[47.1386, 52.8614]
| Margin of error | ±2.861428 |
| Critical t (df = n − 1) | 2.262157 |
| Estimate | 50.000000 |
x̄ ± t(n − 1) × s / √n
Assumptions: independent observations. The z version also needs a known population σ; the t version uses the sample s and needs a roughly normal population or a large n. The proportion versions need independent trials with a fixed success probability.
Get a confidence interval for a mean or a proportion. A confidence interval gives a range of plausible values for a population parameter, based on a sample. This calculator handles three cases: a mean when the population standard deviation is known (z), a mean when you only have the sample standard deviation (t), and a proportion (Wald and Wilson). Pick the case, choose a confidence level, enter the sample numbers, and the interval, margin of error and critical value appear with the formula used.
Choose the right critical value. When you estimate a mean with a sample standard deviation, the correct multiplier comes from the t distribution with n − 1 degrees of freedom, not from z, so the interval is honest about the extra uncertainty in s. 100% free, no registration, and complete privacy — everything runs locally in your browser, so your data never touches a server.
When the population standard deviation is known, the interval is x̄ ± z × σ / √n with the normal critical value for your confidence level.
With a sample standard deviation, the calculator uses the t critical value with n − 1 degrees of freedom. For n = 10 at 95% that is 2.262, not 1.96.
The Wald interval is the textbook normal approximation. The Wilson score interval is shown next to it, since it behaves better for small samples or extreme proportions.
The formula and the assumptions for each method are shown with the result, so you can check the calculation by hand.
Z score, percentile and p-values for a value in a normal distribution.
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One-tailed and two-tailed p-values for a z or t test statistic.
Sample and population standard deviation and variance for a list.
One-sample, Welch and paired t tests with the p-value.
Use t when you estimate the population standard deviation from your sample, which is almost always the case in practice. The t distribution has heavier tails for small samples, so the critical value is larger than z until the sample is large.
The observations should be independent. The z interval also needs a known population standard deviation. The t interval assumes the population is roughly normal or that n is large enough for the sample mean to be close to normal. Proportion intervals assume a fixed success probability across independent trials.
The Wald interval is simple and widely taught, but it can go below 0 or above 1 and often under-covers for small n or p near 0 or 1. The Wilson score interval stays inside 0 and 1 and performs better in those cases, so comparing them is useful.
If you repeated the sampling and built an interval the same way many times, about 95% of those intervals would contain the true parameter. It is a statement about the method, not a 95% probability for the one interval you computed.
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