Confidence Interval Calculator

What It Does

This calculator turns a sample's own figures — its mean x̄, its standard deviation s and its size n — into the interval the population mean is likely to sit in, at one of three fixed confidence levels. It computes the standard error s / √n, multiplies it by the level's critical value z* to get the margin of error, and prints x̄ ± margin with both ends shown. The model is the normal one, with the sample's s standing in for the unknown σ, and the family is scale-free: whatever unit the sample is in, the interval comes back in that same unit with nothing converted.

Real-World Usage

An interval is how one measurement becomes a claim. A poll reports a share plus or minus a margin; a laboratory repeats a measurement and states a mean with a range; a factory samples 30 units and quotes the batch's average strength to within so many units. Reading it correctly is what the margin is for: 100 ± 5 at 95 % is a statement about the method, so two overlapping intervals are not proof that two populations differ, and a margin wider than the effect you are looking for means the sample cannot settle the question yet.

Local Context

The interval carries no unit and no country's convention: x̄ and s are in whatever the reader measured, and the arithmetic is identical on every page this site serves. Only the formatting follows the page, where a French catalogue writes 94,6324 and an English one 94.6324 for the same bound. The levels are deliberately not regional either — the z* values are the standard-normal quantiles published tables use everywhere, so an interval quoted here can be compared with one from a report written in another country.

Does a 95 % interval mean there is a 95 % chance the true mean is inside it?

No, and the difference is worth holding on to. The population mean is a fixed number — it is not redrawn per sample — so one finished interval either contains it or does not, and no probability can be attached to it afterwards. What the 95 % describes is the method: if you drew many samples and built an interval this way from each, about 95 of every 100 intervals would catch the true mean. A higher level buys more of that coverage, not more certainty about this one sample.

What is the margin of error, and how do I make it smaller?

The margin is z* × s / √n — the level's critical value times the standard error — and it is the half-width added and subtracted around the sample mean. Three things move it: a smaller s (a more consistent sample), a larger n, and a lower confidence level. Only n can be raised by collecting more data, and it sits under a square root, so the margin halves only when the sample is four times as large: from 100 readings to 400 halves it, while 100 to 200 cuts it by about 29 %.

Why can I only choose 90, 95 or 99 % — where is 92 %?

The three fixed levels each come with a published standard-normal quantile z* — 1.644854, 1.959964 and 2.575829 — that you can look up in any table and re-derive for yourself. An arbitrary level would need a z* most readers cannot check, and the page would be asserting a figure it does not show the source of. These three are also the levels published tables and reports actually use, which keeps an interval from this page comparable with one quoted elsewhere.

Why does it refuse a spread of zero, or a sample of one?

A standard deviation of zero says every observation is identical, which would put a margin of error of zero on the page — an interval claiming the population mean is known exactly, the one answer this tool must never print. A sample of one has no spread to measure at all, so n has to be at least two. Both refusals leave the figures at zero and mark the field instead of returning an interval that looks usable. The page also uses the normal-approximation interval, with the sample's own s standing in for σ, which is the reading to trust when the sample is reasonably large.