Confidence Interval Calculator

What the 95% is a probability about

Almost everyone reads “95% confidence interval” as “95% chance the true value is in here”. That is not what it says, and the difference is not pedantic.

In the classical framework the population parameter is a fixed number. It is not random. It is either inside your interval or it is not, and there is no probability attached to that — the probability is 1 or 0 and you do not know which.

What is random is the interval. Take a different sample and you get a different one. The 95% describes the long-run behaviour of the procedure: if you repeated the whole experiment many times, about 95% of the intervals you constructed would contain the true value.

So the correct reading is: “this interval was produced by a method that captures the truth 95% of the time.” The confidence is in the method, not in this particular pair of numbers.

Does the distinction change anything practically? Sometimes. It is why you cannot say a value just outside the interval is “5% likely”, why intervals from a badly-designed study are not rescued by being narrow, and why two overlapping intervals do not tell you the difference is non-significant.

If you genuinely want “the probability the parameter lies in this range”, that is a credible interval and it requires Bayesian machinery and a prior. It is a legitimate thing to want; it is a different calculation.

The n greater than 30 rule, and what it costs

Textbooks routinely say: use t for small samples, switch to z once n passes 30. That rule is a convenience from the era of printed tables, not a statistical result, and this page does not use it.

The reason t exists is that you almost never know the population standard deviation. You estimate it from the same sample, and that estimate carries its own error. Student’s t has heavier tails precisely to account for it, with degrees of freedom n − 1.

That extra uncertainty does not switch off at any particular n. It shrinks smoothly and never reaches zero.

At n = 30 the multiplier is 2.045 against z’s 1.960. Using z there makes the interval 4.35% too narrow — measured, and asserted in this page’s verification suite rather than estimated. At n = 100 the gap is about 1.2%, at n = 1000 about 0.1%.

None of those are catastrophic, and all of them are avoidable for free. A computer does not care whether it looks up t or z, and the only reason to prefer z is that a printed t table ran out of rows.

The one case where z is genuinely correct: when σ is known rather than estimated — a manufacturing process with a long calibration history, or a simulation where you set it. The toggle in the tool is for that, and it is rarer than it looks.

Why the textbook proportion formula is broken

The Wald interval — p̂ ± z√(p̂(1−p̂)/n) — is what nearly every course teaches and nearly every calculator uses. It fails in ways that are easy to demonstrate.

At 0 successes out of 20 it returns [0, 0]. A zero-width interval. It is asserting that the proportion is exactly zero, with total certainty, on the strength of twenty observations. Wilson gives 0 to 0.161, which is the honest answer: you have seen no successes, so the rate is low, but twenty trials cannot rule out one in ten.

At 1 success out of 10 it goes below zero. A negative proportion is not a quantity that exists.

Its coverage is wrong even away from the edges. A nominal 95% Wald interval often captures the true value far less than 95% of the time for small n or extreme p — and the shortfall does not shrink smoothly with n, it oscillates.

The cause is the approximation. Wald assumes p̂ is normally distributed and centres the interval on it. Near 0 or 1 the sampling distribution is strongly skewed and bounded, so a symmetric normal interval is the wrong shape.

Wilson inverts the score test instead: it asks which values of p would not be rejected by the data. That formulation is naturally asymmetric, always lands inside [0, 1], and keeps its coverage at small n. It has been the recommended default in the statistical literature since the 1990s and it is what this page returns.

Both are shown, because the gap between them at the edges is more convincing than any description of it.

What actually controls the width

Three things, with very different costs.

Sample size, through √n. This is the one you can buy, and the square root is the catch: halving the width needs four times the data, and a tenth of the width needs a hundred times. That is why survey costs escalate so sharply past a certain precision, and why the jump from 1,000 to 2,000 respondents buys much less than the jump from 250 to 500.

The confidence level. Going from 95% to 99% widens the interval by about 30% on the same data. There is no free certainty: a claim you are more confident about is necessarily a vaguer one. A 100% interval is the entire range of possible values and says nothing.

The variability in the data. Mostly outside your control, though better measurement or a more homogeneous population reduces it.

Worth being explicit about what the width does not include: it covers sampling error only. A biased sample produces a narrow interval around the wrong number, and no amount of extra data fixes that — it just tightens the interval around the same wrong number. Non-response, a bad sampling frame and a leading question are all invisible to this arithmetic.

