Standard deviation measures how spread out a set of numbers is around the mean. A low standard deviation means the values cluster near the average; a high one means they are spread widely. It is the square root of the variance, which is the average of the squared differences from the mean.
There are two flavors. The sample standard deviation divides by n−1 (one fewer than the count) and is used when your numbers are a sample of a larger population — the n−1 correction makes the estimate unbiased. The population standard deviation divides by n and is used when you have every member of the population. This calculator reports both the chosen deviation and the variance, plus the mean, median, sum, min, max, and range.
Enter your numbers separated by commas or spaces and choose sample or population. The calculator parses the list, validates that every entry is a number, and computes the statistics. Sample mode requires at least two numbers because you cannot measure spread from a single value.
Frequently asked questions
What is the difference between sample and population standard deviation?
Sample (n−1) is used when your data is a sample of a larger group — it is an unbiased estimate of the population spread. Population (n) is used when you have the entire population. This calculator defaults to sample mode and reports which one it used.
How is standard deviation calculated?
Take the mean, square the difference of each value from the mean, average those squares to get the variance, then take the square root. For the dataset 2,4,4,4,5,5,7,9 the sample standard deviation is about 2.14.
Why does sample mode need at least two numbers?
With one number the spread is undefined — dividing by n−1 = 0 is impossible. You need at least two values to measure variation, so sample mode requires a minimum of two numbers.
Can I paste numbers separated by spaces or newlines?
Yes. The calculator splits on any whitespace and commas, so "2, 4, 4" and a column of newline-separated numbers both work. Every entry must parse as a number, or it will report an error.
