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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: 1.282 for 80 %, 1.440 for 85 %, 1.645 for 90 %, 1.960 for 95 % and 2.576 for 99 %. Any other confidence value, such as 97, silently falls back to 95 % (z = 1.960). 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.
  • Typing a confidence level that is not in the list (such as 97) and assuming it was applied: the tool then uses 95 %.

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. Enter the sample mean, standard deviation, and sample size to see the interval and margin of error at 90%, 95%, or 99% confidence.

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