Statistics Calculator: Standard Deviation, Variance and Quartiles
Paste a list of numbers to get the standard deviation and variance (sample and population), the standard error, quartiles and IQR, alongside the mean, median and mode.
The standard deviation measures how far values typically sit from the mean. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared distances from it add up to 32. Dividing by one less than the count gives the sample variance, 32 over 7, or 4.5714, and its square root is a sample standard deviation of 2.1381. Treated as a whole population, divide by 8 instead and the standard deviation is exactly 2.
This calculator computes descriptive statistics for the numbers you enter, for general reference. It does not test hypotheses, check whether your data is a fair sample, or judge whether a method suits your study. Check quartile conventions against the method your course or field expects.
Sample standard deviation
2.1381
- Count
- 8
- Mean
- 5
- Median
- 4.5
- Mode
- 4
- Minimum, maximum
- 2, 9
- Range
- 7
- Sample variance
- 4.5714
- Population standard deviation
- 2
- Population variance
- 4
- Standard error of the mean
- 0.7559
- Quartiles Q1, Q3
- 4, 6.5
- Interquartile range
- 2.5
- Sum
- 40
- Sum of squared deviations
- 32
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About the Statistics Calculator
Most spread measures start from the same step: find the mean, subtract it from every value, square those differences and add them up. That sum of squares is 32 for the default list. What you divide it by is the choice that matters. If your numbers are the entire group you care about, such as every score in one class, divide by the count to get the population variance. If they are a sample standing in for a bigger group, divide by the count minus one, which corrects for the fact that a sample tends to sit closer to its own mean than to the true one. The standard deviation is the square root of either, so it comes back in the same units as the data. The standard error divides the sample standard deviation by the square root of the count, and it shrinks as you add data. Quartiles split the sorted list into quarters; this calculator uses the percentile rule in the NIST/SEMATECH engineering statistics handbook, which interpolates between neighbors, so Q3 for the default list is 6.5. Spreadsheet quartile functions use a slightly different rule and can give a nearby number on small lists.
Frequently asked questions
Should I use the sample or the population standard deviation?+
Use the population figure only when your list is every member of the group, like all 30 students in a class. If the numbers are a sample drawn to learn about a larger group, use the sample figure, which divides by one less than the count and comes out a little larger. For the default list that is 2.1381 against exactly 2.
Why don't my quartiles match Excel?+
There are several accepted ways to place a quartile between two data points. This page follows the percentile rule in the NIST/SEMATECH engineering statistics handbook; spreadsheet QUARTILE.INC uses another. On long lists they agree closely, but on a short list like the default one they can differ by a fraction of a step.
What is the interquartile range used for?+
The IQR is Q3 minus Q1, the width of the middle half of the data: 2.5 for the default list. Because it ignores the top and bottom quarters, one extreme value barely moves it, which makes it a steadier measure of spread than the range when a list has outliers.
How many numbers can I enter?+
Up to 10,000, separated by commas, spaces or new lines. Anything that is not a number is skipped, and the count under the box shows how many values were read. Sample variance and the standard error need at least 2 values.
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