Confidence Interval Calculator

A confidence interval is an interval estimate—a lower bound and upper bound around a point estimate. The point estimate is a sample statistic, while the range describes plausible values for the corresponding population parameter. It expresses uncertainty from sampling rather than the probability that this one fixed interval contains the parameter.

A higher confidence level requires a larger critical value, which increases the margin of error. For a two-sided 95% interval, the significance level is 5%, so alpha is 0.05 and 0.025 is placed in each tail of the reference distribution. That tradeoff gives greater confidence but less precision.

When the population standard deviation is unknown, the sample standard deviation is used to estimate it. The resulting t-score comes from a t-distribution with n − 1 degrees of freedom and accounts for the added uncertainty in the standard error. A z-score comes from the standard normal distribution and is appropriate when the population variability is known or for methods that use a normal approximation.

A larger sample size, lower variance, or lower confidence level produces a narrower result. Increasing the sample reduces the standard error because the formula divides variability by the square root of n. Population size usually has little effect unless the observations are drawn without replacement from a relatively small, finite population.

Yes. For paired data, calculate the difference within each matched observation, such as a before-and-after measurement from the same participant. Treat the differences as one dataset, then enter their sample mean, standard deviation, and count in mean mode. This preserves the pairing used by the experiment.

The observations should be independent and come from a representative random sample or a well-designed experiment. For a small-sample mean interval, inspect the data for severe skew and outliers; with a sufficiently large sample, the central limit theorem makes the sampling distribution of the mean approximately normal. Survey weights, clusters, repeated measures, and other complex research methodology may require a statistician or specialized statistical software for valid inference.

The sample proportion is the number of successes divided by the total number surveyed, often reported as a percentage. It is the point estimate of the population proportion. This calculator applies the Wilson methodology, which generally produces more reliable bounds than the basic normal formula when the sample is small or the estimated percentage is close to 0% or 100%.

Standard deviation describes how individual measurements vary within a dataset. Standard error describes how much an estimate, such as the sample mean, would vary across repeated random samples. The calculator combines that error with the appropriate critical value to form the interval.

For a two-sided test using the same assumptions and alpha, a null hypothesis value outside the confidence interval would be rejected at that significance level. A value inside the interval would not be rejected. The interval is often more informative because it shows the estimate, uncertainty, and range of values compatible with the sample rather than only a yes-or-no test result.

For a mean, it uses a t-value based on the selected confidence level and degrees of freedom. For a proportion, it uses the corresponding z-value from the normal distribution. These values are inserted into the relevant statistics formula to convert the standard error into a margin of error.