The Confidence Interval Calculator finds the range likely to contain the true population mean, from a sample size, sample mean, standard deviation and confidence level.
20.6 ± 0.886978
| Confidence interval | 19.713022 to 21.486978 |
| Margin of error | ±0.886978 |
| Z value | 1.9600 |
| Standard error | 0.452548 |
A confidence interval gives a range of plausible values for an unknown population value, based on a sample. A 95% confidence interval means that if you repeated the sampling many times, about 95% of the intervals built this way would contain the true value.
CI = X̄ ± z × σ/√n, where z is the critical value for the confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%) and σ/√n is the standard error.
Worked example (defaults): n = 50, mean 20.6, σ = 3.2, 95%. Standard error = 3.2/√50 = 0.4525; margin = 1.96 × 0.4525 = 0.887; interval 19.713 to 21.487.
Higher confidence makes the interval wider. A larger sample makes it narrower (halving the width needs four times the sample). More variable data makes it wider.
With small samples and an unknown population standard deviation, statisticians use the t-distribution instead of z, which gives slightly wider intervals. For n above about 30 the difference is small.
Does 95% mean a 95% chance the true mean is inside? Strictly, the true mean is fixed; the method captures it 95% of the time.
What confidence level should I use? 95% is the most common.
Can I use this for proportions? Use the Sample Size Calculator's margin of error tool.