Build an interval around a sample mean or a sample proportion, save the exact scenario as a link, and print the result.
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The confidence level dropdown above is not decorative; it is wired directly to a z critical value from the standard normal curve. These are the exact multipliers in the code, not rounded approximations:
Source: standard normal distribution critical values, the same figures published in NIST's Engineering Statistics Handbook.
Pick Mean when you have a numeric average, like a test score or a measurement. Pick Proportion when you have a percentage or a yes/no rate. The default scenario above is a mean of 100 with an SD of 15 across 50 observations at 95% confidence: standard error is 15 divided by the square root of 50, which is 2.121, and the margin of error is 1.96 times that, or 4.158. The interval runs from 95.842 to 104.158. Change the type toggle to Proportion and the same panel switches to p and n instead, using the proportion version of the standard error formula.
This tool uses the Wald interval (z times standard error) for both cases. It does not implement the Wilson score interval, which some statisticians prefer for small samples or proportions near 0% or 100%; see the FAQ below for what that means in practice.
Related: the margin of error calculator isolates just the plus-or-minus figure, and the sample size calculator works the formula backward to tell you how big a sample you would need for a target margin.
The two options compute standard error differently. For a mean, standard error is your sample standard deviation divided by the square root of n. For a proportion, it is the square root of p times one minus p, divided by n. Switch the Type field above and the calculator swaps formulas automatically.
Z-based intervals assume a large-ish sample, informally n of 30 or more for the mean case. Below that, the t-distribution has fatter tails and produces a wider, more honest interval. This tool always uses z; for a small sample, treat the result as optimistic and check the t-test calculator instead.
Wider is what buys the extra confidence. A 99% interval has to be wide enough to catch the true value 99 times out of 100 repeated samples, so it trades precision for certainty. A narrow interval at high confidence is not achievable just by asking for it; it needs a bigger sample.
Yes, with a small n or a p near the edges. This calculator uses the Wald interval, z times standard error, which is simple but can produce a technically impossible bound in that situation. It does not implement the Wilson score interval, which corrects for that edge case; treat any bound outside 0 to 100% as a sign the sample is too small for this method.
They are the z critical values for a two-tailed 90%, 95%, and 99% confidence level on the standard normal curve, the exact numbers this calculator's confidence level dropdown is wired to.