Confidence Interval Calculator

How This Calculator Works

A confidence interval is a range of values, built around a sample’s own mean, that likely contains the true population mean. Enter a sample’s mean, standard deviation, and size, choose a confidence level, and this calculator finds the range within which the actual population mean probably falls.

A “95% confidence interval,” for example, means that if you repeated the same sampling process many times, about 95% of the intervals it produced would contain the true population mean. It does not mean there’s a 95% chance the true mean falls inside this one specific interval — a common, technically-incorrect way people phrase it. A wider interval (higher confidence level) casts a wider net and is more likely to contain the true mean; a narrower interval is more precise but less certain.

The Formula

$$\vE{\text{Standard Error}} = \frac{\vA{s}}{\sqrt{\vB{n}}}$$ $$\vF{\text{Margin of Error}} = \vC{z} \times \vE{\text{Standard Error}}$$ $$\text{Confidence Interval} = \vD{\bar{x}} \pm \vF{\text{Margin of Error}}$$

Where $\vA{s}$ is the sample’s standard deviation, $\vB{n}$ is the sample size, $\vC{z}$ is the critical value for the chosen confidence level (looked up from the standard normal distribution), and $\vD{\bar{x}}$ is the sample mean. A bigger sample size shrinks the standard error — and therefore the whole interval — since more data gives a more precise estimate of the true mean.

Worked Example

A sample of 25 items has a mean of 100 and a standard deviation of 15, at a 95% confidence level (a critical value of z = 1.96):

1. Standard error: $\vE{\text{Standard Error}} = \vA{15} \div \sqrt{\vB{25}} = \vE{3}$.
2. Margin of error: $\vF{\text{Margin of Error}} = \vC{1.96} \times \vE{3} = \vF{5.88}$.
3. Confidence interval: $\text{Confidence Interval} = \vD{100} \pm \vF{5.88} = [94.12, 105.88]$.

So the true population mean is likely (with 95% confidence) somewhere between 94.12 and 105.88.