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Confidence Interval Calculator

Find the confidence interval around a sample mean at 90%, 95%, or 99%.

How it is calculated

The interval is built around the sample mean using the normal (z) approximation:

SE = s / √n, margin = z × SE, interval = x̄ ± margin

x̄ is the sample mean, s the sample standard deviation and n the number of observations (at least 1). The critical value z is fixed per confidence level, and the form offers five: 1.282 for 80 %, 1.440 for 85 %, 1.645 for 90 %, 1.960 for 95 % (the default) and 2.576 for 99 %. No other level, such as 97 %, can be chosen. The tool shows the margin, the lower and upper bounds and the range, with two decimals. The standard deviation cannot be negative.

Worked example

Mean 72, standard deviation 12, sample of 36, 95 % confidence.

  1. Standard error = 12 / √36 = 12 / 6 = 2
  2. Margin = 1.960 × 2 = 3.92
  3. Interval = 72 ± 3.92 = 68.08 – 75.92

At 99 % confidence (z = 2.576) the same data give a wider interval of 66.85 – 77.15.

Try this example →

Common mistakes

  • Entering the standard error where the standard deviation belongs: the tool already divides by √n, so this shrinks the interval twice.
  • Reading the 95 % interval as a 95 % chance that each individual value lies inside it: it is about the mean, not about single observations.
  • Using the z value with a small sample (under about 30) and unknown population SD: a t-based interval would be wider than the one shown.
  • Looking for a confidence level the form does not offer, such as 97 %: only 80 %, 85 %, 90 %, 95 % and 99 % can be selected, so pick the nearest one and say which you used.

About this calculator

This calculator finds the confidence interval for a sample mean using CI = x̄ ± z·(s/√n), where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and z is the critical value for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%). The margin of error shrinks as the sample grows larger or the confidence level drops.

Researchers, market analysts, and statistics students use it to report how much uncertainty surrounds a sample estimate — for example, an average customer satisfaction score or a mean test result.

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Comments (1)

  • Editorial team

    Because the margin shrinks with the square root of n, quadrupling the sample only halves it: with a standard deviation of 10, a mean of 50 and 95 percent confidence, n = 25 gives plus or minus 3.92 and n = 100 gives 1.96. Keep in mind this tool uses fixed z values, so for very small samples a t-based interval would be wider. What sample sizes do you usually work with?

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