MultiConvers

Confidence Interval Calculator

Build a two-sided z-interval for a population mean from a sample mean, a standard deviation, n and a confidence level. The result is a range for that mean, not a forecast for the next person.

Updated

How to use

  1. Pick 80, 85, 90, 95 or 99 percent. The page uses the matching two-sided normal z (about 1.96 at 95%).
  2. Enter the sample mean x̄, a standard deviation s (or a known σ), and the sample size n.
  3. Read the interval, then the margin of error and the standard error s/√n in the details.
  4. Do not treat the bounds as “the next customer will fall here.” That would be a prediction interval, which this tool does not compute.

How it works

A confidence interval for a mean is a statement about a parameter: after you have already collected a sample, where might the population mean sit? The formula is x̄ ± z × (s / √n). s / √n is the standard error — the usual scale for how much a sample mean wobbles. z is a two-sided standard-normal quantile (1.96 at 95%, 1.645 at 90%, 2.576 at 99%). With x̄ = 50, s = 10 and n = 25, the standard error is 2 and the 95% interval is 46.08 to 53.92. n must be an integer of at least 2.

That interval is not a prediction for one new observation. A person drawn tomorrow is scattered on the scale of s, not s/√n. Saying “we are 95% sure this next commuter takes between 46 and 54 minutes” confuses the mean of a process with a single draw. If you need a band for one future value, you want a prediction interval (wider, and not offered here). This page only locates the mean.

The 95% label is also easy to over-read. It is not “there is a 95% chance that this particular interval contains the true mean.” After the sample is in, the mean either sits inside these bounds or it does not. The 95% refers to the method: in repeated sampling, about 95% of such intervals cover the mean. One printed interval is a single draw from that procedure.

The calculator uses a normal z, not Student’s t. That is a large-sample or known-σ approximation. For small n and an estimated s, a t-interval with n − 1 degrees of freedom would be wider. This page stays with z on purpose and shows the z it used. You choose whether the number in the SD box is a sample s or a known population σ; the formula treats it as the s (or σ) in x̄ ± z × (s / √n).

Planning how large n should be before you collect the sample is the sample-size calculator, and that tool is written for a proportion, not a mean. Turning one raw score into a standard score is the z-score page — a different z entirely.

Formula

x̄ ± z × (s / √n)

Worked examples

  • A 95% interval

    x̄ = 50, s = 10, n = 25 → 46.08 to 53.92

Useful notes

  • For small n and unknown σ, a t-interval is more appropriate. This page stays with z and says so.
  • s can be a sample SD or a known population σ — you choose which number to type.
  • The interval estimates a mean. It does not predict one new observation.

FAQ

What is a confidence interval for a mean?

A range built from the sample so that, in repeated sampling, the stated percentage of such ranges contain the true mean.

Why z and not t?

This tool uses the normal approximation. A t-interval would be wider for small n. The formula is spelled out so you can see the assumption.

What is the margin of error?

The term z × (s / √n) — half the width of the interval.

Does this tell me where one new person will land?

No. It is an interval for the population mean. A single new draw is much more variable. This page does not compute a prediction interval.

Why must n be at least 2?

A standard error needs a sample size in the denominator under a square root, and a single observation cannot support this z-interval. n must be an integer of 2 or more.

Related tools

See all Statistics tools.

Sources