The formula
How it works
A confidence interval is a range around your sample mean that likely contains the true average of the whole population. Enter your mean, standard deviation and sample size to see the interval at your chosen confidence level.
FAQ
What does “95% confidence” actually mean?
It means that if you repeated the study many times, about 95% of the intervals you built this way would contain the true population mean. It is a statement about the method, not a probability for one specific interval.
How do I make the interval narrower?
Collect a larger sample. The interval shrinks with the square root of the sample size, so quadrupling n halves the margin. A lower confidence level also narrows it, but at the cost of being less sure.
Where does the z-score come from?
It comes from the standard normal distribution and marks how many standard errors you need either side of the mean to capture the chosen percentage of area. For example, 1.96 is the value that leaves 95% of the area between ±1.96 standard errors.
What is the difference between the margin of error and the interval?
The margin of error is the ± amount added and subtracted from the mean; the interval is the resulting range itself. A margin of error of 5.4 around a mean of 100 gives the interval 94.6 to 105.4.
Does this calculator assume a normal distribution?
Yes, it uses the z-based formula, which relies on the sampling distribution of the mean being approximately normal. This holds well for larger samples thanks to the central limit theorem, but for small samples from a non-normal population a t-distribution is more accurate.
Why does raising the confidence level widen the interval?
A higher confidence level means you want to be more certain the interval captures the true mean, which requires covering more of the distribution and therefore a bigger z-score and a wider range. There is always a trade-off between confidence and precision.
Should I use the population or sample standard deviation?
Use the standard deviation of your sample, since the population value is normally unknown — that is the whole reason you are estimating it. This calculator treats the value you enter as that estimate.
About the confidence interval calculator
This calculator builds a confidence interval — a range that is likely to contain the true average of a whole population, based on a sample you measured. A single sample mean is only an estimate, and the confidence interval puts honest error bars around it. Reporting a result as a range rather than a single number is one of the most important habits in statistics, because it shows how much uncertainty is really there.
How to use it
Enter your sample mean, the standard deviation, and the sample size, then choose a confidence level such as 95%. The calculator returns the margin of error and the lower and upper bounds of the interval. For example, a mean of 100 with a standard deviation of 15 from a sample of 30 gives a 95% interval of roughly 94.6 to 105.4. A larger sample tightens the interval; a higher confidence level widens it.
The formula
The interval is , where is the sample mean, is the standard deviation, is the sample size and is the z-score for your confidence level (1.96 for 95%). The term is the standard error, and multiplying it by gives the margin of error either side of the mean. Larger samples shrink the standard error and so narrow the interval.
Where it is used
Confidence intervals appear wherever a sample stands in for a bigger group. Opinion polls report them as the “margin of error”, medical trials use them to state the effect of a treatment with its uncertainty, and quality control uses them to monitor a production line. Reading a result as an interval rather than a point estimate is what separates a careful conclusion from an overconfident one.