Sample size for a two-sample t-test

When to use it

A randomised trial or comparative study with two independent groups and a continuous outcome.

Not for

Groups expected to vary very differently, especially with unequal group sizes. Use Welch’s t-test.
More than two groups. Use one-way ANOVA, or the multi-arm comparison against a shared control.
Showing that a new treatment is not worse than, or equivalent to, another. Use non-inferiority or equivalence.

Notation

$n_1,\ n_2$: Participants in groups 1 and 2
$r_a$: Allocation ratio n₂ / n₁
$\Delta$: Difference between the group means
$\sigma$: Common standard deviation
$d$: Standardised effect, Δ / σ
$\lambda$: 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
$d_t,\ d_c$: Share of the treatment arm not taking treatment, and of the control arm receiving it
$q$: Expected dropout, overall or for each arm (q₁, q₂)

Formulas

The smallest n with enough power

\Pr\!\left(T'_{\nu,\lambda} > t_{\nu,\,1-\alpha/\tau}\right) \ge 1-\beta, \qquad \lambda = d\sqrt{\frac{n_1 n_2}{n_1 + n_2}}, \quad \nu = n_1 + n_2 - 2, \quad n_2 = r_a\,n_1

Student’s t-test with a pooled standard deviation. When the effect is real its statistic follows a noncentral t distribution, and n₁ is the smallest whole number whose power reaches the target; the lower rejection region is included for a two-sided test [1].

For intuition: the normal approximation

n_1 \approx \left(1 + \frac{1}{r_a}\right)\left(\frac{z_{1-\alpha/\tau} + z_{1-\beta}}{d}\right)^{2}

Not what the calculator uses, but it shows that equal groups are the most efficient, and that halving the effect quadruples the sample.

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].

Non-adherence and crossover

n' = \frac{n}{(1 - d_t - d_c)^{2}}

An intention-to-treat analysis keeps everyone in the arm they were allocated to, so participants who do not receive their allocated treatment dilute the difference. Lachin’s correction restores the power [7].

All adjustments together

n_{\text{adj}} = \left\lceil n (1 - r_b^{2})\,\mathrm{DE}\;F_t\,\frac{1}{(1 - d_t - d_c)^{2}} \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},i} = \left\lceil \frac{n_{\text{adj}}}{1 - q_i} \right\rceil, \qquad i = 1, 2

Applied last, to the participants actually recruited. The rate can differ between the arms; when it does not, q₁ = q₂ = q.

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

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 $\sigma$

Values: Above 0, in the measurement’s units

From a comparable published study, a pooled SD from several studies, or pilot data. The sample size grows with its square, so check a larger value too.

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.

Allocation ratio $r_a$ Optional

Values: Above 0; 1 for equal groups

The planned size of group 2 divided by group 1. Equal groups need the fewest participants in total; unequal allocation can be worth it when one arm is cheaper or more acceptable.

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.

Dropout in each arm $q_1,\ q_2$ Optional

Values: 0% up to, but not including, 100%, for each arm

When dropout is expected to differ between arms, for example higher with an unpleasant treatment. Each arm is then inflated by its own rate.

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 [8]; 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.

Non-adherence and crossover $d_t,\ d_c$ Optional

Values: 0% up to, but not including, 100%; the two together below 100%

The share of the treatment arm expected not to take it, and of the control arm expected to receive it, from comparable trials.

Assumptions

The groups are independent, and participants are independent of one another apart from any clustering specified.
The outcome is roughly normal in each group with the same standard deviation. With unequal spread, and especially unequal group sizes, use Welch’s test.
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.
Non-adherence is corrected for an intention-to-treat analysis, with treatment received having its full effect and no effect otherwise.

Worked examples

Two equal groups

Difference 5, SD 10, so d = 0.5; α = 0.05 two-sided, 80% power: 64 per group, 128 in total.

Group 1: 64 · Group 2: 64 · Total: 128

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a two-sample t-test, and assuming a standard deviation of 10, the required sample size is 64 per group (128 in total). Calculated with StatWisely (statwisely.com) using: n₁ and n₂: solve noncentral-t power for d, α, 1-β, and allocation ratio. These are planning estimates; confirm the final design with a qualified statistician.

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Twice as many in group 2

With a 1 : 2 allocation, 48 and 96: 144 in total, against 128 with equal groups.

Group 1: 48 · Group 2: 96 · Total: 144

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Adjusting for the baseline

With a baseline correlation of 0.5: 64 × (1 − 0.5²) = 48, plus 1 for the covariate, so 49 per group.

