Confidence intervals: a complete guide

A confidence interval is a range of plausible values for something you cannot observe directly. You measured a sample; you want to say something about the population. The sample mean is your best single guess, but it is certainly a little wrong. The interval quantifies how wrong it might plausibly be.

Reporting a mean of 24.8 says nothing about precision. Reporting 24.8, 95% CI [22.1, 27.5] says the data are consistent with a population mean anywhere from about 22 to about 27.5 — and are not consistent with 30. That second sentence is a finding. The first is a number.

Why examiners care about it

A p-value answers one narrow question: could this result have arisen by chance if the null were true? A confidence interval answers the question you actually had — how big is the effect, and how sure are we? Journals in medicine, psychology and education increasingly require intervals precisely because a p-value alone hides the difference between a precisely estimated null result and a study that was simply too small to tell.

This is the part that is most often stated incorrectly, including in published work. The confidence is a property of the procedure, not of any single interval you happen to have calculated.

Imagine repeating your study 100 times, each time drawing a fresh sample and computing a 95% interval. About 95 of those intervals would contain the true population value and about 5 would not. Your study produced one of them. You cannot tell which.

Almost every confidence interval has the same three-part shape: an estimate, plus and minus a critical value multiplied by a standard error.

CI = estimate ± (critical value × standard error)

Choosing the critical value

There is a genuine trade-off here and it is not free. A 99% interval is more likely to contain the true value, but it is wider — so it rules out less. Raising your confidence level buys certainty with precision. Use 95% unless your field or your registered protocol says otherwise.

A study measures wellbeing scores in 50 participants. The sample mean is 24.8 and the sample standard deviation is 9.6. We want a 95% confidence interval for the population mean.

Step 1 — the standard error

SE = s / √n = 9.6 / √50 = 9.6 / 7.071 = 1.358

The standard error is the standard deviation of the sample mean across hypothetical repeated samples. It is much smaller than the SD of the raw scores, because averaging cancels out individual variation.

Step 2 — the critical value

We estimated the SD from the sample, so we use t with df = n − 1 = 49. For a 95% interval, t = 2.010.

Step 3 — the margin of error

margin = t × SE = 2.010 × 1.358 = 2.73

Step 4 — the interval

CI = 24.8 ± 2.73 = [22.07, 27.53]

Proportions need different arithmetic because the standard error depends on the proportion itself. Suppose 38 of 120 respondents agreed with a statement, so p̂ = 38/120 = 0.317.

The standard approach (Wald)

SE = √(p̂(1 − p̂) / n) = √(0.317 × 0.683 / 120) = √0.001804 = 0.0425

CI = 0.317 ± 1.96 × 0.0425 = [0.234, 0.400] — that is 23.4% to 40.0%.

When to use Wilson instead

Three things, and only one of them is usually under your control.

The square-root relationship is the one that surprises people. Halving the width of your interval requires quadrupling the sample, not doubling it. This is why post-hoc requests to “just collect a few more” rarely rescue an imprecise study, and why the sample size should be settled before data collection begins.

Both come from the same arithmetic, so they never contradict each other: if a 95% interval excludes the null value, the two-tailed p is below .05. But they carry very different amounts of information.

Rows two and four both produce a non-significant p-value, and reporting them the same way — “no significant difference” — is actively misleading. The interval separates them immediately: one study found a possibly large effect it could not pin down, the other found nothing much. Only the interval tells you which.

Most research questions are about a comparison, not a single group. “Is the intervention better than the control?” is a question about a difference, and the interval you should report is the interval for that difference — not one interval per group.

Worked example: the difference between two means

An intervention group (n = 40) scores 28.4 (SD = 8.9). A control group (n = 38) scores 24.2 (SD = 9.4). The observed difference is 4.2 points.

First the pooled standard deviation, weighting each group by its degrees of freedom:

sₓ = √[((39 × 8.9²) + (37 × 9.4²)) / 76] = √[(3089.2 + 3269.3) / 76] = √83.7 = 9.15

Then the standard error of the difference:

SE = sₓ × √(1/n₁ + 1/n₂) = 9.15 × √(1/40 + 1/38) = 9.15 × 0.2265 = 2.073

With df = 76, t = 1.992, so the margin is 1.992 × 2.073 = 4.13:

95% CI for the difference = 4.2 ± 4.13 = [0.07, 8.33]

Why two separate intervals mislead

Had you instead reported a 95% interval for each group — [25.6, 31.2] and [21.1, 27.3] — you would have seen them overlap and might have concluded there was no difference. That conclusion would be wrong. Overlap between separate intervals is not a significance test, and the arithmetic above shows exactly why: the standard error of a difference is not the sum of the two individual standard errors.

Intervals around effect sizes

An effect size without an interval is as incomplete as a mean without one. Cohen's d of 0.62 sounds like a medium effect; d = 0.62, 95% CI [0.19, 1.05] tells you the data are consistent with anything from a small to a large effect. That interval is what stops a single study being over-read, and it is what meta-analyses eventually pool.

1. Saying there is a 95% probability the mean is in the interval

The population mean is a fixed number. Once the interval is calculated, it either contains it or does not. Write “we are 95% confident”.

2. Treating overlap between two intervals as a significance test

Two 95% intervals can overlap while the difference between the groups is still significant. If you want to test a difference, compute an interval for the difference, not two separate intervals and a visual comparison.

3. Using z when you estimated the SD

With n = 20, using 1.96 instead of t = 2.093 makes the interval about 7% too narrow. Use t unless the population SD is genuinely known.

4. Reporting the interval but interpreting only the p-value

Putting the CI in the table and then writing “the difference was significant” wastes it. Interpret the range: what does the lower bound mean in practice? What does the upper bound rule out?

5. Using Wald intervals for small or extreme proportions

It produces impossible bounds and under-covers. Use Wilson.

6. Computing intervals on skewed data without checking

The formula assumes the sampling distribution of the mean is roughly normal. With n above about 30 the central limit theorem usually handles this, but with small samples and severe skew, bootstrap the interval instead.

APA 7 requires square brackets, a comma between bounds, and the confidence level stated on first use.