Sample size to estimate a difference between two means

When to use it

Estimating how much two groups differ on a continuous measurement, with the size of the difference as the result rather than a significance test.
Pilot and feasibility studies, and comparisons that aim to estimate the size of a difference rather than test it, whose objective is stated as a precision.

Not for

Showing that one group differs from the other. Use a powered two-sample t-test under “Detect an effect”.
The same people measured twice. Use the paired mean difference.

Notation

$n_1,\ n_2$: Sample sizes of groups 1 and 2
$\sigma_1,\ \sigma_2$: Standard deviations in groups 1 and 2
$r_a$: Allocation ratio n₂ / n₁
$E$: Margin of error for the difference, in the measurement’s units
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α (1.960 for 95%)
$N_1,\ N_2$: Population sizes of the two groups
$\mathrm{DE}$: Design effect
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Usable observations per cluster
$k$: Number of clusters in each group
$\mathrm{CV}$: Coefficient of variation of cluster size
$L,\ U$: Low and high ends of the range of cluster values
$q$: Non-response rate

Formulas

Simple random sampling

n_1 = \left\lceil \frac{z_{1-\alpha/2}^{2}\left(\sigma_1^{2} + \sigma_2^{2}/r_a\right)}{E^{2}} \right\rceil, \qquad n_2 = \lceil r_a\, n_1 \rceil

The variance of a difference between independent means is the sum of the two groups’ variances, so the groups’ SDs need not be equal [1]. Left empty, σ₂ is taken to equal σ₁.

Finite population correction

n_i' = \left\lceil \frac{n_i}{1 + (n_i - 1)/N_i} \right\rceil, \qquad i = 1, 2

Applied to each group against its own population when population sizes are given [2]. It matters only when a group is a sizeable share of its population, and is not offered with cluster sampling.

Design effect from an ICC

\mathrm{DE} = 1 + (m - 1)\,\rho

For clusters of equal size [3]. A design effect can also be entered directly, in which case no ICC is used.

Unequal cluster sizes

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho, \qquad \mathrm{CV} = \frac{\operatorname{SD}(s_j)}{\bar{s}}

When each cluster is given its own share s_j of the sample, the CV of those shares enlarges the design effect [4]. With a design effect entered directly the shares only divide the sample, since a measured design effect already reflects its survey’s cluster sizes.

Number of clusters for a chosen cluster size

k = \max\!\left(2,\ \left\lceil \frac{n \cdot \mathrm{DE}}{m} \right\rceil\right), \qquad n_{\text{clustered}} = k\,m

Applied to each group separately, with distinct clusters in each. The requirement is rounded up to whole clusters, so the total can exceed n × DE slightly, and rounding can shift the allocation ratio a little.

Cluster size for a fixed number of clusters

m = \left\lceil \frac{n\,(1-\rho)}{k - n\,\rho} \right\rceil, \qquad k > n\,\rho

Solving k·m = n·DE for m. Below that number of clusters no cluster size reaches the margin, because each extra person per cluster adds less and less information; the calculator reports the minimum number of clusters instead. With two groups, k clusters are taken in each.

ICC from a range across clusters

\rho \approx \frac{\left[(U - L)/4\right]^{2}}{\sigma_1^{2}}

When no ICC is available: the range within which most clusters’ values are expected to fall is taken as about four between-cluster standard deviations, and the ICC is their share of the first group’s outcome variance σ₁². An elicitation aid, not a measurement. With two groups, the range describes the first group’s clusters.

Few clusters

n \;\leftarrow\; n \left(\frac{t_{k-1,\,1-\alpha/2}}{z_{1-\alpha/2}}\right)^{2}

Optional. With few clusters the interval is built on k − 1 degrees of freedom, so the normal critical value is replaced by t. Because k depends on n, this is iterated until it settles.

Non-response

n_{\text{approach}} = \left\lceil \frac{n}{1 - q} \right\rceil

The number to approach in each group so that n respond. Applied last, after any clustering.

Parameters

Confidence level $1-\alpha$

Values: 90%, 95% or 99%

95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

Margin of error $E$

Values: Above 0, in the measurement’s units

The largest distance between the estimated and true difference you can accept, such as a fraction of the smallest difference that would matter clinically.

Standard deviation in group 1 $\sigma_1$

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 the square of the SD.

Standard deviation in group 2 $\sigma_2$ Optional

Values: Above 0; leave empty if the same as group 1

Give it when the groups are expected to vary differently, for example a treated group whose response is more variable.

Allocation ratio $r_a = n_2/n_1$ Optional

Values: Above 0; 1 for equal groups

The planned size of group 2 divided by group 1. Equal groups give the most precision for a given total; use another ratio when one group is easier or cheaper to recruit.

Population sizes $N_1,\ N_2$ Optional

Values: Whole numbers of at least 2; leave the second empty if both groups come from populations of the same size

Leave empty for large populations. They matter only when a group would be more than about 5% of its population. Not available with cluster sampling.

Expected non-response rate $q$ Optional

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

The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

The share of the outcome’s variation that lies between clusters. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [6], but depend on the outcome and the kind of cluster; prefer an estimate from a comparable study.

Range of values across clusters $L,\ U$ Optional

Values: Two values in the outcome’s units, the first below the second

An alternative to the ICC: the range within which you expect most clusters’ values to fall. It is converted to an ICC by the formula above.

Design effect $\mathrm{DE}$ Optional

Values: Above 0; usually between 1 and 3

An alternative to the ICC, when a previous study of the same design reports its design effect.

