Z-score Calculator

A mean summarizes total magnitude per observation, while a median locates the middle of an ordered data set. A few extreme values can affect them differently. Standard deviation describes spread in the original units; variance uses squared units. A frequency chart helps show distribution shape that one summary cannot capture.

Keep sample and population formulas distinct. A sample-variance calculation typically uses n−1 in its denominator. Repeated observations and units must be handled consistently, and a list of numbers does not by itself establish independent random sampling.

Confidence intervals, sample sizes and p-values depend on a specified model, confidence or tail convention, and data assumptions. A narrow calculated interval is not evidence that the assumptions were met. A p-value is not the probability that a hypothesis is true.

Use the page’s formula and worked example to check whether the statistic, sample size and degrees of freedom match the intended analysis. Do not substitute a population standard deviation for a sample value, or a one-sided test for a two-sided one, without changing the model.