How to Read a Confidence Interval (and What 95% Means)

The structure: estimate plus or minus a margin

Nearly every confidence interval has the same shape. You take your estimate and add and subtract a margin of error, which is a critical value multiplied by the standard error.

interval = estimate ± critical value × standard errorFor a mean, the standard error is the standard deviation divided by √n.

The standard error shrinks as the sample grows, but only with the square root of n. Four times the data makes the interval half as wide, not a quarter as wide.

Worked example: the mean

A sample of 100 measurements has a mean of 120 and a standard deviation of 15. The standard error is 15 ÷ √100 = 1.5. For 95% confidence with a large sample the critical value is 1.96, so the margin of error is 1.96 × 1.5 = 2.94.

The 95% confidence interval is 120 ± 2.94, or 117.06 to 122.94.

With a small sample, the standard deviation is itself uncertain, so a t critical value is used instead of 1.96. For a sample of 9 with a mean of 50 and a standard deviation of 6, the standard error is 6 ÷ 3 = 2. The t value for 8 degrees of freedom at 95% is 2.306, giving 50 ± 4.61, or 45.39 to 54.61. Using 1.96 here would give 46.08 to 53.92, which is too narrow.

Look up critical values with the critical value calculator, or get the whole interval from the confidence interval calculator.

Worked example: a proportion

In a survey of 100 people, 56 say yes, so the sample proportion is 0.56. The standard error is √(0.56 × 0.44 ÷ 100) = √0.002464 ≈ 0.0496.

At 95% confidence the margin is 1.96 × 0.0496 ≈ 0.0973, so the interval runs from 0.463 to 0.657, or roughly 46.3% to 65.7%.

That range includes 50%, so this survey cannot say whether yes is the majority view. A reader seeing only "56% said yes" would miss this entirely, which is the practical value of reporting the interval.

What the 95% actually means

The 95% describes the method, not one particular interval. If you repeated the whole study many times and built an interval each time, about 95% of those intervals would contain the true value. Any single interval either contains it or does not, and you do not know which.

So the common summary "there is a 95% probability the true value is in this range" is a loose paraphrase rather than a strict statement. A safer wording is: "we are 95% confident, using this method, that the true value lies between 117 and 123".

An interval for a mean is also not a range for individual values. With a standard deviation of 15, individual measurements around a mean of 120 spread much wider, roughly 120 ± 29, than the 117 to 123 range for the mean itself. The interval describes where the average plausibly lies, not where most observations fall.

• Higher confidence means a wider interval. For large samples, 90% uses 1.645, 95% uses 1.96 and 99% uses 2.576.
• A larger sample or less variable data gives a narrower interval.
• The interval only reflects random sampling error. Bias, such as a non-random sample, is not covered.

Using intervals to compare results

When two groups' intervals do not overlap at all, the difference is usually clear. Overlap alone does not prove there is no difference, however, because the proper comparison is an interval for the difference itself.

An interval for a difference that excludes zero corresponds to a two-sided test with p below 0.05 at the same level, so intervals and p-values carry related information. The interval has the advantage of showing the size of the effect, which a p-value alone does not.