Observations per cluster $m$ Optional

Values: A whole number of at least 1

Usable responses per cluster. Larger clusters are cheaper to reach, but when the ICC is above zero each additional person in a cluster adds less information.

Number of clusters in each group $k$ Optional

Values: A whole number of at least 2

Instead of a cluster size, when the clusters are fixed: the emirates of a country, the schools of a district. The calculator then finds the cluster size, or reports how many clusters the margin needs.

Share of the sample for each cluster $s_j$ Optional

Values: One positive value per cluster, on any scale

When clusters will contribute unequally, for example in proportion to their population. With an ICC, unequal shares enlarge the design effect [4].

Correction for few clusters $t_{k-1}$ Optional

Values: On or off

Uses t on k − 1 degrees of freedom instead of the normal distribution. Worth applying when there are few clusters, roughly fewer than 30.

Assumptions

The two groups are independent random samples, and participants respond independently apart from the clustering specified.
Each group’s mean is approximately normal; the analysis uses an interval that allows unequal variances (Welch).
The standard deviations are planning values; if they are larger in truth, the margin is wider.
With cluster sampling: the two groups have distinct clusters, with the same ICC and cluster size in each; clusters are of equal size unless shares are given; the sampled clusters are a small fraction of all clusters; and the analysis accounts for the clustering, for example with survey-design standard errors [3, 5].
Non-response is unrelated to the outcome.

Worked examples

Two equal groups

SD 10 in both groups, margin ±3: n₁ = 1.96² × (10² + 10²) / 3² = 85.37, rounded up to 86 per group.

Group 1: 86 · Group 2: 86 · Total: 172

Protocol statement
To estimate the difference between two means with 95% confidence and a margin of error of ±3, assuming a standard deviation of 10 in both groups, the required sample size is 86 participants per group, 172 in total. Calculated with StatWisely (statwisely.com) using: n₁ = z_(1-α/2)² × (σ₁² + σ₂²/r) / E²; n₂ = r × n₁. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Unequal spread and twice as many in group 2

SD 10 and 15, with group 2 twice the size of group 1: n₁ = 1.96² × (10² + 15²/2) / 3² = 90.70, so 91 in group 1 and 182 in group 2.

Group 1: 91 · Group 2: 182 · Total: 273

Open this example in the calculator

Cluster sampling with an ICC

ICC 0.05 and 10 per cluster: DE = 1 + 9 × 0.05 = 1.45. Each group needs 86 × 1.45 = 124.7, so 13 clusters of 10 and 130 participants per group.

Group 1: 130 · Group 2: 130 · Total: 260

Protocol statement
To estimate the difference between two means with 95% confidence and a margin of error of ±3, assuming a standard deviation of 10 in both groups, 172 participants would be required under simple random sampling. Allowing for a design effect of 1.45 (ICC 0.05, 10 per cluster) for cluster sampling, the required sample size is 130 participants per group, 260 in total, in 13 clusters of 10 in each group. This assumes clusters of equal size, distinct between the two groups, with a negligible sampling fraction. Calculated with StatWisely (statwisely.com) using: n₁ = z_(1-α/2)² × (σ₁² + σ₂²/r) / E²; n₂ = r × n₁; clustered: n x DE, DE = 1 + (m-1)ρ. 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 procedure, with and without cluster sampling, every night and before each production release, and the results are compared with the engine’s.
The basic result agrees with the closed form: 1.96² × 200 / 9 = 85.37, rounded up to 86 per group. In R, ceiling(qnorm(0.975)^2 * (10^2 + 10^2) / 3^2) returns 86.

How to cite

StatWisely. Sample size to estimate a difference between two means. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/precision-mean-difference/

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

References

1.Julious SA. Sample Sizes for Clinical Trials. Boca Raton: Chapman & Hall/CRC; 2010.
2.Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977.
3.Kish L. Survey Sampling. New York: John Wiley & Sons; 1965.
4.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
5.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
6.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

Simple random sampling

Finite population correction

Design effect from an ICC

Unequal cluster sizes

Number of clusters for a chosen cluster size

Cluster size for a fixed number of clusters

ICC from a range across clusters

Few clusters

Non-response

Parameters

Confidence level1−α1-\alpha1−α

Margin of errorEEE

Standard deviation in group 1σ1\sigma_1σ1​

Standard deviation in group 2σ2\sigma_2σ2​Optional

Allocation ratiora=n2/n1r_a = n_2/n_1ra​=n2​/n1​Optional

Population sizesN1, N2N_1,\ N_2N1​, N2​Optional

Expected non-response rateqqqOptional

Intracluster correlation (ICC)ρ\rhoρOptional

Range of values across clustersL, UL,\ UL, UOptional

Design effectDE\mathrm{DE}DEOptional

Observations per clustermmmOptional

Number of clusters in each groupkkkOptional

Share of the sample for each clustersjs_jsj​Optional

Correction for few clusterstk−1t_{k-1}tk−1​Optional

Assumptions

Worked examples

Two equal groups

Unequal spread and twice as many in group 2

Cluster sampling with an ICC

Validation

How to cite

References

Confidence level $1-\alpha$

Margin of error $E$

Standard deviation in group 1 $\sigma_1$

Standard deviation in group 2 $\sigma_2$ Optional

Allocation ratio $r_a = n_2/n_1$ Optional

Population sizes $N_1,\ N_2$ Optional

Expected non-response rate $q$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Range of values across clusters $L,\ U$ Optional

Design effect $\mathrm{DE}$ Optional

Observations per cluster $m$ Optional

Number of clusters in each group $k$ Optional

Share of the sample for each cluster $s_j$ Optional

Correction for few clusters $t_{k-1}$ Optional