Statistics Calculator: Mean, Median and Standard Deviation

Mean, median and repeated values

The mean is the sum divided by the count. The median is the middle value after sorting; with an even count, average the two middle values. The NIST handbook on measures of location explains these measures and why they can give different descriptions of the same data.

This tool reports the most frequent exact values when at least one repeats. If several values tie for the highest frequency, it lists each tied value. If every value occurs once, it displays “No repeated values” rather than picking an arbitrary winner. Values that differ only after rounding are still treated as different inputs.

Population versus sample standard deviation

Variance summarizes squared distances from the mean. Population variance divides their total by n, the number of values. Sample variance divides by n − 1 and requires at least two observations. Standard deviation is the square root of the selected variance. The NIST measures-of-scale reference describes sample variance, standard deviation and range.

Choose the formula that matches the question you are answering. If your data are every expense in a particular month's completed ledger, population spread describes that full set. If you are using a sample to estimate variation in a larger group, a sample calculation is appropriate. Neither choice fixes a sample that was collected badly or excludes important observations.

Worked example

For 2, 4, 4, 4, 5, 5, 7 and 9, the count is 8 and the sum is 40. The mean is 5. The two central sorted values are 4 and 5, so the median is 4.5. The most frequent value is 4. The range is 9 − 2 = 7.

The squared deviations sum to 32. Population variance is 32 ÷ 8 = 4, with population standard deviation 2. Sample variance is 32 ÷ 7, about 4.571429, giving sample standard deviation about 2.13809. Selecting the other formula changes the spread results, not the mean or median.

Check your data before interpreting the result

Keep all observations in the same units. Combining dollars with cents, or minutes with hours, makes a summary misleading even when the arithmetic is correct. Include zero when it is an actual observation; do not use zero to stand in for a missing value without explaining that decision.

The calculator rejects text, infinity and invalid numbers instead of silently discarding them. Decimal commas and thousands separators are not supported because commas separate observations. Scientific notation such as 1e3 is accepted. Results are rounded for display and use the browser's floating-point arithmetic, so this is not an exact-decimal accounting system.

A large mean does not prove most values are large. Check the median, range and original values together, especially when one entry is much larger than the rest. These are descriptive summaries, not hypothesis tests, forecasts or evidence of a causal relationship. Your list is calculated in this browser page and is not submitted or stored by the calculator.