š Statistics Calculator
Paste a list of numbers (separated by commas, spaces or new lines) to get the core descriptive statistics ā including both population and sample variance/standard deviation.
Where the twelve figures come from
The list is sorted for the order statistics, then measured against its own mean. Variance is the only place two answers exist:
mean = Σx / n
population variance = Σ(x − mean)² / n
sample variance = Σ(x − mean)² / (n − 1)
standard deviation = √varianceThe default list sums to 57 across ten values, so the mean is 5.7 and the squared deviations total 48.1. Divide by 10 for a population variance of 4.81 and a standard deviation of 2.1932; divide by 9 and you get 5.3444 and 2.3118.
Reading the output
- Use n − 1 when your numbers are a sample of something larger, n when they are the whole group. The sample figure is always bigger.
- The median ignores outliers. Change the 9 in that list to 90 and the mean jumps from 5.7 to 13.8, while the median stays at 5.5.
- Ties in the mode are all listed. Here 5 and 8 each appear twice; when nothing repeats you get "no mode".
- Commas, spaces or line breaks all separate values, so a spreadsheet column pastes in as it is.
Frequently asked questions
What is the difference between sample and population standard deviation?
The divisor. Population uses n, sample uses n - 1, correcting for the mean being estimated from the same data. Here that is 2.1932 against 2.3118.
How do you find the median of an even number of values?
Average the two middle ones after sorting. The defaults sort to 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, putting the median halfway between 5 and 6.
Can a set of numbers have two modes?
Yes, and that is called bimodal. The default list has both 5 and 8 appearing twice, so the tool reports both rather than picking one.