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Confidence Interval Calculator

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A single sample average is a point estimate and almost certainly not the true figure. A confidence interval says how much room to leave around it. Enter your sample and this works out the interval, the margin of error, and the critical value behind both.

Interval around a

Applies the finite population correction when the sample is a large share of it

How to use the confidence interval calculator

  1. 1Choose whether you are estimating a mean or a proportion.
  2. 2For a mean, enter the sample mean, the sample standard deviation and the sample size — or paste the raw values and let them be worked out.
  3. 3For a proportion, enter how many in the sample had the characteristic, and how many were sampled.
  4. 4Pick a confidence level, and add the population size if you sampled a large share of a small population.

Examples

A mean from 40 measurements

Input
Mean 68.4, standard deviation 5.2, n = 40, at 95%
Result
68.4 ± 1.66, giving 66.74 to 70.06. Standard error 0.822, critical t 2.023 on 39 degrees of freedom.

The t distribution is used rather than z, because the population standard deviation was estimated from the sample.

A proportion from a survey

Input
520 of 1,000 respondents said yes, at 95%
Result
52.0% ± 3.10 points, giving 48.9% to 55.1%. Critical z 1.960 — an interval that straddles half, so the survey does not establish a majority.

About the confidence interval calculator

The interval is about the method

The single most common misreading of a confidence interval is treating it as a probability statement about the true value. Under the frequentist framework these intervals come from, the true value is a fixed number rather than a random one, so it makes no sense to say there is a 95% chance it sits in a particular range. What varies between repetitions is the interval, not the target.

The reading that does work is about the recipe: a procedure that captures the truth 95% of the time was applied here. If you want a statement of the form 'given this data, there is a 95% probability the value lies between these bounds', you want a Bayesian credible interval, which is a different calculation resting on different assumptions.

What the interval does not cover

Everything on this page addresses sampling error alone — the variation you would see from drawing a different random sample of the same size from the same population. That is the only source of uncertainty the arithmetic knows about.

It says nothing about a sampling frame that missed part of the population, a question worded so that people answer a different one, respondents who declined in a pattern related to the answer, or a measuring instrument reading consistently high. Those are biases, not noise: they shift the centre of the interval rather than widening it, and no sample size fixes them. A tight interval around a biased estimate is a precise wrong answer.

Frequently asked questions

What does 95% confidence actually mean?
It describes the procedure, not this particular interval. If you repeated the whole sampling exercise many times and built an interval the same way each time, about 95 of every 100 such intervals would contain the true value. The one in front of you either contains it or it does not — there is no 95% probability attached to it, however naturally that reading suggests itself.
Why does it use t for a mean and z for a proportion?
Because a mean uses a standard deviation estimated from the same small sample, and that estimate carries its own uncertainty. The t distribution widens the interval to account for it, by an amount that shrinks as the sample grows — at n = 40 the critical value is 2.02 against z's 1.96. A proportion's spread follows from the proportion itself, so there is no separate quantity being estimated and the normal approximation is used directly.
When is a proportion interval not trustworthy?
When successes or failures are scarce. The normal approximation used here needs at least five of each, and below that it can produce a lower bound under zero or an upper bound over one — nonsense on the face of it. The page warns when you cross that line. A Wilson or Clopper-Pearson interval is the right instrument at the extremes.
What is the finite population correction for?
Ordinary interval formulas assume you are sampling from an effectively infinite population, so sampling 500 people tells you the same amount whether the population is a million or a thousand. It does not: at some point you are measuring most of the population directly. Enter a population size and the correction is applied, narrowing the interval — noticeably once your sample passes about 5% of it.
How do I make the interval narrower?
Collect more data, or accept less confidence. Width scales with one over the square root of the sample size, so halving it takes four times as many observations — the reason precision gets expensive quickly. Dropping from 95% to 90% narrows the interval too, but it is a real trade: you are stating a tighter range and being wrong more often.

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