Confidence Interval Calculator

Sample mean

Standard deviation

Sample size

Confidence level, %

Result

Interval94.6324 … 105.3676
Lower bound94.632418
Upper bound105.367582
Margin of error5.367582
Standard error2.738613
z value1.96

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. The confidence interval gives the range that contains the true value with a stated probability, conventionally 95 percent.

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.

What this calculator shows

The lower and upper bounds of the interval
The margin of error and the standard error of the mean
The z value corresponding to your confidence level

What to keep in mind

The normal approximation is used, sound from about thirty observations; below that the t-distribution is correct.
The standard deviation is treated as known; estimating it from the same sample makes the interval slightly too narrow.

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?

Below about thirty observations, or when the deviation is estimated from the sample itself. The t interval is wider, reflecting that extra uncertainty.

How does this relate to a p-value?

Directly: if a 95 percent interval excludes the null value, p is below 0.05. They are two views of the same result.

Worked example

A sample of thirty

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

Output: Interval 94.63 to 105.37

Note: The margin is 5.37. Halving it would require growing the sample from thirty to a hundred and twenty.

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Confidence Interval Calculator

Result

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P-value Calculator

Z-score Calculator

Standard Deviation Calculator

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