Standard Deviation Calculator
Paste a list of numbers to get the standard deviation. Choose sample (divide by n − 1) or population (divide by n); the chart shows the mean and median.
Last updated: October 2026
Standard deviation
2.03
Formula below · 2 sources (itl.nist.gov, Wikipedia) · Updated Oct 2026
Compare with similar
About this calculator
Standard deviation measures how far values typically sit from their mean. A small standard deviation means the numbers cluster tightly; a large one means they are spread out. It is in the same units as the data, so a test with mean 72 and standard deviation 8 points is easy to read.
How it is calculated
1. Find the mean: add the values and divide by the count, n. 2. Subtract the mean from each value and square the difference. 3. Add up the squared differences. 4. Divide by n − 1 for a sample, or by n for a population. This is the variance. 5. Take the square root of the variance.
Sample or population?
Use the sample formula (s, divide by n − 1) when your numbers are a subset of a larger group you want to describe: survey answers, lab measurements, a month of sales. Dividing by n − 1 (Bessel's correction) corrects for the fact that values sit a little closer to their own sample mean than to the true population mean. Use the population formula (σ, divide by n) only when the list holds every member of the group, such as the scores of everyone in one class. The difference matters most for short lists: with 5 values the sample variance is 25% larger than the population variance; with 100 values the gap is about 1%.
The chart under the result marks the mean and median and reports the population standard deviation, so with Sample selected its figure is slightly smaller than the result. In Excel or Google Sheets, STDEV.S gives the sample value and STDEV.P the population value.
How to use
Type or paste your numbers separated by commas, without thousands separators (write 1200, not 1,200). Choose Sample or Population. The result is the standard deviation; square it for the variance. The default list 4, 8, 6, 5, 3, 7, 9, 5 has n = 8 and mean 47 ÷ 8 = 5.875. The squared differences from the mean add up to 28.875, so the sample variance is 28.875 ÷ 7 = 4.125 and s = √4.125 = 2.03. Switch to Population and the variance is 28.875 ÷ 8 = 3.609, so σ = 1.9. A second check: 2, 4, 4, 4, 5, 5, 7, 9 has mean 5 and squared differences 9, 1, 1, 1, 0, 0, 4 and 16, which add to 32. The population standard deviation is √(32 ÷ 8) = 2 and the sample standard deviation is √(32 ÷ 7) = 2.138.
Frequently asked questions
Should I use the sample or the population standard deviation?
Use the sample standard deviation unless your list contains every member of the group you are describing. Most real data are samples: a poll, a batch of measurements, last quarter's orders. The population formula fits only complete data, such as the heights of all 24 players on one team when you care about that team alone.
Why does the sample formula divide by n − 1?
The deviations are measured from the sample mean, which is calculated from the same numbers. That makes the squared deviations slightly smaller on average than they would be around the true population mean. Dividing by n − 1 instead of n makes the sample variance an unbiased estimate of the population variance. The correction shrinks as n grows.
What does the standard deviation tell me about normally distributed data?
For data that follow a bell-shaped normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three. The rule does not hold for skewed data such as incomes or waiting times, where the median and interquartile range often describe the data better.
How should I enter my numbers?
Separate values with commas, use a period for decimals and leave out thousands separators and units. Negative numbers are fine. If an entry is not a number, the calculator names it instead of guessing. You need at least two values.
Related calculators
Sources & references
Think a result is wrong? Report a problem with this calculator.