Population vs Sample Standard Deviation
Standard deviation measures spread around the mean. The population and sample formulas use the same squared deviations but a different divisor.
Method
- Calculate the mean of the data.
- Subtract the mean from each observation and square the differences.
- Add the squared differences.
- Divide by n for a population or by n−1 for a sample, then take the square root.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the population variance is 4 and population standard deviation is 2.
The sample standard deviation for the same values is slightly larger because it divides by n−1.
Notes
- Use the formula required by the statistical context.
- Sample standard deviation requires at least two observations.
- Standard deviation is in the original measurement units; variance is in squared units.
- Choose the denominator from the role of the data, not from which result looks better. Use population SD when the entered observations are the complete population of interest; use sample SD when they are a sample used to describe broader variability.
The denominator reflects whether you are describing a population or estimating from a sample
Population variance divides the sum of squared deviations by N because every member of the defined population is included. Sample variance commonly divides by n-1 when the sample is being used to estimate population variance, which corrects the average downward bias created by estimating the mean from the same data.
The square root of variance returns the standard deviation to the original measurement units. That makes it easier to compare spread with the scale of the observed values.
Decide what your dataset represents before selecting the formula
A dataset can be the complete population for one question and a sample for another. Scores for every student in one class are a population if the question is only about that class, but they may be a sample if you want to generalize to a larger group.
Neither formula fixes outliers, dependence or measurement problems. Use the population/sample choice together with a clear definition of the target group.
Authoritative references
Primary or authoritative references for the external standard, scoring rule or identifier convention used on this page.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- Mean Median Mode Calculator — Calculate mean, median, mode, minimum and maximum from a list of numeric values.
- Five Number Summary & Outlier Calculator — Calculate min, Q1, median, Q3, max, IQR, 1.5×IQR fences, box-plot whiskers and flagged outliers from numeric data.
- Standard Error Calculator — Calculate the standard error of a sample mean from standard deviation and sample size.