Single mean — Sample Size Calculator

Determines the sample size required to test whether a population mean differs from a specified hypothesized or reference mean, given a chosen significance level and statistical power.

Key takeaways

What is calculated: Find the smallest integer n at which the exact power of the one-sample t-test reaches the target.
Quantities you must supply: Reference mean; Anticipated mean; Standard deviation; Significance level; Power.
Type of calculation: Power-based: it answers how much data is needed to detect a stated effect. Power 0.80 at α = 0.05 is the usual target; confirmatory studies use 0.90.
Calculation routes: 2 are available for this design; the one selected determines the formula, the quantities requested and the result.

Methods available for Single mean

Normal approximation (z-based) Closed-form sample size treating the population standard deviation as known. Chosen for transparency and comparability: the formula is the one tabulated in most textbooks, so a reviewer recomputing the figure will reproduce it exactly.
Noncentral t (exact, one-sample t-test) Iterative solution using the exact distribution of the one-sample t statistic under the alternative. Chosen because it accounts for the additional uncertainty from estimating σ, which the normal approximation ignores. This is not merely a small-sample correction: the normal formula approximates this calculation at every n.

When to use

Use when the primary objective is to determine whether a population mean differs from a specified reference or hypothesized value with adequate statistical power.

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Sample size

Calculation method

Reference mean (μ₀)

Anticipated mean (μ₁)

Standard deviation (σ)

Significance level (α)

Type I error rate.

Power (1 − β)

Probability of detecting the specified effect.

Alternative

A two-sided alternative splits α between both tails; a one-sided alternative places all of α in one tail, needs a smaller n, and must be pre-specified.

Anticipated dropout / non-response (proportion)

Final n is inflated by 1/(1 − dropout).

Reset to defaults

Awaiting inputs

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Set the parameters and calculate.

Methodology — One-sample mean — noncentral t (exact)

Formula used One-sample mean — noncentral t (exact)

1 - β = \Pr (| t_{n - 1, λ} | > t_{1 - α / 2, n - 1})

Exact power under the noncentral t; the requirement is the smallest n reaching the target

λ = \frac{δ \sqrt{n}}{σ}

Noncentrality parameter

δ = | μ_{1} - μ_{0} |

Minimum detectable difference

$z_{1 - α / 2} (two-sided) in place of z_{1 - α} (one-sided)$ Two-sided alternative selected: α is split between both tails, so the larger quantile is used and the requirement rises

where —

μ₀: Hypothesised reference mean under H₀. Any value on the outcome scale
μ₁: Anticipated mean under the alternative. Must differ from μ₀
σ: Population standard deviation of the outcome. σ > 0
α: Probability of rejecting a true null hypothesis (Type I error). 0.0001 – 0.5; conventionally 0.05, or 0.025 one-sided for regulatory non-inferiority
1 − β: Probability of rejecting the null hypothesis when the specified alternative is true. 0.50 – 0.9999; conventionally 0.80 or 0.90
ℓ: Expected proportion of enrolled units lost before analysis. 0 – 0.95

Formulas for the 2 methods

Calculation method	Formula
Normal approximation (z-based)	$n = \frac{{(z_{1 - α / 2} + z_{1 - β})}^{2} σ^{2}}{δ^{2}}$
Noncentral t (exact, one-sample t-test)	$1 - β = \Pr (\| t_{n - 1, λ} \| > t_{1 - α / 2, n - 1})$

How this method works

When σ is estimated from the sample, the test statistic (x̄ − μ₀)/(s/√n) does not follow a normal distribution under the alternative; it follows a noncentral t distribution with n − 1 degrees of freedom and noncentrality λ = δ√n/σ. Power is computed from that distribution and n is found by search.

The test is two-sided: H₀: μ = μ₀ against H₁: μ ≠ μ₀. α is split between the two tails, so the α-quantile is z₁₋α/₂.

Selected method — noncentral t. When the population standard deviation is unknown and the planned analysis is a one-sample t-test, the test statistic follows a noncentral t distribution under the alternative, with noncentrality λ = δ√n/σ on n − 1 degrees of freedom. The smallest n attaining the target power is found by iteration.

This is the exact calculation and is the appropriate choice whenever σ is estimated from the data. It is not merely a small-sample correction: the normal formula is an approximation to it at every n, and the two converge as n grows.

Null hypothesis. H₀: μ = μ₀

Alternative hypothesis. Two-sided H₁: μ ≠ μ₀. One-sided H₁: μ > μ₀ or μ < μ₀.

Calculation procedure

State $μ_{0}$ , $μ_{1}$ and $σ$ , giving the standardised effect $d = (μ_{1} - μ_{0}) / σ$ .
For a candidate $n$ , form the noncentrality $λ = d \sqrt{n}$ ; the test statistic follows a noncentral t on $n - 1$ degrees of freedom.
Compute exact power as $\Pr (| t_{n - 1, λ} | > t_{1 - α / 2, n - 1})$ .
Search upward for the smallest $n$ whose power reaches the target.
Round up, then inflate for dropout. This is the correct calculation whenever $σ$ is estimated rather than known.

Exact or approximate

Exact, to the numerical tolerance of the distribution routines.

The formula shown always matches what was computed: when a one-sided test is selected the rendered quantile changes from z_1−α/2 to z_1−α, and where several methods exist, the formula follows the method selected in the calculator.

Advantages & limitations

Advantages

Correct when σ is estimated, which is the normal case in practice.
Accounts for degrees of freedom explicitly.
Agrees with G*Power and the R pwr package.

Limitations

No closed form; the figure cannot be reproduced by hand.
More sensitive to the normality assumption than the large-sample approximation.

Assumptions

Observations are independent and identically distributed.
The outcome is normally distributed — this matters more here than for the normal approximation, since the t distribution depends on it.
σ is unknown and estimated by the sample standard deviation.
Simple random sampling with no clustering.

Applicable adjustments

Supports dropout inflation. Finite population correction and clustering as for the z-based method.

Conclusion

This is the defensible figure for a protocol whose analysis is a one-sample t-test. Report it together with σ, the difference sought, α and power, since the result is conditional on all four.

References

Guenther WC. Sample size formulas for normal theory t tests. The American Statistician. 1981;35(4):243–244. Link
Chow S-C, Shao J, Wang H, Lokhnygina Y. Sample Size Calculations in Clinical Research. 3rd ed. Boca Raton: Chapman & Hall/CRC; 2017.
Rosner B. Fundamentals of Biostatistics. 8th ed. Boston: Cengage; 2015.