Standard Deviation Calculator

Calculate mean, variance, and standard deviation for both population and sample data sets instantly.

Standard Deviation Calculator tool

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Example

Input: Data set 2, 4, 4, 4, 5, 5, 7, 9

Result: Population SD 2, sample SD 2.13809 (mean 5, population variance 4, sample variance 4.571429)

The squared deviations from the mean of 5 add up to 32. Population: 32 / 8 = 4 and sqrt(4) = 2. Sample: 32 / 7 = 4.571429 and its square root is 2.13809.

How Much Larger the Sample SD Is

Number of values (N)Sample SD / population SDSample SD is larger by
21.414241.4%
31.224722.5%
51.118011.8%
101.05415.4%
201.02602.6%
301.01711.7%
501.01021.0%
1001.00500.5%
10001.00050.1%

Both are computed from the same data, so the ratio is the square root of N / (N - 1).

  1. Enter your data set as a list of numbers separated by commas, spaces, or line breaks.
  2. Review the computed mean, variance, and standard deviation, shown for both a population (entire group) and a sample (subset of a larger group).
  3. Pick the population or sample figures depending on whether your data covers the whole group or only part of it.
  4. Examine the sorted data and, if any values repeat, the frequency table with each value count and percentage.
  5. Copy the results or use them as inputs in related tools like the normal distribution calculator.

Population vs Sample Standard Deviation

Standard deviation measures how far values typically sit from their average. A small value means the data is tightly bunched, and a large one means it is spread out. Paste a list of numbers and this calculator gives the mean, variance and standard deviation. It works them out both for a population and for a sample.

The two versions differ in one step. Population standard deviation divides by N, the number of values. Sample standard deviation divides by N - 1. The smaller divisor corrects for the fact that a sample tends to understate the spread of the group it came from. Use population when your list is the whole group, such as every student in one class. Use sample when your list stands in for a bigger group, such as 50 customers out of thousands. For 2, 4, 4, 4, 5, 5, 7, 9 the population figure is 2 and the sample figure is 2.14. The gap is large for small lists and shrinks as N grows, as the table on this page shows.

For data shaped roughly like a bell curve, about 68% of values fall within one standard deviation of the mean, and about 95% within two. So if test scores have a mean of 70 and a standard deviation of 10, most students scored between 60 and 80. The Normal Distribution Calculator turns this into exact percentages. Keep in mind that one extreme value can inflate the result a lot.

You can separate numbers with commas, spaces, semicolons or new lines. Anything that is not a number is skipped, and a count shows how many values were read so you can check. You also get the sum, minimum, maximum and range, the data in sorted order, and a frequency table when values repeat. For the median and mode, use the Mean Median Mode Calculator.

Formula / How It Works

Population SD: sigma = sqrt(sum((xi - mu)^2) / N) | Sample SD: s = sqrt(sum((xi - x-bar)^2) / (N - 1)) | Variance = SD^2

Sources

Frequently Asked Questions

Use population standard deviation when your data covers every member of the group you care about. Use sample standard deviation when your data is a subset used to describe a larger group, which is the usual case in surveys and experiments. If in doubt, the sample version is the safer choice. The calculator shows both side by side.
Compare it with the mean. A standard deviation of 2 on a mean of 100 means values barely vary. The same 2 on a mean of 5 means they vary a lot. For bell-shaped data, about 68% of values lie within one standard deviation of the mean and about 95% within two.
Find the mean. Subtract it from each value and square the results. Add the squares, then divide by N for a population or by N - 1 for a sample. That gives the variance. Take the square root to get the standard deviation. For 2, 4, 4, 4, 5, 5, 7, 9 the squares add to 32, so the population variance is 4 and the SD is 2.
Variance is the average squared distance from the mean, and standard deviation is its square root. Variance is in squared units, such as square centimeters, which are hard to picture. Standard deviation is in the same units as your data, so it is the one people usually report.
Yes, when every value is the same, such as 5, 5, 5, 5. There is no spread at all. It can never be negative, because it is built from squared distances. With a single value, the sample standard deviation cannot be worked out, so the calculator shows N/A for it.
When the data is strongly skewed or has outliers. A few very large values, as with incomes or house prices, pull the standard deviation up. In those cases the interquartile range, the spread of the middle half of the data, often describes a typical value better.

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