Confidence Interval for a Mean

An interval for the population mean

A sample mean does not identify the population mean exactly. A confidence interval describes uncertainty due to sampling. Divide the standard deviation by the square root of the sample size to get the standard error. Multiply by the critical value, then subtract and add this margin to the sample mean.

Choosing the distribution

Use the normal method only when the population standard deviation is known. When deviation is estimated from the same sample, use a t distribution with n − 1 degrees of freedom. With four observations and known deviation 2, the standard error is 1. Independence, random sampling and normality for small samples require separate assessment.

Follow a small example by hand

To illustrate the method, use mean 20, sample standard deviation 2 and sample size 2. Choose 50 percent confidence for this teaching example only. There is one degree of freedom. Its t value at cumulative probability 75 percent is 1, so the standard error is 2 / √2 ≈ 1.414214 and the margin is also 1.414214. The bounds are approximately 18.585786 and 21.414214. This low confidence level is not a recommendation for practical use.

Standard deviation describes the spread of observations. Standard error describes the sampling variability of the estimated mean. Do not enter one in place of the other. The mean and standard deviation must use the same measurement unit. When counting repeated measurements, consider whether those measurements are truly independent.

A confidence interval does not predict the range of a single future observation. More observations do not automatically correct systematic measurement error or biased sampling. Raising the confidence level for the same data widens the interval. Report the sample size, selected method and confidence level alongside the bounds so that the result can be interpreted correctly.

Frequently asked questions

How do you calculate a confidence interval from the standard error?

For example, 195 observations with sample standard deviation 0.022789 give standard error 0.001632 and a 95 percent confidence interval from 9.25824135 to 9.26467865 around mean 9.26146. Multiply the standard error by the critical t value, then subtract and add that margin to the mean, assuming independent random sampling and excluding systematic measurement error.

Calculate with these values