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Confidence Interval Calculator

Calculate a normal-distribution confidence interval from a sample size, sample mean, standard deviation, and confidence level.

ExampleSample values — edit any field to see your result.

Results update as you type.

Confidence Interval

20.6 ±0.887 (±4.3%)

Estimated result

Interval Range
[19.713 – 21.487]
Margin of Error
0.887
Lower Bound
19.713
Upper Bound
21.487
Critical Z-score
1.9600
Calculation
CI = 20.6 ± 1.9600 × 3.2 / √50
Confidence Interval20.6 ±0.887 (±4.3%)

Calculate the confidence interval around a sample mean. Enter the sample size, sample mean, standard deviation, and confidence level to see the critical Z-score, margin of error, and lower and upper bounds.

Formula

The normal confidence interval is:

CI = x̄ ± Z × s / √n

Here, x̄ is the sample mean, s is the standard deviation, n is the sample size, and Z is the two-sided critical value for the chosen confidence level.

A higher confidence level produces a wider interval. A larger sample size produces a narrower interval, all else equal.

Examples

95% interval for the default sample

With n = 50, mean = 20.6, and standard deviation = 3.2, the 95% critical value is 1.9600. The result is 20.6 ± 0.887, or [19.713, 21.487].

Frequently asked questions

What is a confidence interval?
It is a range around a sample estimate produced by a stated statistical procedure. Across repeated samples, a 95% procedure captures the population parameter about 95% of the time under its assumptions.
How is the margin of error calculated?
The calculator multiplies the critical Z-score by the standard deviation divided by the square root of the sample size.
Does a larger sample make the interval narrower?
Yes. Because standard error divides by the square root of n, increasing sample size reduces the margin of error when the other inputs stay fixed.
When is the Z method appropriate?
It is appropriate when the population standard deviation is known or when a normal approximation is reasonable, commonly with a sufficiently large sample.

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