Confidence Interval Calculator

A confidence interval is an estimate with its uncertainty attached

A sample mean is a guess at the population mean, and a guess without a range is a guess you cannot check. The confidence interval is that range: the estimate, plus and minus a margin of error. The margin is a critical value times the standard error, and each of those two factors answers a different question. The standard error is how much the estimate would move from one sample to the next; the critical value is how many of those standard errors the interval reaches to hit the confidence level you asked for.

For a mean the page follows the standard recipe. From raw data it computes the mean and the sample standard deviation exactly, the standard deviation kept as a radical; from a summary it takes your numbers as given. With the standard deviation estimated from the sample, the critical value comes from the t curve with n minus 1 degrees of freedom, which is the honest choice and the default here; the z version is a switch away for a known sigma or a large n. For a proportion the page gives two intervals: the Wald interval, which is the one in most textbooks, and the Wilson score interval, which is the one to report, because Wald misbehaves when the proportion is near 0 or 1 or the sample is small.

The interval is drawn on a number line under the answer, with the estimate marked, because a picture of an interval says something a pair of numbers does not: how wide it is relative to where it sits. An interval for a mean that runs from 4.9 to 5.1 and one that runs from 2 to 8 are both intervals; only one of them is worth reporting to three decimals.

What to know about confidence intervals

• "95% confident" is a statement about the method, not this interval. Build intervals this way many times and about 95% of them will contain the true value; any single one either does or does not.
• Wider level, wider interval. A 99% interval reaches about 2.6 standard errors instead of 2, so it is a third wider than a 95% one for the same data.
• The width shrinks with the square root of n. Four times the data halves the margin of error; a hundred times the data cuts it to a tenth.
• Use t when sigma is estimated from the sample. With n = 10 the t critical value at 95% is 2.26, not 1.96, and pretending otherwise makes the interval too narrow.
• For a proportion, report Wilson. The Wald interval can run below 0 or above 1 and is unreliable with fewer than about 10 successes or failures; Wilson stays inside the possible range and keeps its promised coverage.

How to build a confidence interval

1. 1

   Compute the estimate

   The sample mean, or the sample proportion. Keep it exact if you can; the page does.

2. 2

   Compute the standard error

   For a mean, the standard deviation over root n. For a proportion, the root of p(1 − p) over n.

3. 3

   Look up the critical value

   For a mean with estimated sigma, from the t curve with n minus 1 degrees of freedom at your level. For a known sigma or a proportion, from the normal curve: 1.645, 1.960 or 2.576 for 90, 95 or 99%.

4. 4

   Multiply to get the margin, then add and subtract

   Margin of error equals critical value times standard error. The interval is the estimate minus the margin to the estimate plus the margin.

Critical values for a confidence interval

How many standard errors the interval reaches either side of the estimate.

Worked examples

x̄ = 50, s = 10, n = 25, 95%

2, 4, 4, 4, 5, 5, 7, 9

40 successes in 100 trials

Confidence interval mistakes

• Saying there is a 95% chance the true mean is in this interval. The probability statement is about the procedure across repeated samples, not about one fixed interval.
• Using z with a small sample. Under about n = 30 with sigma estimated from the data, the t critical value is noticeably larger and the z interval is too narrow.
• Reporting a Wald interval for a rare proportion. With 2 successes in 20 the Wald interval dips below zero; the Wilson interval does not, and is the one to write down.
• Dividing by n instead of root n for the standard error. The margin would be far too small.
• Reading a narrow interval as a true one. Width is precision, not accuracy; a biased sample gives a narrow interval around the wrong number.
• Comparing two overlapping intervals and concluding no difference. Two 95% intervals can overlap while the difference between the means is significant; test the difference directly.

Confidence interval FAQ