Paste a dataset to calculate population and sample standard deviation, variance, centre, quartiles, range and relative spread without hiding the working.
Statistics & probability calculator
Paste values once to compare population and sample standard deviation, variance, quartiles and coefficient of variation.
Your values stay in this browser and are used only for this calculation.
Dataset summary
Population and sample measures plus a value-by-value deviation table will appear here.
Methodology
The calculator subtracts the arithmetic mean from each value, squares those deviations and totals them. Population variance divides that total by n; sample variance divides by n − 1. Standard deviation is the positive square root of the chosen variance.
Worked example
For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5. The squared deviations total 32, so population variance is 32 ÷ 8 = 4 and population standard deviation is 2. Sample standard deviation is about 2.138.
Interpretation
Variance uses squared units, while standard deviation returns to the original unit. That makes standard deviation easier to compare with the mean and individual observations. The deviation table lets you audit the intermediate values instead of accepting a single unexplained answer.
Reference method
NIST describes standard deviation as the square root of variance and distinguishes the sample calculation using n − 1. The calculator reports both so you can choose the measure that matches your data.
NIST: measures of scaleRelated statistics tools
Common questions
Use population standard deviation when the values are the entire group you want to describe. Use sample standard deviation when the values are a sample used to estimate variability in a larger population.
Sample variance divides the squared-deviation total by n − 1 rather than n. This degrees-of-freedom correction reduces systematic underestimation of population variance.
No. Deviations are squared before averaging and the final square root cannot be negative. A standard deviation of zero means every entered value is identical.
It means the values are more dispersed around their mean in the dataset's original unit. Whether that is large in context depends on the scale and subject.
Coefficient of variation divides standard deviation by the absolute mean and expresses the result as a percentage. It can help compare relative spread across positive ratio-scale datasets, but it is undefined when the mean is zero.
The calculator sorts the values and uses linear interpolation at the 25th, 50th and 75th percentiles. Other software may use a different quartile convention and return slightly different values for small datasets.
These descriptive statistics summarize the values entered. They do not prove that a sample is representative, normally distributed or free of data-quality problems. This calculator is for planning and education, not personal financial advice.