Paste a list of numbers to get the sample and population standard deviation, variance, mean, count and range, or to check your work when you calculate sample standard deviation by hand.
Use sample when your numbers are a subset of a larger group (divides by n-1); use population when they’re the entire group (divides by n).
Standard deviation measures how spread out your numbers are around the mean. A larger value means more spread. The calculator also shows variance (the square of the standard deviation).
How to Calculate Sample Standard Deviation by Hand
Find the mean, square each value’s distance from it, add those squares, divide by n − 1, and take the square root. For 1, 2, 3, 4, 5 the mean is 3, the squared deviations are 4, 1, 0, 1 and 4, which sum to 10, so the sample standard deviation is √(10 ÷ 4) = 1.58.
- Mean: (1 + 2 + 3 + 4 + 5) ÷ 5 = 3.
- Squared deviations: 4, 1, 0, 1, 4.
- Sum them: 10.
- Divide by n − 1: 10 ÷ 4 = 2.5 (the sample variance).
- Square root: √2.5 = 1.58.
Formula: Mean = (sum of all numbers) ÷ n. For each number, find its deviation from the mean and square it: (x − mean)^2. Population variance = (sum of all squared deviations) ÷ n. Sample variance = (sum of all squared deviations) ÷ (n − 1). Standard deviation = √variance, using whichever variance (sample or population) matches the data set type you selected.
- Numbers: your data set, entered separated by commas, spaces or new lines
- Data set type: “a sample” (divides by n − 1) if your numbers are a subset of a larger group, or “the whole population” (divides by n) if they’re the entire group
How to Calculate Variance and Standard Deviation: Examples
Variance comes first, and the standard deviation is its square root. Take 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5 and the squared deviations add to 32, so the population variance is 32 ÷ 8 = 4 (standard deviation 2) and the sample variance is 32 ÷ 7 = 4.57 (standard deviation 2.14).
| Data set | Type | Standard deviation |
|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | Population | 2 |
| 2, 4, 4, 4, 5, 5, 7, 9 | Sample | 2.14 |
| 1, 2, 3, 4, 5 | Population | 1.41 |
| 1, 2, 3, 4, 5 | Sample | 1.58 |
| 10, 10, 10, 10 | Population | 0 |
| 5, 7, 9 | Sample | 2 |
Sample vs Population Standard Deviation: When to Use Each
Choose population when your list covers every member of the group you care about. Choose sample when the numbers stand in for a larger group, such as 30 survey replies; it divides by n – 1 instead of n, so it comes out slightly larger.
- Clean the data first. A stray zero or a typo like 1000 for 100 inflates the result sharply, because deviations are squared.
- Keep units consistent. Standard deviation is in your data’s units, while variance is in squared units.
- Judge spread against the mean: 5 is large for values near 10 and tiny for values near 1,000.
So what does standard deviation tell you about a data set? It is the typical distance of a value from the mean. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two. Skewed data breaks that rule, and a figure from five values is only a rough estimate. The standard deviation article on Wikipedia covers the theory.
Common questions about standard deviation
How do you calculate sample standard deviation by hand?
Find the mean, square each value’s distance from it, add the squares, divide by n − 1, then take the square root. For 1, 2, 3, 4, 5 that is 10 ÷ 4 = 2.5, and √2.5 = 1.58.What is the standard deviation of 1 2 3 4 5?
The mean is 3 and the squared deviations add to 10. The population standard deviation is √(10 ÷ 5) = 1.41, and the sample standard deviation is √(10 ÷ 4) = 1.58.Sample vs population standard deviation: when to use each?
Use population only if your numbers cover every member of the group you care about. Use sample when they are a subset standing in for a larger group, which is the usual case for surveys, experiments and most real data.Why is sample standard deviation bigger than population standard deviation?
Sample variance divides by n − 1 instead of n, which gives a slightly larger number. That correction makes up for a sample tending to underestimate the true spread of the full population.What does standard deviation tell you about a data set?
It tells you how far values typically sit from the mean, in the data’s own units. A standard deviation of 0 means every value is identical, while a large one relative to the mean means the values are widely scattered.Can I paste numbers directly from a spreadsheet?
Yes. The Numbers field accepts values separated by commas, spaces or new lines, so a column or row copied from a spreadsheet works without reformatting, and the order doesn’t change the result.Related Calculators
Standard deviation is easier to interpret alongside the center of your data, so try the average calculator for a quick mean, or the weighted average calculator if some values in your set should count more than others. To see how much a value has shifted over time rather than how spread out it is, check the percentage change calculator. For a related estimate, try the Comma to New Line Converter.