Math
Mean, variance, and standard deviation from any list of numbers — sample or population.
Choose "Sample" if your numbers are a subset of a larger group; choose "Population" if they're the entire group.
Standard deviation
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Enter a list of numbers
Mean
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Variance
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Count
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For the data set 10, 12, 23, 23, 16, 23, 21, 16, the mean is 18. The population standard deviation comes out to approximately 4.9, meaning most values sit within about 5 points of the mean. Treating the same data as a sample instead of the full population gives a slightly higher standard deviation of about 5.24, reflecting the extra uncertainty from working with a subset.
Using the wrong formula — population when you actually have a sample, or vice versa — systematically under- or overestimates the true spread of your data, since the sample formula divides by one less than the count specifically to correct for that extra uncertainty. For anything used to draw conclusions about a larger group from a subset, picking the correct mode matters.
It measures how spread out your numbers are from the average. A small standard deviation means values cluster tightly around the mean; a large one means they're spread wide, with some far from typical.
Use "Population" when your numbers are the entire group you care about — every student in one class, for instance. Use "Sample" when your numbers are a subset used to estimate a larger group — a survey of 200 people out of a city's population. Sample standard deviation divides by one less than the count, which corrects for that extra uncertainty.
Standard deviation is simply the square root of variance. Variance is useful mathematically but is in squared units (like "dollars squared"), which is hard to interpret — taking the square root brings it back to the original units, which is why standard deviation is the number most people actually look at.