How do I interpret a 95% confidence interval?

If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability that this particular interval contains it.

Use t when the standard deviation comes from the sample, which is nearly always. Use z only when the population σ is genuinely known. For n above 100 the two are practically identical.

What is the margin of error?

Half the width of the interval: critical value × standard error. The interval is the sample mean plus and minus the margin of error.

Why does the interval get wider at 99% confidence?

To be more confident of capturing the true mean you need a larger critical value (2.58 instead of 1.96 for z), which widens the interval.

Confidence Interval for a Mean Calculator

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Confidence Interval for a Mean Calculator

A confidence interval for the mean gives a range of plausible values for the population mean based on one sample: x̄ ± (critical value × s/√n). Paste raw data or enter the sample mean, standard deviation and size, choose whether the standard deviation is the sample’s (use t) or a known population σ (use z), and pick the confidence level.

The calculator shows the standard error, the critical value from the t distribution with n − 1 degrees of freedom or from the normal distribution, the margin of error and the final interval, followed by a correctly worded interpretation. The t version is the right one in almost every real problem because σ is rarely known.

A wider interval comes from a higher confidence level, a larger standard deviation or a smaller sample. To find the sample size that keeps the margin of error under a target, use the sample size calculator for a mean. For a hypothesis test on the same data see the one-sample t-test.

Sample data (optional)

Separate with commas, spaces or new lines. Leave blank to use the summary boxes.

Sample mean (x̄)

Standard deviation

Sample size (n)

Standard deviation is

Confidence level

Result and step-by-step solution

95% confidence interval

69.0327 to 75.9673

x̄ ± margin of error = 72.5 ± 3.4673

Sample statistics
Sample: n = 25, x̄ = 72.5, s = 8.4.
Standard error
SE = s ÷ √n = 8.4 ÷ √25 = 1.68
Critical value
t_{0.025, df = 24} = 2.0639 (t distribution)
Margin of error
ME = critical value × SE = 2.0639 × 1.68 = 3.4673
Interval
x̄ ± ME = 72.5 ± 3.4673 = (69.0327, 75.9673)

Interpretation: we are 95% confident that the population mean lies between 69.0327 and 75.9673. If we repeated the sampling many times, about 95% of intervals built this way would contain the true mean.

Frequently asked questions

How do I interpret a 95% confidence interval?: If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability that this particular interval contains it.
Should I use t or z?: Use t when the standard deviation comes from the sample, which is nearly always. Use z only when the population σ is genuinely known. For n above 100 the two are practically identical.
What is the margin of error?: Half the width of the interval: critical value × standard error. The interval is the sample mean plus and minus the margin of error.
Why does the interval get wider at 99% confidence?: To be more confident of capturing the true mean you need a larger critical value (2.58 instead of 1.96 for z), which widens the interval.

Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

Confidence Interval for a Mean Calculator

Result and step-by-step solution

Frequently asked questions

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