Comparing two intervals, which is where it goes wrong

A tempting shortcut: two confidence intervals overlap, so the difference is not significant. It is not reliable in either direction.

Overlapping intervals can still be a significant difference. Two 95% intervals can overlap by a noticeable margin while a proper two-sample test returns p below 0.05. Roughly, overlap up to about a quarter of the average interval length is still compatible with significance at 0.05.

Non-overlapping intervals do imply significance at that level, so the test is conservative in one direction only. If they clearly do not touch, the difference is real at 0.05.

The reason is that the comparison has its own standard error. The difference between two independent estimates has variance equal to the sum of their variances, so its standard error is √(SE₁² + SE₂²) — smaller than SE₁ + SE₂, which is effectively what eyeballing the overlap uses.

The right move is to build an interval for the difference itself and see whether it contains zero. That answers the actual question, and it also gives you the size of the difference, which comparing two separate intervals never does.

The interval and the test are the same statement

A confidence interval and a two-tailed hypothesis test are not two methods. They are one, presented differently.

A value falls outside the 95% interval exactly when a two-tailed test of that value rejects at 0.05. Not approximately: exactly, when both are built from the same standard error and the same distribution. This page’s suite asserts it across a grid of sample sizes and hypothesised values.

So an interval carries all the information a p-value carries, plus two things a p-value does not: the size of the estimate, and how precisely it was pinned down.

That is the whole argument for reporting intervals in preference to bare p-values. “p = 0.03” tells you a null value was rejected. “3.2 points, 95% CI 0.4 to 6.0” tells you that too — zero is outside — and adds that the effect might plausibly be as small as 0.4, which may well be too small to act on.

It also makes a non-significant result readable. A wide interval spanning zero says you learned very little. A narrow interval tightly around zero says you have real evidence the effect is small. Both give p above 0.05 and they are completely different findings.

What has to be true for any of this to hold

The arithmetic always produces an interval. Whether it means anything depends on conditions the calculator cannot check.

The sample has to be random. This is the assumption that fails most often and matters most. An interval computed from a convenience sample, a self-selected poll or whoever happened to reply describes that group and nothing wider, however impressive the arithmetic looks.

Observations have to be independent. Repeated measurements on the same people, students within the same class, or readings from one instrument over time are correlated, and treating them as independent makes the interval far too narrow.

For a mean, either normality or enough n. The central limit theorem does most of the work here, and for mildly skewed data a few dozen observations is usually plenty. For strongly skewed data — incomes, waiting times — it can take hundreds, and a bootstrap interval or one built on the median is a better instrument.

For a proportion, no minimum count with Wilson. The familiar “at least 10 successes and 10 failures” rule is a patch for Wald’s failures. Wilson does not need it, which is a large part of why it is the default here.

Sources and methodology

The methodological references behind the two defaults on this page.

• Brown, Cai & DasGupta, “Interval Estimation for a Binomial Proportion”, Statistical Science 16(2), 2001 — the paper that documents Wald’s coverage failures and recommends Wilson
• Wilson, “Probable Inference, the Law of Succession, and Statistical Inference”, JASA 22, 1927 — the original score interval
• NIST/SEMATECH e-Handbook — confidence limits for the mean, and the t multiplier
• p-value Calculator — the same statement expressed as a tail probability
• Margin of Error Calculator — the half-width on its own, and what it excludes
• How we calculate — methodology

Method. The mean interval uses Student-t with n − 1 degrees of freedom whenever the standard deviation was estimated, at every sample size, and switches to z only when you assert that σ is known. The proportion interval returns Wilson and Wald together and flags Wald when it produces a zero-width or out-of-range result, so the comparison is on the page rather than in a footnote. The suite asserts the duality directly — that a hypothesised value lies inside the 95% interval exactly when a two-tailed test fails to reject it at 0.05 — across a grid of sample sizes and null values, which is a check neither the interval nor the test could pass alone if either were wrong. It also asserts that quadrupling n halves the margin of error, that a higher confidence level always widens the interval, and that Wilson never leaves [0, 1] for any count. That engine is verified on every change against 155 assertions. The count and the per-case breakdown are published on the formula verification page.

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