Confidence interval calculator for a mean

Interval = x̄ ± z × σ ÷ √n. Standard error = σ ÷ √n. A confidence interval gives a range that is likely to contain the true mean. Enter the sample mean, the standard deviation and the number of observations, and choose the confidence level. Larger samples shrink the standard error, so the interval narrows in proportion to the square root of the sample size. This calculator uses the normal (z) distribution, which is appropriate when the standard deviation is known or the sample is large.

With sample mean (x̄) 72, standard deviation (σ) 10, sample size (n) 64, confidence level 95%, the margin of error (±) is 2.449954981. Lower bound: 69.55004502; Upper bound: 74.44995498; Standard error: 1.25; Z-score for the confidence level: 1.959963985.

This is an estimate based on the inputs and assumptions shown on this page, not professional advice. Check anything with financial, health or safety consequences independently.

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If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true mean. Any single interval either contains it or does not.

Increase the sample size or accept a lower confidence level. Quadrupling n halves the margin of error, because it depends on √n.

When the sample is small and the standard deviation is estimated from the same sample. The t distribution has heavier tails, which widens the interval to account for the extra uncertainty.