Sample Size for Mean Calculator

Formula

How this is calculated

The required sample size for estimating a mean comes from inverting the confidence-interval margin of error. A confidence interval for a mean has half-width E = z* × σ / √n, where z* is the standard-normal critical value for your confidence level, σ is the population standard deviation, and n is the sample size. Solving for n gives n = (z* × σ / E)².

The confidence level selects z*: 90% uses z* = 1.645, 95% uses z* = 1.96, and 99% uses z* = 2.576 (two-sided values). A larger σ widens the natural spread and increases the sample needed, while a smaller margin of error E demands a sharply larger sample because n scales with 1/E². Because n must be a whole number and you can never under-sample, the raw value is always rounded up (ceiling) to the next integer.

This formula assumes you know (or can reasonably estimate) σ and that the sample mean is approximately normally distributed, which holds for large samples by the Central Limit Theorem. If σ is unknown and the sample is small, a t-based approach is more accurate, but the z formula is the standard planning tool. E and σ must be expressed in the same units and both must be positive.