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Confidence Interval Calculator for a Mean

This free calculator finds the confidence interval for a mean from raw data or from summary numbers. It uses Student’s t distribution, so it is accurate for small samples as well as large ones.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

A confidence interval turns a sample average into a range for the real average. Researchers, quality teams and students use it to show how precise an estimate is.

Paste your numbers, or give the mean, standard deviation and sample size, and choose a confidence level.

How to Use This Tool

  1. Choose list or summary input.
  2. Enter the numbers or the summary values.
  3. Choose the confidence level.
  4. Press Calculate.

How the Calculation Works

From a list, the tool finds the mean and the sample standard deviation. The standard error is s ÷ √n. The critical value t comes from the t distribution with n − 1 degrees of freedom, found numerically. The margin is t times the standard error, and the interval is the mean plus and minus the margin.

Worked Example

For the data 10, 12, 23, 23, 16, 23, 21, 16, the mean is 18 and s is 5.237. With 95% confidence and t = 2.365, the margin is 4.38 and the interval is 13.62 to 22.38.

Common Uses

  • Reporting experiment results.
  • Quality control.
  • Student projects.
  • Business metrics.

Tips and Common Mistakes

  • You need at least two numbers.
  • Check the data for typing errors first.
  • Intervals assume the data is roughly bell-shaped for small samples.

Next, you may need the margin of error calculator, sample size calculator, normal distribution calculator, percentile calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

A range that is likely to contain the true mean of the population. A 95% interval means that if you repeated the study many times, about 95% of such intervals would contain the true mean.
When the population standard deviation is unknown, which is usual, the t distribution gives wider, more honest intervals for small samples.
mean ± t × s ÷ √n, where t is the critical value for n − 1 degrees of freedom.
The interval gets narrower as n grows. Very small samples give very wide intervals.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.