Calculate how many observations you need to test whether the population mean differs from a reference value with the desired power. Unlike the precision-based approach, here the goal is to detect a minimum relevant difference.
Calculator
Calculate the sample size needed to test a mean with a target power.
Explanation
This calculator estimates the sample size for testing a population mean with null hypothesis \(H_0: \mu=\mu_0\), significance level \(\alpha\), power \(1-\beta\) and minimum detectable difference \(\Delta=|\mu-\mu_0|\).
You can configure a two-tailed alternative (\(H_a: \mu \neq \mu_0\)) or a one-tailed one (\(H_a: \mu \geq \mu_0\) or \(H_a: \mu \leq \mu_0\)). For a one-tailed test, the critical quantile is smaller and the sample size is usually reduced.
Sample size formula
\( n = \left(\frac{(Z_{\alpha^*}+Z_{\beta})\sigma}{\Delta}\right)^2 \)
- \(\alpha^*\): \(\alpha/2\) for a two-tailed alternative, \(\alpha\) for one-tailed.
- \(\sigma\): expected standard deviation.
- \(\Delta\): minimum detectable difference \(|\mu-\mu_0|\).
- \(1-\beta\): target power of the test.
Quick setup
- σ: use historical data, a pilot study, or the literature.
- Δ: define the minimum relevant difference in the original units.
- α: 0.05 is common; 0.01 for confirmatory scenarios.
- Power: 0.80 is a common minimum; 0.90 when you want to reduce false negatives.
- H0 and alternative: specify \(\mu_0\) and the direction of the test according to the scientific objective.
Simple example
With \(\sigma=10\), \(\Delta=2\), \(\alpha=0.05\), power 0.80 and a two-tailed alternative, approximately \(n \approx 197\) is required.
Worked example
A quality control technician at a packaging plant wants to verify whether the mean fill volume of the containers is still at the nominal 50 ml. Process specifications indicate a standard deviation of \(\sigma = 6\) ml. The technician wants 90% power to detect any deviation of at least \(\Delta = 2\) ml from the nominal value, using a two-tailed test with \(\alpha = 0.05\) (\(z_{\alpha/2} = 1.960\), \(z_\beta = 1.282\)).
The hypothesis to test is \(H_0: \mu = 50\) ml versus \(H_a: \mu \neq 50\) ml. Applying the formula:
\( n = \left(\frac{(z_{\alpha/2}+z_\beta)\,\sigma}{\Delta}\right)^2 = \left(\frac{(1.960+1.282)\times 6}{2}\right)^2 = \left(\frac{3.242 \times 6}{2}\right)^2 = \left(\frac{19.452}{2}\right)^2 = (9.726)^2 = 94.6 \rightarrow n = 95 \)
You need to measure 95 containers from the production line for the test to have a 90% probability of detecting deviations of 2 ml or more, with a 5% type I error rate. If the total production batch has fewer than 1,000 units, it would also be worth applying a finite population correction.
Practical interpretation: with this sample, if the process is centered at 48 ml or 52 ml (a 2 ml deviation), the test will detect the problem in 90% of inspections. Larger deviations will be detected with an even higher probability.
Variant with \(\alpha = 0.01\): if the packaging process has regulatory consequences and tighter control over false positives is required, you can use \(\alpha = 0.01\) (\(z_{\alpha/2} = 2.576\)): \( n = ((2.576 + 1.282) \times 6 / 2)^2 = (3.858 \times 3)^2 = (11.574)^2 = 133.97 \rightarrow n = 134 \). The stricter statistical rigor requires 41% more measurements to maintain the same 90% power.
Model assumptions
- Random sample and independence between observations.
- Approximately normal distribution or large n (CLT).
- Reasonably estimated expected standard deviation.
How to interpret the result
The value \(n\) is the minimum number of valid observations needed for the test \(H_0\!: \mu = \mu_0\) versus \(H_1\!: \mu = \mu_1\) to have the specified power \((1-\beta)\) at significance level \(\alpha\). Always round up. If you anticipate data loss or subjects who won't complete the study, the number to recruit is \(\lceil n / (1 - \text{dropout rate}) \rceil\); with an expected 15% dropout rate, recruit \(\lceil n / 0.85 \rceil\) individuals.
The parameters that most influence \(n\) are the effect size \(|\mu_1 - \mu_0| / \sigma\) and the power. Halving the minimum detectable difference \(\delta = |\mu_1 - \mu_0|\) quadruples \(n\); increasing power from 80% to 90% increases \(n\) by roughly 30%. Error in \(\sigma\) propagates quadratically: if the true standard deviation is 20% larger than assumed, you'll need 44% more subjects. For that reason, always run a sensitivity analysis calculating \(n\) for \(\sigma - 25\%\), \(\sigma\) and \(\sigma + 25\%\), and plan using the largest \(n\).
When the resulting \(n\) is impractical, the available levers are: (1) review whether \(\delta\) can be larger without losing practical relevance, (2) accept 80% power instead of 90%, or (3) use a one-tailed design if the context justifies it (which requires a somewhat smaller \(n\)). If \(n\) is very small (< 20), check that the data distribution is approximately normal or apply a nonparametric test (one-sample Wilcoxon test) and adjust \(n\) with the corresponding relative efficiency factor. Once the data are collected, run the test with the hypothesis test calculator for one mean.