Sample size for a one-sample t-test

When to use it

Comparing a group’s mean with a fixed reference value, such as a target level, a guideline threshold or a published norm.

Not for

Comparing two groups. Use the two-sample t-test.
Before-and-after measurements on the same people. Use the paired t-test.
Estimating the mean rather than testing it. Use the mean under “Estimate precisely”.

Notation

$n$: Number of participants
$\Delta$: Difference between the true mean and the reference value
$\sigma$: Standard deviation of the measurement
$d$: 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
$q$: 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}

The test statistic follows a noncentral t distribution with n − 1 degrees of freedom when the effect is real, and n is the smallest whole number whose power reaches the target. For a two-sided test the lower rejection region is included too [1].

For intuition: the normal approximation

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

Not what the calculator uses, but it shows the shape of the answer: halving the effect quadruples the sample. The exact t calculation adds a couple of participants, because σ is estimated from the data.

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

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.

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

Participants are a random sample, independent of one another apart from any clustering specified.
The measurement is roughly normal, or the sample is large enough for its mean to be.
The reference value is fixed and known, not itself estimated from data.
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

A medium effect

Difference 5, SD 10, so d = 0.5; α = 0.05 two-sided, 80% power. The normal approximation gives (2.80 / 0.5)² = 31.4; the exact t calculation gives 34.

Sample size: 34

Protocol statement
To detect a difference of 5 from the reference value with 80% power at a two-sided significance level of 0.05, using a one-sample t-test, and assuming a standard deviation of 10, the required sample size is 34 participants. Calculated with StatWisely (statwisely.com) using: n: 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

One-sided

The same effect tested in one direction only needs 27.

Sample size: 27

Open this example in the calculator

Allowing for dropout

Expecting 10% to provide no outcome: 34 / (1 − 0.10) = 37.8, so 38 are recruited.

Needed for analysis: 34 · To recruit: 38

Protocol statement
To detect a difference of 5 from the reference value with 80% power at a two-sided significance level of 0.05, using a one-sample t-test, and assuming a standard deviation of 10, the required sample size is 34 participants. Allowing for 10% dropout, 38 participants should be recruited. Calculated with StatWisely (statwisely.com) using: n: 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 results agree with G*Power’s one-sample t-test: 34 two-sided and 27 one-sided for d = 0.5, α = 0.05 and 80% power [2].

How to cite

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

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

Expected dropoutqqqOptional

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

Assumptions

Worked examples

A medium effect

One-sided

Allowing for dropout

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 $\tau$

Expected dropout $q$ 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