Confidence Intervals, Explained

After reading this you can build and read a confidence interval for a mean, a proportion, or a standard deviation, pick the right method, and avoid the errors that turn a correct interval into a misleading one.

What a confidence interval tells you

A confidence interval is a range built from your data that is designed to cover the unknown population value at a stated rate over many repeated samples. The stated rate is the confidence level, usually 95%.

Here is the hook. Suppose you measure the fill volume of 25 bottles and get a sample mean of 50 ml with a sample standard deviation of 10 ml. The 95% t interval for the true mean is 50 ± 4.128, that is 45.87 to 54.13 ml. The single best guess is 50, but the data are consistent with any true mean between about 46 and 54.

The interval width, 8.256 ml here, is the honest statement of how much the mean could move if you sampled again. A point estimate alone hides that uncertainty.

When to use each mode

This calculator covers four situations. Match the mode to your data.

Mean, sample sd (t interval)

You have continuous data, you estimated the standard deviation from the sample, and you want the mean. This is the common case. Use it whenever you computed s from your own numbers.

Mean, known σ (z interval)

You want the mean and the population standard deviation is genuinely known ahead of time, for example from a long production history. This is rare in practice.

Proportion

Your data are yes or no, and you want the fraction of yeses. You supply the number of successes and the sample size.

Standard deviation (χ² interval)

You want a range for the spread itself, not the mean. This assumes the data are normally distributed, and it is sensitive to that assumption.

Do not use a confidence interval to describe where individual observations fall. That is a prediction interval or a tolerance interval, and it is much wider. A 95% confidence interval for the mean answers a question about the mean, not about a single new bottle.

The formulas and the intuition

Every interval here has the same shape: estimate, plus or minus a critical value times a standard error.

This is the t interval for a mean. \bar{x} is the sample mean, s is the sample standard deviation, n is the sample size, and t_{n-1,\, 1-\alpha/2} is the critical value from the t distribution with n-1 degrees of freedom. The term s/\sqrt{n} is the standard error of the mean: it shrinks as n grows, which is why bigger samples give tighter intervals.

The z interval replaces t with a standard normal quantile z_{1-\alpha/2} and uses the known \sigma instead of s. For 95% confidence, z = 1.960. The t value is always a little larger because estimating s adds uncertainty. At n=25 the t value is 2.064.

This is the Wald interval for a proportion, where \hat{p} is successes divided by n. It is simple but poor when \hat{p} is near 0 or 1, or when n is small. The Wilson interval corrects that by centering the interval differently:

The Wilson interval pulls the center toward 0.5 and never runs below 0 or above 1. Prefer it in almost every proportion problem.

Worked example with the demo data

A worked proportion, both ways

Reading and interpreting the result

Say the correct sentence. For the demo mean interval: "with 95% confidence, the true mean fill volume is between 45.87 and 54.13 ml." That describes the procedure's long-run coverage.

Avoid the wrong sentence: "there is a 95% probability the mean is in this interval." Once the interval is computed, the mean is either in it or not; the 95% describes the method across many samples, not this one number.

Two practical readings follow from the width. First, if the interval is too wide to be useful, you need more data; recall that halving the margin takes four times the sample size. Second, if the interval excludes a value of interest, say a target of 60 ml, that is evidence against the mean being 60. The demo interval 45.87 to 54.13 excludes 60, so 60 is implausible at the 5% level.

Common mistakes

These errors are frequent and each one changes the numbers.

• Using z when you estimated the standard deviation. At n=25 the z margin is 3.920 but the correct t margin is 4.128. The z interval is too narrow, so it overstates precision. Use z only for a truly known \sigma.
• Wald near the boundary. As shown above, Wald gave a negative lower bound. Prefer Wilson for proportions.
• Confusing the standard deviation with the standard error. The sd is 10; the standard error is 2. The interval uses the standard error. Dividing by \sqrt{n} is the step people skip.
• Treating the χ² interval as symmetric. The interval for a standard deviation is skewed. For n=25 and s=10, the 95% interval runs about 7.81 to 13.91, wider on the high side. Do not write it as s \pm something.
• Assuming normality without checking. The t interval is robust for large n, but the χ² interval for the sd is not; heavy tails or skew break it.

Related tools

To plan a study rather than analyze one, use the Sample Size Calculator to find the n that hits a target margin of error. For designs framed around detecting a difference, the Power and Sample Size for t-Tests calculator gives the n for a chosen power.

When you report a difference between groups, pair the interval with an effect size from the Cohen's d Effect Size Calculator, and convert between measures with the Effect Size Converter.

Related tools

Sample Size Calculator

Required n to estimate a mean or proportion within a margin of error.

Cohen's d Effect Size Calculator

Effect size from group means and SDs, or from a t statistic.

Power & Sample Size for t-Tests

Required n for a target power, or the power of a given design.

Effect Size Converter

Convert between Cohen's d, r, η² and odds ratios.

Binomial Distribution Calculator

Probabilities of k successes in n independent trials.

Chi-Square Test of Independence

Test association between two categorical variables from a contingency table.