Confidence Interval Calculator

Written byOleh Soloninupdated 27 September 2026

Sample mean

Standard deviation

Sample size

Confidence level, %

Where the deviation comes from

A deviation worked out from the same sample calls for the t-distribution at any n; z only when the whole population's deviation is known in advance.

Result

Interval94.3989 … 105.6011
Lower bound94.398908
Upper bound105.601092
Margin of error5.601092
Standard error2.738613
Critical value2.0452
Distributiont with 29 degrees of freedom

How precisely a sample pins down the population

Interval bounds from the sample mean, standard deviation and sample size, plus the margin of error and the standard error.

A sample mean almost never equals the true population mean. A confidence interval is built so that, over many repeated samples, a stated share of such intervals — 95 percent, say — covers the true mean. It gives no probability for one particular interval: that one either covered the mean or it did not.

Interval width falls with the square root of sample size. Halving the width therefore needs four times the observations — the fundamental constraint that makes large surveys expensive.

How to use

Enter the sample mean, the standard deviation and the sample size.
Choose the confidence level: 90, 95 or 99 percent.
Read the interval bounds, the margin of error and the standard error of the mean.

Why the interval narrows only as a square root

The standard error is the deviation divided by the square root of the sample size. With a deviation of 15 and 30 observations that is 2.74. The deviation was estimated from the sample, so the multiplier is t with 29 degrees of freedom, 2.045, and the 95% margin of error is 5.60 — an interval from 94.40 to 105.60. Had the population deviation been known, z = 1.96 would have given 5.37.

The square root is why precision is expensive. Halving the interval takes four times the sample: 100 observations give half the error of 25, not a quarter of it, and that arithmetic sets the price of every serious survey.

Mean 100, sample standard deviation 15, t-distribution. Worked out with this calculator.
Sample size	Standard error	95% margin	Interval
30	2.74	5.60	94.40 … 105.60
100	1.50	2.98	97.02 … 102.98
400	0.75	1.47	98.53 … 101.47

What this calculator shows

The lower and upper bounds of the interval
The margin of error and the standard error of the mean
The critical t or z value for your confidence level

What to keep in mind

When the deviation comes from the sample itself, the interval uses Student's t-distribution with n − 1 degrees of freedom at any n; z is used only when the population deviation is known in advance.
An interval for a mean leans on normality: with a small sample the data themselves should be close to normal, while with a large one the mean is close to normal regardless, provided the observations are independent and their variance is finite. Heavy tails or dependent data need more care.

FAQs

What does 95 percent confidence mean?

That repeating the experiment many times, 95 percent of such intervals would contain the true mean. It describes the procedure, not the probability for one particular interval.

How do I narrow the interval?

Increase the sample — width falls as the square root of n. Lowering the confidence level also narrows it, at the cost of being wrong more often.

When should I use the t-distribution?

Whenever the deviation is estimated from the sample itself, at any sample size — which is what the calculator does by default. z fits only when the population deviation is known in advance. The t interval is wider: at n = 30 the multiplier is 2.045 rather than 1.96.

How does this relate to a p-value?

Through the matching two-sided test on the same model and assumptions: if the 95 percent interval excludes the null value, that test gives p below 0.05. They are two views of the same result.

What is a 95 percent confidence interval?

An interval built so that, over many repetitions of the experiment, 95 percent of such intervals contain the true mean. For a mean of 100, a sample deviation of 15 and 30 observations it runs from 94.40 to 105.60.

How do I halve the interval?

Quadruple the sample. The error falls as the square root of the size, so moving from 30 observations to 120 cuts the margin from 5.60 to 2.71 — slightly more than half, because the t multiplier falls too.

Can I say the true mean lies inside with 95 percent probability?

Strictly, no. The true mean is a fixed quantity; it is the interval that is random. The probability belongs to the procedure, not to any single pair of numbers it produced.

Worked example

A sample of thirty

Input: Mean 100, deviation 15, n = 30, 95 percent confidence

Output: Interval 94.40 to 105.60

Note: The margin is 5.60, on t with 29 degrees of freedom. A hundred and twenty observations would bring it to 2.71 — a little under half, because the t multiplier shrinks as the sample grows.

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Confidence Interval Calculator

Result

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