Plot No. 59 · Education
Standard Deviation Calculator
Paste a list of numbers to get the mean, variance and both the population and sample standard deviation. Standard deviation measures how spread out the values are: it is the square root of the variance, where variance is the average of the squared distances of each value from the mean.
Enter at least two numbers to get a sample standard deviation.
Sample standard deviation divides by n minus 1 and is the right choice when your data is a sample of a larger group. Population standard deviation divides by n.
The link saves your inputs so you can bookmark or share this exact result.
How the standard deviation is worked out
Standard deviation measures how far a set of numbers typically sits from their average. The calculator first finds the mean, then for every value subtracts the mean and squares the difference. Averaging those squared differences gives the variance, and the square root of the variance is the standard deviation.
Two versions are returned because the divisor differs. The population standard deviation divides the sum of squared differences by n, the count of values, and assumes your list is the whole group. The sample standard deviation divides by n minus 1, a correction (Bessel's correction) that compensates for a sample tending to understate the spread of the larger population it came from.
How to use the standard deviation calculator
- Paste or type your numbers into the box, separated by commas, spaces or new lines.
- Read the sample standard deviation at the top, which is the right choice for most survey or experimental data.
- Check the population standard deviation, mean, count and variance below if you need the full breakdown.
Worked examples
A spread of six values
Inputs: 4, 8, 15, 16, 23, 42
Result: The mean is 18 and the population variance is about 151.6667, giving a population standard deviation of about 12.3153 and a sample standard deviation of about 13.4907.
Tightly clustered values
Inputs: 10, 10, 11, 9, 10
Result: The mean is 10 and the spread is small, so the population standard deviation is about 0.6325 and the sample standard deviation about 0.7071.
Limitations and common mistakes
Edge cases and limitations
- Text that is not a number is ignored, so a stray label in your paste is dropped rather than flagged.
- A single value has no sample standard deviation because dividing by n minus 1 would be dividing by zero; the sample figure only appears with two or more numbers.
- Results are rounded to four decimal places for display.
Common mistakes
- Using the population formula (divide by n) on what is really a sample, which slightly understates the true spread.
- Reporting the variance as if it were the standard deviation; the standard deviation is the square root of the variance and is in the same units as the data.
Frequently asked questions
What is the difference between population and sample standard deviation?
Population standard deviation divides by the number of values (n) and assumes you have every value in the group. Sample standard deviation divides by n minus 1, which corrects for the fact that a sample tends to underestimate the spread of the wider population it came from.
Should I use population or sample standard deviation?
Use the sample version (n minus 1) when your numbers are a sample drawn from a larger group, which is the most common case in surveys and experiments. Use the population version (n) only when your list contains every member of the group you care about.
How is standard deviation calculated?
Find the mean, subtract it from each value and square the result, average those squared differences to get the variance, then take the square root. For 4, 8, 15, 16, 23, 42 the mean is 18 and the population standard deviation is about 12.32.
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