Direct answer
Variance is the average squared distance from the mean under a stated sample or population convention; standard deviation is its square root and therefore returns spread to the original measurement unit.
Visual explanation
From deviations to squared distance
What this calculation tells you
Variance and standard deviation quantify how far observations tend to lie from their arithmetic mean. They use every value and react strongly to observations far from the center.
Choose the population result when the entered observations are the entire finite population being described. Choose the sample result when the values are used to estimate wider population variability under an appropriate sampling model.
Where it is used
Process control
Describe production variation relative to a documented process.
Laboratories
Summarize repeated numerical measurements while keeping method and precision visible.
Finance
Describe historical dispersion without calling it a complete risk measure.
Education
Connect algebraic deviations with geometric distance from the mean.
Common situations
- Comparing variability after units and means are considered.
- Choosing between sample and population formulas.
- Explaining why variance has squared units.
- Checking how one extreme observation changes spread.
Start with the statistical question
Choose the population result when the entered observations are the entire finite population being described. Choose the sample result when the values are used to estimate wider population variability under an appropriate sampling model.
Vertical segments connect observations to the mean; each deviation becomes a square whose area represents its squared contribution. A final square root converts the average area back into a length-like spread.
Worked example
For 2, 4, and 6, the mean is 4 and squared deviations are 4, 0, and 4. Population variance is 8/3, while sample variance is 8/2=4; the corresponding standard deviations are about 1.633 and 2.
Assumptions that carry the result
The calculation assumes numerical observations with meaningful differences. The familiar sample correction addresses estimation under independent sampling; it does not repair clustered, weighted, autocorrelated, or otherwise complex data.
Interpret the result without overreaching
A small standard deviation is not automatically good, and a large one is not automatically bad. Interpret spread in the variable’s unit, process context, distribution shape, and measurement resolution.
- Mixing sample and population conventions silently.
- Comparing standard deviations measured in incompatible units.
- Assuming standard deviation alone establishes normality.
Practical questions
Frequently asked questions
Why square deviations?
Squaring prevents cancellation, weights larger departures more strongly, and supports important algebraic and probabilistic properties.
Why divide sample variance by n−1?
Estimating the mean from the same sample consumes one degree of freedom; the correction makes the variance estimator unbiased under its model.
Does one standard deviation always contain 68% of observations?
No. That approximate rule requires a bell-shaped normal model; it is not a universal property of standard deviation.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.