A Confidence Interval Calculator

A confidence interval turns a single sample estimate into a range, and the width of that range is the honest measure of how much the sample actually tells you.

Sample meanThe centre of the interval.Standard deviationStandard deviation of the sample, in the same unit as the mean.Sample sizeNumber of observations. The standard error divides by its square root.Critical value1.645 for 90 percent, 1.96 for 95 percent, 2.576 for 99 percent.

Margin of error

4.9

Standard error: 2.5
Lower bound: 95.1
Upper bound: 104.9

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

confidence interval = mean +/- z x (standard deviation / sqrt(n))

x bar: Sample mean, the centre of the interval
s: Sample standard deviation
n: Sample size
z: Critical value for the chosen confidence level

How to check the result by hand

1
Work out the standard error
Divide the standard deviation by the square root of the sample size. A standard deviation of fifteen with thirty six observations gives an error of 2.5.
2
Choose the z score for your confidence level
1.645 for ninety percent, 1.96 for ninety five and 2.576 for ninety nine. Rounding 1.96 to 2 is common and makes the interval slightly wider than stated.
3
Multiply to get the margin of error
1.96 times 2.5 is 4.9. This is the half width, not the full width, and confusing the two halves or doubles the reported interval.
4
Add and subtract from the sample mean
A mean of a hundred gives a lower bound of 95.1 and an upper bound of 104.9. The interval is centred on the sample mean by construction.
5
Check the assumptions before trusting it
Random sampling, independent observations and either a roughly normal distribution or a large enough sample. A biased sample gives a precise but meaningless interval.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate A Confidence Interval page.