How to calculate a confidence interval
A confidence interval for a mean is mean ± z × (sd ÷ √n). The term z × (sd ÷ √n) is the margin of error. With mean 100, sd 15, n = 36 and z = 1.96, the margin is 1.96 × 15 ÷ 6 = 4.9, giving a 95% interval of 95.1 to 104.9. The half-width comes from the standard error of the mean multiplied by a critical value.
How to use this calculator
Enter your sample mean, standard deviation, sample size, and the z-score for your confidence level. The result shows the lower and upper bounds plus the margin of error. To estimate how large a sample you need first, use the sample size calculator.
The interval is reported in the same units as your data. Widening or narrowing it is mostly a question of sample size, since n sits under a square root: quadrupling the sample halves the margin of error.
Common z-scores
The z-score depends on your confidence level: 90% uses 1.645, 95% uses 1.96, and 99% uses 2.576. A higher confidence level widens the interval. For a proportion-based estimate, see the margin of error calculator.
Interpret the output carefully: the interval describes the uncertainty around the mean, not the range in which individual observations fall. Most of the data will lie outside 95.1 to 104.9 in the example above, because the standard deviation of 15 is far larger than the margin of error of 4.9.
Frequently asked questions
- How do I calculate a confidence interval?
- Use mean ± z × (sd ÷ √n). With mean 100, sd 15, n 36 and z 1.96, the 95% interval is 95.1 to 104.9.
- Which z-score should I use?
- Use 1.645 for 90%, 1.96 for 95%, or 2.576 for 99% confidence.
- What does a 95% confidence interval mean?
- If you repeated the sampling many times, about 95% of the intervals built this way would contain the true mean.
- Why does a higher confidence level widen the interval?
- A larger z-score increases the margin of error, so the interval must be wider to be more confident.
- Should I use a z-score or a t-score for a small sample?
- Use a t-score when the sample is small (roughly under 30) and the population standard deviation is unknown. The t distribution has heavier tails, so it produces a slightly wider interval; with large samples the two converge and z is fine.