StatReason

Mean Confidence Interval Calculator

Enter your sample values to estimate the population mean with a stated margin of uncertainty, using the t distribution rather than assuming a large-sample normal shortcut.

What this answers

This calculator answers "given my sample, what range of values plausibly contains the true population mean?" A single sample mean is only ever an estimate. This interval expresses how precise that estimate is by giving a range that, under repeated sampling using this same method, would contain the true mean the stated percentage of the time.

How it is calculated

The interval is your sample mean plus and minus a margin of error, where the margin of error is a t critical value (based on your confidence level and n minus 1 degrees of freedom) multiplied by the standard error of the mean (the sample standard deviation divided by the square root of n). The t distribution is used rather than the normal distribution because it correctly accounts for the extra uncertainty of estimating the standard deviation from a finite sample, which matters more the smaller your sample is.

Worked example

For the values 1, 2, and 3 at 95% confidence: the mean is 2, the sample standard deviation is 1, so the standard error is 1 divided by the square root of 3, about 0.577. With 2 degrees of freedom, the t critical value for 95% confidence is about 4.303. The margin of error is 4.303 times 0.577, about 2.485, giving an interval of roughly negative 0.485 to 4.485. That wide interval reflects how little a sample of only 3 values can pin down a true mean, not a flaw in the method.

Assumption audit

Calculated from your data: sample size, whether it meets the minimum of 2 values the t interval requires, and whether the data is constant (giving a zero-width interval).
Evidence to review: whether your sample looks roughly symmetric rather than heavily skewed. The t interval is fairly robust to mild departures from normality, especially as n grows, but a strongly skewed small sample can make the stated coverage less reliable than advertised.
You must verify: that your sample was drawn in a way that makes each observation representative and independent of the others. No amount of correct arithmetic on the numbers you entered can establish that on its own.

What this interval does not mean

A 95% confidence interval does not mean there is a 95% probability the true mean falls in this specific computed range. Under the standard frequentist interpretation, the true mean is a fixed (if unknown) number, not a random variable, so it either is or is not in this particular interval. The 95% describes the long-run behavior of the method across many hypothetical samples, not a probability statement about this one result.

Limitations

A wider confidence level always produces a wider interval, all else equal, trading precision for confidence. This calculator cannot correct for a biased sampling method, and a technically correct interval built on unrepresentative data is still an estimate of the wrong thing.