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How to use
Standard deviation measures how widely values spread around their mean. Mihsaba shows population and sample versions together so the effect of dividing by N versus n−1 is explicit, along with mean, variance and squared-deviation sum. For 10, 12, 14 and 16, the mean is 13 and the squared-deviation sum is 20. Describing these observations as the full population gives variance 20/4=5 and deviation √5≈2.2361; treating them as a sample gives variance 20/3 and deviation ≈2.5820. Choose the version from the data definition, rather than the smaller result.
- Enter the data set.
- Calculate to find the mean and the sum of squared deviations from it.
- Compare population variance/deviation with the sample versions.
How it is calculated
Example
Example: for 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5, population variance is 4 and population standard deviation is 2. The sample version divides by 7 instead of 8, so it is slightly larger.
Important notes
Use population statistics when the data contain every item in the group you are describing. Use the sample version when the observations are a sample from a larger population. Standard deviation alone does not make a value good, bad or risky without context.
Worked examples and interpreting results
For 2, 4, 6, the mean is 4 and the sum of squared deviations is 8. The result depends on whether the values represent the full population or a sample, because the divisor changes.
| Case | Calculation | Result |
|---|---|---|
| Population standard deviation | √(8 ÷ 3) | 1.632993 |
| Sample standard deviation | √(8 ÷ (3 − 1)) | 2 |
| Population deviation for 10,12,14,16 | √(20 ÷ 4) | 2.236068 |
| Sample deviation for 10,12,14,16 | √(20 ÷ (4 − 1)) | 2.581989 |
How to check the result
The sample formula needs at least two values. Standard deviation uses the original data unit, while variance uses its square. Understand an outlier’s origin before deciding whether to exclude it.
Population vs sample standard deviation calculator
Standard deviation measures how widely values spread around their mean. Sum the squared deviations, divide by n for a population or n−1 for a sample, then take the square root.
- Population deviation for 2, 4, 6: √(8 ÷ 3) = ≈ 1.633
- Sample deviation for 2, 4, 6: √(8 ÷ 2) = 2
- Deviation of three equal values: √(0 ÷ 3) = 0
Calculation caveat
Use the n−1 denominator for the sample standard deviation, not for a population. Sample standard deviation is undefined for fewer than two observations.
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Frequently asked questions
What is the difference between population and sample standard deviation?
Population divides squared deviations by N; sample divides by n−1 when estimating a larger population.
Why is variance the square of standard deviation?
Standard deviation is defined as the square root of variance, so variance equals standard deviation squared.
Can sample standard deviation be calculated from one value?
No. With n=1, n−1 is zero, so at least two sample observations are required.
Does adding the same number to every value change standard deviation?
No. The mean shifts but deviations from it stay the same. Multiplying all values by a factor a multiplies standard deviation by |a|.
What units do standard deviation and variance use?
Standard deviation has the original data unit; variance has its square. Metre measurements give deviation in metres and variance in square metres.