Confidence Interval for a Mean Calculator

Enter Calculation

Sample mean (x̄)

Sample standard deviation (s)

Sample size (n)

Confidence level 90% 95% 99%

Results

Confidence Interval

94.3989  to  105.6011

margin of error ± 5.6011

Margin of error	5.601092
Critical t value	2.04523
Standard error (s/√n)	2.738613
Degrees of freedom	29

What this calculator does

This tool builds a confidence interval for a population mean from sample data. When the population standard deviation is unknown — which is almost always the case in practice — the correct approach uses the Student-t distribution rather than the normal (z) distribution. The interval gives you a plausible range for the true mean at your chosen confidence level.

How to use it

Enter the sample mean (\(\bar{x}\)), the sample standard deviation (\(s\)), the sample size (\(n\)), and pick a confidence level of 90%, 95%, or 99%. The calculator returns the lower and upper bounds, the margin of error, the critical t value, the standard error, and the degrees of freedom (\(n - 1\)).

The formula explained

The interval is $$\text{CI} = \bar{x} \pm t_{\alpha/2,\,df}\cdot\frac{s}{\sqrt{n}}$$ Here \(s/\sqrt{n}\) is the standard error of the mean, which measures how much the sample mean is expected to vary from the true mean. The critical value \(t\) depends on the degrees of freedom (\(n - 1\)) and the chosen confidence level. Multiplying the standard error by \(t\) gives the margin of error, which is added to and subtracted from the sample mean.

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The confidence interval captures the central area under the t-distribution, leaving \(\alpha/2\) in each tail.

Worked example

Suppose \(\bar{x} = 100\), \(s = 15\), \(n = 30\), at 95% confidence. The standard error is $$\frac{15}{\sqrt{30}} \approx 2.7386$$ With 29 degrees of freedom the critical \(t \approx 2.0452\), so the margin of error is about \(5.601\). The 95% confidence interval is roughly \(94.40\) to \(105.60\).

A confidence interval is centered on the sample mean and extends by the margin of error in both directions.

FAQ

When should I use t instead of z? Use the t distribution whenever the population standard deviation is unknown and you estimate it from the sample — true for most real datasets. For large \(n\) the t and z values nearly coincide.

What does 95% confidence mean? If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population mean.

Does this assume normal data? The t interval assumes the data are approximately normal or the sample is large enough for the central limit theorem to apply.