adj}}}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Clusters to recruit

C = \left\lceil \frac{n_{\text{recruit}}}{m} \right\rceil, \qquad C_{\text{recruit}} = \left\lceil \frac{C}{1 - \ell} \right\rceil

Per group. Losing a whole cluster loses every participant in it, so an expected loss of clusters inflates the clusters recruited rather than the participants.

Parameters

Mean difference to detect $\Delta$

Values: Not zero, in the measurement’s units

The smallest difference that would matter, clinically or practically. Not the difference you hope for: a study sized for an optimistic effect is underpowered for a real but smaller one.

Standard deviation of the differences $\sigma_d$

Values: Above 0, in the measurement’s units

Of the within-pair differences, not of either measurement. If only the two SDs and their correlation ρ₁₂ are known, σ_d = √(σ₁² + σ₂² − 2ρ₁₂σ₁σ₂).

Cohen’s d, instead of the two above $d$ Optional

Values: Above 0

The difference in standard-deviation units. Cohen’s benchmarks of 0.2, 0.5 and 0.8 for small, medium and large are a last resort; a difference and SD from your own field are better [1].

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

One- or two-sided $\tau$

Values: 2 (two-sided) or 1 (one-sided)

Two-sided is the default. A one-sided test is justified only when an effect in the other direction would be treated exactly as no effect, and should be stated in advance.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Baseline correlation $r_b$ Optional

Values: Between −1 and 1

The correlation between the outcome and its baseline measurement, from previous data. Only use it if the analysis will adjust for the baseline.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

For clustered designs only. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [7]; prefer an estimate from a comparable trial.

Average cluster size $m$ Optional

Values: At least 1

The average number of participants per cluster. Given with the ICC.

Variation in cluster size $\mathrm{CV}$ Optional

Values: 0 or more; 0 for equal clusters

The SD of cluster sizes divided by their mean. Unequal clusters cost power on their own account [5].

Expected loss of whole clusters $\ell$ Optional

Values: 0% up to, but not including, 100%

Clusters, such as practices or schools, expected to withdraw entirely.

Correction for few clusters $F_t$ Optional

Values: On or off

Worth applying when there are few clusters per group, roughly fewer than 15–20.

Assumptions

Pairs are independent of one another apart from any clustering specified, and both measurements are available for every pair counted.
The differences are roughly normal, or there are enough pairs for their mean to be.
With clustering: clusters are independent, with the same ICC throughout, and the analysis accounts for the clustering, for example with a mixed model or cluster-level summaries [4].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Before and after

Mean change 5, SD of the changes 10, so d = 0.5; α = 0.05 two-sided, 80% power: 34 pairs.

Sample size: 34

Protocol statement
To detect a mean within-pair difference of 5 with 80% power at a two-sided significance level of 0.05, using a paired t-test, and assuming a standard deviation of the differences of 10, the required sample size is 34 pairs. Calculated with StatWisely (statwisely.com) using: n_pairs: solve noncentral-t power for d, α, and 1-β. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for missing second measurements

Expecting 15% not to complete the second measurement: 34 / (1 − 0.15) = 40.0, so 40 are enrolled.

Needed for analysis: 34 · To recruit: 40

Protocol statement
To detect a mean within-pair difference of 5 with 80% power at a two-sided significance level of 0.05, using a paired t-test, and assuming a standard deviation of the differences of 10, the required sample size is 34 pairs. Allowing for 15% dropout, 40 pairs should be recruited. Calculated with StatWisely (statwisely.com) using: n_pairs: solve noncentral-t power for d, α, and 1-β. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
The basic result agrees with G*Power’s matched-pairs t-test: 34 pairs for d = 0.5, α = 0.05 two-sided and 80% power [2].

How to cite

StatWisely. Sample size for a paired t-test. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/means-paired/

Cite the original references for the formulas your design uses as well.

References

1.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.
2.Faul F, Erdfelder E, Lang AG, Buchner A. G*Power 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods. 2007;39(2):175–191.
3.Borm GF, Fransen J, Lemmens WAJG. A simple sample size formula for analysis of covariance in randomized clinical trials. Journal of Clinical Epidemiology. 2007;60(12):1234–1238.
4.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
5.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
6.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
7.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013