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Confidence interval calculator

Estimate an interval around a sample mean when the population standard deviation is known.

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Give your estimate some context.

This is a z-interval. It is not a substitute for a t-interval when only the sample standard deviation is known.

Sample size (n)

Independent observations; at least 2.

Sample mean

Zero and negative means are valid.

Known population standard deviation (σ)

A positive value, not the sample standard deviation.

Confidence level

Population assumption

Small samples require a justified normal-population assumption.

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BEHIND THE RESULT

Know what goes into it.

This estimates a population mean from independent observations, using a known population standard deviation. For a non-normal population, the large-sample option requires at least 30 observations; strongly skewed data can require more. If σ is unknown, use an appropriate t-interval instead.

NIST on known-sigma confidence intervals

THE FORMULA

Confidence interval = sample mean ± z × σ / √n

A quick example

With n = 100, mean = 50, known σ = 20, and 95% confidence, the interval is approximately 46.08 to 53.92.

Keep in mind

The population standard deviation is genuinely known.
Observations are independent and representative of the target population.
The sampling distribution of the mean is approximately normal.

Confidence interval, margin of error, and standard deviation

The confidence interval gives lower and upper bounds around the sample mean. The margin of error is the distance from that mean to either bound for this symmetric method. Increasing the confidence level widens the interval when other inputs stay the same.

This calculator uses a z interval with a known population standard deviation. If you only have a sample standard deviation, a t interval is generally the appropriate mean-interval method. Having 30 observations does not make the population standard deviation known.

Spreadsheet functions distinguish sample standard deviation (STDEV.S) from population standard deviation (STDEV.P). Calculating STDEV.P on a sample does not turn that sample into the population. Choose a method that matches how your data was collected.

GOOD QUESTIONS

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context.

The useful details behind the numbers and links.

What does 95% confidence mean?

Under the method’s assumptions, about 95% of intervals constructed this way over repeated random samples would contain the fixed population mean. It is not a 95% probability statement about this particular interval.

Can my mean be zero or negative?

Yes. Means can be zero or negative depending on the measurement. Counts and standard deviations have different constraints.

Does this use z-scores or t-scores?

It uses z-scores and assumes the population standard deviation is known. A mean interval based on an estimated sample standard deviation generally uses a t distribution instead.

How does sample size affect interval width?

Under the same assumptions, larger samples reduce the standard error and narrow the interval. Representativeness and independence still matter; a narrow interval can miss a population value when sampling is biased.

Give your estimate some context.

Your result

Ready when you are.

Good answers start with a better survey.

Know what goes into it.

A quick example

Keep in mind

Confidence interval, margin of error, and standard deviation

A little morecontext.

While you’re here.

Sample size calculator

Statistical significance calculator

Chi-square calculator

A little more
context.