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Standard error &
95% confidence interval.

From a few sample statistics to a clearer picture. Calculate standard error and a two-sided 95% confidence interval for a population mean.

02 / MEAN CONFIDENCE INTERVALSTUDENT’S t

Start with your sample.

SUMMARY DATA
A whole number from 2 to 1,000,000,000.
The average of your observations.
Use sample SD, calculated with n − 1.

Use summary statistics from a numeric sample. This calculator is for a mean, not a proportion, a prediction interval, or a comparison between two groups.

A range, with context.

95% INTERVAL
45.8722 – 54.1278
Two-sided confidence interval for the mean
Sample mean50
Standard error2
Margin of error4.1278
t critical value2.0639
Degrees of freedom24
Sample size / sample SD25 / 10

Display rounded to 6 significant digits; calculation uses full precision. The line above is an illustration, not a density plot.

Results reflect the last calculation. Recalculate after editing inputs.

Calculated with TinkerMint · An estimate based on the inputs shown.

What does 95% mean?

If you repeatedly took samples in the same way and constructed an interval each time, about 95% of those intervals would contain the true population mean, when the method’s assumptions hold.

It doesn’t mean that 95% of individual observations lie inside this interval, or that there is a 95% probability that the fixed population mean is inside this particular interval.

Try the exampleFor n = 25, mean = 50, and sample SD = 10, the standard error is 2. With 24 degrees of freedom, t ≈ 2.0639, so the interval is approximately 45.8722 to 54.1278.

Check the assumptions.

  • Observations should come from an appropriate random sample and be independent. A larger sample doesn’t fix selection bias or dependence.
  • The t interval is exact for a normal population. With small samples, strong skew or outliers can make it unreliable; larger samples may help under suitable conditions.
  • Summary statistics can’t reveal outliers, skew, clusters, or a flawed sampling process. Check the original data and study design.

This is an educational calculator, not a substitute for a study-specific statistical analysis.

Under the hood.

Which formula does this use?

Standard error = s ÷ √n. The interval is x̄ ± t(0.975, n − 1) × s ÷ √n, where s is the sample standard deviation and x̄ is the sample mean. The two-sided 95% interval leaves 2.5% in each tail of the Student-t distribution.

Why Student-t instead of 1.96?

When the population standard deviation is unknown and is estimated from the sample, Student-t accounts for that extra uncertainty. Its critical value is larger for small samples and approaches the standard-normal value as the degrees of freedom grow.

What if my sample standard deviation is zero?

The formula returns a zero-width interval. That is an arithmetic result, not proof that the population has no variability. Check measurement resolution, data quality, and whether the model is appropriate.

Method references: NIST: confidence limits for the mean and NIST: Student-t critical values.