Group 1: 49 · Group 2: 49 · Total: 98

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a two-sample t-test, and assuming a standard deviation of 10, the test itself needs 64 per group. With adjustment for a baseline covariate (r = 0.5), the required sample size is 49 per group (98 in total). Calculated with StatWisely (statwisely.com) using: n₁ and n₂: solve noncentral-t power for d, α, 1-β, and allocation ratio; ANCOVA: n x (1 - r^2) + 1. These are planning estimates; confirm the final design with a qualified statistician.

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A cluster-randomised trial

Clusters of 10 with ICC 0.05: DE = 1 + 9 × 0.05 = 1.45, so 64 × 1.45 = 92.8, rounded up to 93 per group, in 10 clusters per group.

Group 1: 93 · Group 2: 93 · Total: 186

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a two-sample t-test, and assuming a standard deviation of 10, the test itself needs 64 per group. With clustering (ICC 0.05, an average of 10 per cluster; design effect 1.45), the required sample size is 93 per group (186 in total), in 10 clusters per group. Calculated with StatWisely (statwisely.com) using: n₁ and n₂: solve noncentral-t power for d, α, 1-β, and allocation ratio; clustered: n x DE, DE = 1 + (m-1)ρ. These are planning estimates; confirm the final design with a qualified statistician.

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Allowing for non-adherence

If 10% of the treatment arm will not take it and 5% of the control arm will: 64 / (1 − 0.10 − 0.05)² = 88.6, so 89 per group.

Group 1: 89 · Group 2: 89 · Total: 178

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a two-sample t-test, and assuming a standard deviation of 10, the test itself needs 64 per group. With non-adherence of 10% in the treatment arm and crossover of 5% in the control arm, the required sample size is 89 per group (178 in total). Calculated with StatWisely (statwisely.com) using: n₁ and n₂: solve noncentral-t power for d, α, 1-β, and allocation ratio; non-adherence: n / (1 - d_t - d_c)^2. These are planning estimates; confirm the final design with a qualified statistician.

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Different dropout in each arm

Expecting 20% dropout in group 1 and 10% in group 2: 64 / 0.8 = 80 and 64 / 0.9 = 71.1, so 80 and 72 are recruited.

Group 1: 64 · Group 2: 64 · Total: 128 · To recruit: 80 and 72

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a two-sample t-test, and assuming a standard deviation of 10, the required sample size is 64 per group (128 in total). Allowing for dropout of 20% in the first group and 10% in the second, 80 in the first group and 72 in the second group (152 in total) should be recruited. Calculated with StatWisely (statwisely.com) using: n₁ and n₂: solve noncentral-t power for d, α, 1-β, and allocation ratio. 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 two-sample t-test: 64 per group for d = 0.5, α = 0.05 two-sided and 80% power [2].

How to cite

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

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.Lachin JM. Biostatistical Methods: The Assessment of Relative Risks. New York: John Wiley & Sons; 2000.
8.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

When to use it

Not for

Notation

Formulas

The smallest n with enough power

For intuition: the normal approximation

Adjusting for a baseline covariate

Clustering

Few clusters

Non-adherence and crossover

All adjustments together

Dropout

Clusters to recruit

Parameters

Difference to detectΔ\DeltaΔ

Standard deviationσ\sigmaσ

Cohen’s d, instead of the two abovedddOptional

Significance levelα\alphaα

Power1−β1-\beta1−β

One- or two-sidedτ\tauτ

Allocation ratiorar_ara​Optional

Expected dropoutqqqOptional

Dropout in each armq1, q2q_1,\ q_2q1​, q2​Optional

Baseline correlationrbr_brb​Optional

Intracluster correlation (ICC)ρ\rhoρOptional

Average cluster sizemmmOptional

Variation in cluster sizeCV\mathrm{CV}CVOptional

Expected loss of whole clustersℓ\ellℓOptional

Correction for few clustersFtF_tFt​Optional

Non-adherence and crossoverdt, dcd_t,\ d_cdt​, dc​Optional

Assumptions

Worked examples

Two equal groups

Twice as many in group 2

Adjusting for the baseline

A cluster-randomised trial

Allowing for non-adherence

Different dropout in each arm

Validation

How to cite

References

Difference to detect $\Delta$

Standard deviation $\sigma$

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

Significance level $\alpha$

Power $1-\beta$

One- or two-sided $\tau$

Allocation ratio $r_a$ Optional

Expected dropout $q$ Optional

Dropout in each arm $q_1,\ q_2$ Optional

Baseline correlation $r_b$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Average cluster size $m$ Optional

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

Expected loss of whole clusters $\ell$ Optional

Correction for few clusters $F_t$ Optional

Non-adherence and crossover $d_t,\ d_c$ Optional