Compute the confidence interval for a population mean from the sample mean, standard deviation and sample size.
A confidence interval gives a range of values that is likely to contain the true population mean, based on a sample. The 95% confidence interval means that if you repeated the sampling process many times, about 95% of the intervals computed this way would contain the true mean. It is not a probability statement about this particular interval — it is a statement about the method.
The calculation uses the normal approximation: margin of error = z × (σ / √n), where z is the critical value for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%). The interval is the sample mean ± that margin. The standard error (σ / √n) shrinks with larger samples, which is why a larger n gives a narrower interval — more data means more precision.
This tool uses the z-distribution, which assumes the standard deviation is known and the sample is large enough for the central limit theorem to apply (roughly n ≥ 30). For small samples with an unknown σ, the t-distribution would be more appropriate — the difference is small for n above 30 and negligible above 100. Everything runs in your browser.
CI = x̄ ± z × (σ / √n). At 95%: z = 1.96. Margin of error = z × SE, where SE = σ / √n.
It means that if you drew 100 different samples and computed the interval each time, about 95 of those intervals would contain the true population mean. It is not a 95% probability that this specific interval contains it.
Increase the sample size (n). Doubling n cuts the margin of error by about 30%. You can also lower the confidence level (from 99% to 95%), but that increases the chance of missing the true mean.
When the sample size is small (n < 30) and the population standard deviation is unknown. For n ≥ 30, the z and t distributions are nearly identical.
No — this tool is for means. A proportion confidence interval uses a different formula (p̂ ± z × √(p̂(1−p̂)/n)).
No. The calculation runs locally in JavaScript.
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