Statistics
Confidence interval calculator
Turn a sample mean into a range of plausible values. Enter the mean, standard deviation, and sample size to get a confidence interval and its margin of error.
- 95% confidence interval
- 48.04 – 51.96
- Margin of error
- ± 1.96
Uses the normal (z) approximation — accurate for larger samples. For very small samples a t-interval is more precise. It does not correct for biased sampling.
A single number is never the whole story
A sample mean is an estimate, not the exact truth. A confidence interval puts honest bounds around it: instead of “the average is 50,” you can say “the average is 50, give or take about 2.” That margin shrinks as your sample grows.
Reading the margin of error
With a mean of 50, a standard deviation of 10, and 100 observations, the 95% interval is about 48 to 52 — a margin of error near ±2. Widen the confidence level and the interval grows; collect more data and it tightens. Learn what these numbers do and don’t promise in the Statistics subject.
Frequently asked questions
- What is a confidence interval?
- A confidence interval is a range around a sample estimate that’s likely to contain the true value. A 95% confidence interval means that if you repeated the sampling many times, about 95% of the intervals produced this way would capture the real population mean.
- How is it calculated?
- For a mean, interval = sample mean ± z × (standard deviation ÷ √n), where z is 1.645, 1.96, or 2.576 for 90%, 95%, or 99% confidence. The second part, z × (s ÷ √n), is the margin of error.
- Does a 95% interval mean 95% probability the value is inside?
- Not exactly. The true value is fixed; it’s the interval that’s random. The 95% describes the long-run success rate of the method, not the probability for any single interval. In practice it’s a reasonable range of plausible values.
- What are the limits of this calculator?
- It uses the normal (z) approximation, which suits larger samples; small samples are better handled with a t-interval. And no interval fixes a biased sample — it only quantifies random sampling error.