Statistics Calculators

How a Confidence Interval and Margin of Error Are Calculated

A confidence interval's margin of error is a Z-value times the sample's standard deviation divided by the square root of the sample size — and because sample size sits under a square root, doubling the sample only shrinks the margin by about 30%, not by half.

How the Geometric Mean Is Calculated

The geometric mean multiplies all the numbers together and takes the nth root of the product — the correct way to average growth rates, ratios, and returns that compound over time.

How the Harmonic Mean Is Calculated

The harmonic mean divides the count of numbers by the sum of their reciprocals — the correct way to average rates, like speeds traveled over equal distances, where a simple average would overstate the true result.

How the Required Sample Size for a Survey Is Calculated

Cochran's formula sets the sample size a survey needs from three choices — confidence level, target margin of error, and an assumed proportion — and assuming a 50/50 split when the true proportion is unknown always demands the largest, safest sample.

How to Calculate Mean, Median, and Mode

Mean is the sum divided by the count, median is the middle value once the list is sorted (or the average of the two middle values for an even-sized list), and mode is whichever value appears most often.

Mean vs Median vs Mode: Which Average Should You Use?

Mean is the best "typical value" when data has no extreme outliers, median holds up better when a few extreme values would drag the mean off course, and mode is the only one of the three that makes sense for "which value shows up most" questions.