Sample size for Welch’s t-test

When to use it

Two independent groups whose standard deviations are expected to differ, for example when a treatment makes responses more variable.
Any two-group comparison where equal spread cannot be assumed; Welch’s test is the default two-sample test in much current practice.

Not for

Groups with the same spread and equal sizes, where Student’s test needs the same sample and is simpler to describe.

Notation

$n_1,\ n_2$: Participants in groups 1 and 2
$r_a$: Allocation ratio n₂ / n₁
$\Delta$: Difference between the group means
$\sigma_1,\ \sigma_2$: Standard deviations in groups 1 and 2
$\nu_S$: Satterthwaite degrees of freedom
$\alpha$: 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 n₁ with enough power

\Pr\!\left(T'_{\nu_S,\lambda} > t_{\nu_S,\,1-\alpha/\tau}\right) \ge 1-\beta, \qquad \lambda = \frac{\Delta}{\sqrt{\sigma_1^{2}/n_1 + \sigma_2^{2}/n_2}}, \quad n_2 = \lceil r_a n_1 \rceil

Welch’s test uses each group’s own variance in the standard error rather than pooling them [1], and its power is computed on the noncentral t distribution with the Satterthwaite degrees of freedom below.

Satterthwaite degrees of freedom

\nu_S = \frac{\left(\sigma_1^{2}/n_1 + \sigma_2^{2}/n_2\right)^{2}}{\dfrac{(\sigma_1^{2}/n_1)^{2}}{n_1 - 1} + \dfrac{(\sigma_2^{2}/n_2)^{2}}{n_2 - 1}}

Fewer than n₁ + n₂ − 2 when the variances differ, which is the price of not assuming they are equal [2].

The most efficient allocation

r_a^{*} = \frac{n_2}{n_1} = \frac{\sigma_2}{\sigma_1}

Allocating in proportion to the standard deviations needs the fewest participants in total; the calculator reports it alongside the result.

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.

Standard deviation in group 1 $\sigma_1$

Values: Above 0

From comparable studies or pilot data.

Standard deviation in group 2 $\sigma_2$

Values: Above 0

From comparable studies or pilot data. If it equals σ₁, Student’s test is equivalent.

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. σ₂ / σ₁ minimises the total.

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 apart from any clustering specified.
The outcome is roughly normal in each group; the standard deviations may differ.
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

Equal groups, unequal spread

Difference 5, SDs 10 and 15; α = 0.05 two-sided, 80% power: 104 per group, 208 in total, with about 179 Satterthwaite degrees of freedom.

Group 1: 104 · Group 2: 104 · Total: 208

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using Welch’s t-test, and assuming standard deviations of 10 and 15 in the two groups, the required sample size is 104 per group (208 in total). Calculated with StatWisely (statwisely.com) using: n₁: solve noncentral-t power with Var=s₁²/n₁+s₂²/n₂ and Satterthwaite degrees of freedom. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allocating by the standard deviations

With n₂ / n₁ = 15 / 10 = 1.5: 80 and 120, 200 in total, fewer than the 208 needed with equal groups.

Group 1: 80 · Group 2: 120 · Total: 200

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.

How to cite

StatWisely. Sample size for Welch’s t-test. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/means-welch/

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

References

1.Welch BL. The generalization of “Student’s” problem when several different population variances are involved. Biometrika. 1947;34(1–2):28–35.
2.Satterthwaite FE. An approximate distribution of estimates of variance components. Biometrics Bulletin. 1946;2(6):110–114.
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