Sample size to estimate a mean paired difference

When to use it

The same people measured twice, before and after, or under two conditions, when the average change is to be estimated.
Matched pairs, such as two eyes of one person or siblings, measured on a continuous scale.

Not for

Two independent groups. Use the difference between two means.
Showing that the change differs from zero. Use the powered paired t-test under “Detect an effect”.

Notation

$n$: Number of complete pairs
$\sigma_d$: Standard deviation of the within-pair differences
$E$: Margin of error for the mean difference, in the measurement’s units
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α (1.960 for 95%)
$N$: Population size
$\mathrm{DE}$: Design effect
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Usable observations per cluster
$k$: Number of clusters
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters in the population
$W_h$: Population share of stratum h
$a_h$: Sample share of stratum h, set by the allocation
$L,\ U$: Low and high ends of the range of cluster values
$q$: Non-response rate

Formulas

Simple random sampling

n = \left\lceil \left(\frac{z_{1-\alpha/2}\;\sigma_d}{E}\right)^{2} \right\rceil

A paired difference is a one-sample estimate on a derived variable, the within-pair difference, so its precision depends on the spread of the differences rather than of either measurement [1].

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 the SD of each measurement (σ₁, σ₂) and the correlation between them (ρ₁₂) are known. The stronger the correlation, the smaller σ_d, which is why pairing saves participants.

Finite population correction

n' = \left\lceil \frac{n}{1 + (n - 1)/N} \right\rceil

Applied when a population size is given [2]. It matters only when the sample is a sizeable share of the 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

The requirement is rounded up to whole clusters, so the total can exceed n × DE slightly. At least two clusters are needed to estimate between-cluster variation.

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.

ICC from a range across clusters

\rho \approx \frac{\left[(U - L)/4\right]^{2}}{\sigma_d^{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 variance of the differences σ_d². An elicitation aid, not a measurement.

Sampling a share of a finite set of clusters

\mathrm{DE} = 1 + (m - 1)\,\rho - \frac{k}{C}\, m\,\rho, \qquad k = \left\lceil \frac{n\,[1 + (m-1)\rho]}{m\,(1 + n\rho/C)} \right\rceil

Optional. When the clusters sampled are a sizeable share of the C there are, the between-cluster part of the variance shrinks by 1 − k/C, and the number of clusters has the closed form shown [2]. People within a cluster are taken as a small share of it, the cautious side. The people-level correction does not apply to a cluster sample.

Stratified sampling

n_{\text{stratified}} = \left\lceil n\,\mathrm{DE}_w\,\mathrm{DE} \right\rceil, \qquad \mathrm{DE}_w = \sum_h \frac{W_h^{2}}{a_h}, \qquad n_h = \lceil a_h\, n_{\text{stratified}} \rceil

Optional. When every stratum (region, urban or rural area) is sampled, the strata add no design effect of their own; an allocation away from the population shares weights respondents unequally, which costs the combined estimate Kish’s weighting effect DE_w, 1 for proportional allocation [3]. It multiplies the cluster design effect when clusters are sampled within the strata. Every stratum can instead be sized for its own margin, and finite strata carry their own corrections: see “Sample size for a stratified survey” for the allocations, those options and worked examples.

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 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 mean change you can accept.

Standard deviation of the differences $\sigma_d$

Values: Above 0, in the measurement’s units

Not the SD of either measurement. Take it from a study reporting the SD of the change, or compute it from the two SDs and their correlation with the formula above; for example SDs of 10 and a correlation of 0.68 give σ_d = 8.

Population size $N$ Optional

Values: A whole number of at least 2

Leave empty for a large population. It matters only when the sample would be more than about 5% of the 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 $k$ 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].

Population share of each stratum $W_h$ Optional

Values: Positive, adding to 100%, for at least two strata

Tick “Stratified sampling” to use it. From a census or sampling frame.

Allocation across strata $a_h$ Optional

Values: Proportional, square root, equal, Neyman or custom

Proportional costs nothing; move towards equal when small strata must be reported on.

What the margin is for $\text{goal}$ Optional

Values: The combined estimate, or every stratum on its own

Every stratum on its own usually costs far more people.

Clusters in the population $C$ Optional

Values: A whole number, at least the clusters sampled

Leave empty when the clusters sampled are a small share of all there are. Give it when they are not, say 20 of 40 districts; the design effect then falls.

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

Pairs are selected at random and are independent of one another apart from the clustering specified; both measurements are available for every pair counted.
The mean of the differences is approximately normal.
With clustering, the ICC is that of the within-pair differences among participants in the same cluster, not the correlation between the two measurements; the cluster size counts complete pairs.
With cluster sampling: one ICC applies to every cluster; 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

Before and after

SD of the differences 8, margin ±2: n = (1.96 × 8 / 2)² = 61.46, rounded up to 62 pairs.

Sample size: 62

Protocol statement
To estimate a mean paired difference with 95% confidence and a margin of error of ±2, assuming a standard deviation of the paired differences of 8, the required sample size is 62 pairs. Calculated with StatWisely (statwisely.com) using: n_pairs = ((z_(1-α/2) × σ_d) / E)². These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Allowing for missing second measurements

Expecting 10% not to complete the second measurement: 62 / (1 − 0.10) = 68.9, so 69 are enrolled for 62 complete pairs.

Needed for analysis: 62 · To approach: 69

Open this example in the calculator

Cluster sampling with an ICC

ICC of the differences 0.05, 10 pairs per cluster: DE = 1 + 9 × 0.05 = 1.45. Then 62 × 1.45 = 89.9, so 9 clusters and 90 pairs.

Sample size: 90

Protocol statement
To estimate a mean paired difference with 95% confidence and a margin of error of ±2, assuming a standard deviation of the paired differences of 8, 62 pairs 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 90 pairs, in 9 clusters of 10. This assumes clusters of equal size, with a negligible sampling fraction. Calculated with StatWisely (statwisely.com) using: n_pairs = ((z_(1-α/2) × σ_d) / E)²; 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: (1.96 × 8 / 2)² = 61.46, rounded up to 62 pairs. In R, ceiling((qnorm(0.975) * 8 / 2)^2) returns 62.

How to cite

StatWisely. Sample size to estimate a mean paired difference. Method document, engine version 1.0.0, reviewed 2026-10-08. https://statwisely.com/methods/precision-paired-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

SD of the differences from the two measurements

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

Sampling a share of a finite set of clusters

Stratified sampling

Few clusters

Non-response

Parameters

Confidence level1−α1-\alpha1−α

Margin of errorEEE

Standard deviation of the differencesσd\sigma_dσd​

Population sizeNNNOptional

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 clusterskkkOptional

Share of the sample for each clustersjs_jsj​Optional

Population share of each stratumWhW_hWh​Optional

Allocation across strataaha_hah​Optional

What the margin is forgoal\text{goal}goalOptional

Clusters in the populationCCCOptional

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

Assumptions

Worked examples

Before and after

Allowing for missing second measurements

Cluster sampling with an ICC

Validation

How to cite

References

Confidence level $1-\alpha$

Margin of error $E$

Standard deviation of the differences $\sigma_d$

Population size $N$ 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 $k$ Optional

Share of the sample for each cluster $s_j$ Optional

Population share of each stratum $W_h$ Optional

Allocation across strata $a_h$ Optional

What the margin is for $\text{goal}$ Optional

Clusters in the population $C$ Optional

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