Standard Deviation Calculator
Calculate the standard deviation, mean and variance of a data set.
Updates as you type.
What Standard Deviation Measures
Standard deviation tells you how spread out a set of numbers is around their average. A small standard deviation means the values cluster tightly near the mean, while a large one means they are scattered widely. It is the most common measure of dispersion in statistics because it is expressed in the same units as the original data, making it easy to interpret.
Two data sets can share the same mean yet behave very differently. The sets {48, 50, 52} and {10, 50, 90} both average to 50, but the second is far more variable. Standard deviation captures exactly that difference in one number, which is why it underpins everything from quality control and finance to test-score analysis and scientific research.
This calculator computes the mean, the variance, and both the population and sample standard deviation, so you can pick the version that matches your situation without doing the arithmetic by hand.
Population vs. Sample Standard Deviation
The choice between the population and sample formula depends on whether your numbers represent an entire group or just a subset of it. The two formulas differ only in the denominator.
The population standard deviation uses every member of the group and divides by N:
σ = √( Σ(x − μ)² / N )
The sample standard deviation uses a sample drawn from a larger population and divides by n − 1:
s = √( Σ(x − x̄)² / (n − 1) )
Dividing by n − 1 instead of n is called Bessel's correction. It slightly increases the result to compensate for the fact that a sample tends to underestimate the true variability of the full population. Use the population formula when you genuinely have all the data (for example, the test scores of every student in one class). Use the sample formula when you are estimating a larger population from a subset, which is the more common case in research and surveys.
The Formula Step by Step
Whichever version you need, the procedure is the same until the final division:
- Find the mean by adding all values and dividing by the count.
- Subtract the mean from each value to get each deviation.
- Square every deviation so negatives and positives do not cancel.
- Add the squared deviations together. This sum is the numerator.
- Divide by N (population) or n − 1 (sample) to get the variance.
- Take the square root of the variance to get the standard deviation.
The variance is simply the standard deviation before the square root is taken. It is useful in many calculations, but because it is in squared units, the standard deviation is usually easier to interpret.
Worked Example
Consider the data set: 4, 8, 6, 5, 3, 7.
Step 1 — Mean: (4 + 8 + 6 + 5 + 3 + 7) / 6 = 33 / 6 = 5.5
Step 2 & 3 — Squared deviations:
(4−5.5)² = 2.25, (8−5.5)² = 6.25, (6−5.5)² = 0.25, (5−5.5)² = 0.25, (3−5.5)² = 6.25, (7−5.5)² = 2.25
Step 4 — Sum: 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 = 17.5
Population (divide by N = 6): variance = 17.5 / 6 = 2.9167, so σ = √2.9167 ≈ 1.708.
Sample (divide by n − 1 = 5): variance = 17.5 / 5 = 3.5, so s = √3.5 ≈ 1.871.
As expected, the sample standard deviation (1.871) is slightly larger than the population value (1.708) because of the n − 1 correction.
Frequently Asked Questions
Use the population formula (divide by N) only when your data includes every member of the group you care about. Use the sample formula (divide by n − 1) when your data is a subset used to estimate a larger population, which covers most surveys, experiments, and real-world studies. When in doubt, the sample version is the safer default.
This is Bessel's correction. A sample's spread tends to be smaller than the full population's spread, so dividing by n would underestimate the true variability. Dividing by the smaller number n − 1 nudges the estimate upward, making it an unbiased estimator of the population variance.
Variance is the average of the squared deviations from the mean, while standard deviation is the square root of the variance. Standard deviation is expressed in the same units as the original data, so it is easier to interpret. Variance is in squared units but is convenient in further statistical calculations.
Standard deviation can be zero, which happens only when every value in the data set is identical, so there is no spread at all. It can never be negative, because it is the square root of a sum of squared numbers, both of which are always zero or positive.
Outliers have a large impact because each deviation is squared before summing. A single value far from the mean contributes a very large squared term, inflating the standard deviation noticeably. For data sets with extreme values, you may also want to report the median and interquartile range as more robust measures.
Standard deviation carries the same units as your original data. If you measure heights in centimeters, the standard deviation is also in centimeters. This is one of its main advantages over variance, which is in squared units and therefore harder to relate back to the data.