Find a 95% Confidence Interval for a Population Mean

Start with the mean and its sampling design

The target is a population mean μ, while the observed sample supplies a mean x̄ and a sample size n. Independent random sampling is a common starting model. Clustered observations, repeated measurements and survey weights can require a different standard error. A confidence interval cannot compensate for a sample drawn from the wrong population. Define what was measured and how observations were selected before deciding which familiar formula to apply.

Decide whether sigma is actually known

When the population SD σ is known and the mean has a normal sampling distribution, the two-sided interval is x̄±1.96σ/√n. When σ is unknown and estimated by the sample SD s, the usual normal-population method is x̄±t0.975,n−1s/√n. The symbols refer to different sources of information. A number printed in a problem is not automatically a known population parameter; the text must say whether it describes the population or was calculated from the sample.

Calculate a known-sigma interval

Take x̄=50, σ=10 and n=100. The standard error is 10/10=1 and the 95% normal margin is 1.96. The endpoints are 48.04 and 51.96. Enter these values in the local form with confidence 95 to verify the calculation. Check that the midpoint equals 50 and that each endpoint is 1.96 away. The form's SD field must receive 10, not the variance 100 and not the already-divided standard error one.

Use a second worked example

For mean 20, known SD 4 and n=25, the standard error is 4/5=0.8. Multiplying by 1.96 gives margin 1.568 and interval [18.432,21.568]. These are the normal-method results the local form should return. If the value four were instead a sample SD from those 25 observations, use 24 degrees of freedom. A t critical value about 2.0639 gives a wider margin about 1.6511, so the corresponding interval is approximately [18.3489,21.6511].

Understand what a larger sample changes

For a fixed SD, the margin falls as 1/√n. Quadrupling n halves the margin; merely doubling n multiplies it by about 0.7071. A large sample can support an approximate normal sampling distribution under suitable conditions, but there is no universal sample-size threshold that cures severe skew, dependence or influential extremes. If the population SD is estimated, t and normal critical values become close as the degrees of freedom grow, while remaining conceptually different procedures.

Check finite populations and dependent records

Sampling a substantial fraction of a finite population without replacement can require a finite-population correction. Clustered or serially related records can have less information than the same number of independent observations. The local form uses SD/√n without either adjustment. Do not treat the number of rows in a file as an automatically valid independent sample size. Follow the sampling design when estimating uncertainty, especially when observations are nested within households, classrooms, machines or repeated time periods.

Distinguish confidence from prediction

The interval estimates an average, not where a future individual observation will fall. A prediction interval includes individual variation and is generally wider under the corresponding model. A population tolerance interval answers yet another question about covering a specified fraction of the population. For a 95% confidence interval, the repeated-sampling coverage pertains to the estimation procedure for the fixed mean. It is not a statement that 95% of the observed or future measurements must lie inside these endpoints.

Document the method and precision

Report the confidence level, point estimate, endpoints, sample size, source of the SD and chosen z or t method. Include units and any design adjustment. If the original measurements have limited precision, avoid reporting excessive endpoint digits while retaining precision during the computation. A result from another package may differ slightly because this form uses the rounded constant 1.96. A substantially wider t result or a design-adjusted interval, however, should not be dismissed as a rounding difference.

Related guides

Confidence Interval Calculation: Two Mean Examples

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