Confidence interval for the population mean, using the t-distribution
Enter your sample data and choose a confidence level to get a confidence interval for the mean. The calculator uses the t-distribution and also reports the margin of error and standard error.
Two-sided t-interval: mean ± t(n−1) × s/√n.
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A confidence interval is a range of plausible values for an unknown population parameter — here, the population mean — estimated from a sample. A 95% confidence interval means that if you repeated the sampling many times, about 95% of the intervals constructed this way would contain the true mean. It expresses the precision of your estimate: narrower intervals indicate more precise estimates.
For the mean of a sample, the interval is x̄ ± t × (s ÷ √n), where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t is the critical value from the t-distribution with n − 1 degrees of freedom at your chosen confidence level. The quantity s ÷ √n is the standard error of the mean, and t × (s ÷ √n) is the margin of error.
When the population standard deviation is unknown and estimated from the sample — the usual case — the t-distribution gives wider, more honest intervals than the normal distribution, especially for small samples. As the sample size grows, the t-distribution approaches the normal distribution and the two give almost identical results.
It means the method produces an interval that would capture the true population mean about 95% of the time over repeated sampling. It is a statement about the procedure, not a 95% probability that this particular interval contains the mean.
The margin of error is the half-width of the interval: t × (s / √n). The interval is the sample mean plus or minus this amount.
This calculator uses the t-distribution with n − 1 degrees of freedom, which is appropriate when the population standard deviation is estimated from the sample. For large samples the result is nearly identical to a z-interval.