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Sample Variance (s²)
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Mean (x̄)
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Std. Deviation
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Count (n)
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Sum (Σx)
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Minimum
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Maximum
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Range
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Sum of Squares
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Coeff. of Variation
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What Is a Population Measures Tool?
A standard deviation and variance calculator gives you both core measures of data spread in a single operation. Variance (σ² or s²) quantifies dispersion in squared units, while standard deviation (σ or s) converts it back to the original data units by taking the square root. Together, they provide a complete picture: variance is used in mathematical statistics (ANOVA, regression, portfolio theory), while standard deviation is the go-to for intuitive interpretation (the 68-95-99.7 rule, z-scores, confidence intervals). Enter your data once and get both measures for both population and sample formulas.
Formula
Variance and standard deviation are computed together — SD is simply the square root of variance.
Population Measures
σ² = Σ(xᵢ − μ)² / N → σ = √σ²
σ²Population variance
σPopulation standard deviation
μPopulation mean
NTotal values in the population
Sample Measures
s² = Σ(xᵢ − x̄)² / (n−1) → s = √s²
s²Sample variance
sSample standard deviation
x̄Sample mean
n−1Degrees of freedom
How to Calculate Standard Deviation and Variance Together
Compute the mean.x̄ = Σxᵢ / n.
Subtract the mean from each value.Get deviations: (xᵢ − x̄).
Square each deviation.Compute (xᵢ − x̄)².
Sum all squared deviations.Σ(xᵢ − x̄)² — this is the Sum of Squares (SS).
Divide by the appropriate denominator.Divide SS by N for population variance or by n − 1 for sample variance.
Take the square root for standard deviation.SD = √variance. You now have both measures.
Enter your data values separated by commas, spaces, or line breaks.
The calculator instantly returns population and sample versions of both variance and standard deviation.
Expand the step-by-step section to see the full derivation from mean through to final results.
Copy results for reports, academic submissions, or further statistical analysis.
Benefits of Calculating SD & Variance Together
One input, four key results — population and sample variance plus both standard deviations.
Saves time compared to running two separate calculations.
Step-by-step breakdown helps you understand how SD derives from variance.
Complementary statistics (mean, range, CV, sum of squares) included at no extra effort.
Perfect for assignments that require reporting both measures.
Where SD & Variance Are Used Together
Academic Statistics
Textbook problems and exams routinely ask for both measures.
ANOVA & Regression
Variance is used in model computations; SD is reported for interpretability.
Risk Management
Portfolio variance determines combined risk; SD communicates it in the asset's units.
Process Control
Control chart limits use SD, while capability indices use variance components.
Research Papers
Journals often require both measures in descriptive statistics tables.
Machine Learning
Feature scaling and normalization use SD; variance informs feature selection.
Choosing Between Standard Deviation and Variance
Both measure spread, but they serve different purposes. Knowing when to use each one avoids confusion and strengthens your analysis:
Use SD for communication
Standard deviation is in the same units as your data, making it immediately meaningful. Saying "the average is 50 ± 5 units" is intuitive; "the variance is 25 squared-units" is not.
Use variance for mathematical operations
Variances of independent variables add directly: Var(X + Y) = Var(X) + Var(Y). Standard deviations do not add this way, making variance essential in ANOVA, regression, and portfolio theory.
Use SD for the empirical rule
The 68-95-99.7 rule uses standard deviation to define data intervals. Approximately 68% of data falls within one SD of the mean.
Use variance for comparing model fits
R² (coefficient of determination) is based on variance decomposition — it tells you what proportion of total variance is explained by the model.
Report both when thoroughness matters
In scientific papers, dissertations, and quality reports, reporting both demonstrates statistical rigour and satisfies different readers' preferences.
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Frequently Asked Questions — Standard Deviation & Variance Calculator
Everything you need to know about variance.
What is the difference between standard deviation and variance?
Variance measures spread in squared units — it's the average of squared deviations from the mean. Standard deviation is the square root of variance, returning the measure to the original units. For example, if data is in meters, variance is in m² while SD is in meters. Both measure the same concept (data spread), but SD is easier to interpret because it's on the same scale as the data.
How to calculate standard deviation and variance together?
The process is shared: 1. Find the mean. 2. Subtract the mean from each value. 3. Square each deviation. 4. Sum the squared deviations. 5. Divide by n (population) or n−1 (sample) — this is the variance. 6. Take the square root — this is the standard deviation. Our calculator computes both simultaneously.
When should I report standard deviation vs variance?
Report standard deviation when communicating with a general audience or when you need a measure in the original units (e.g., "heights vary by ±2.5 inches"). Report variance in technical contexts where its mathematical properties matter — ANOVA, regression analysis, portfolio optimization, and combining variances of independent variables (variances add; standard deviations don't).
What are the formulas for standard deviation and variance?
Population variance: σ² = Σ(xᵢ − μ)² / N. Sample variance: s² = Σ(xᵢ − x̄)² / (n−1). Population SD: σ = √σ². Sample SD: s = √s². The only difference between population and sample versions is the divisor: N vs n−1. The square root relationship means SD = √variance and variance = SD².
Can standard deviation be larger than variance?
Yes, when variance is between 0 and 1. Since SD = √variance, if variance = 0.25 then SD = 0.5, which is larger. When variance = 1, SD = 1 (they're equal). When variance > 1, SD < variance. For example, variance = 4 gives SD = 2. This is a purely mathematical relationship — the square root of a number less than 1 is greater than the number itself.
Why do we need both standard deviation and variance?
Each has unique advantages. Variance: additive for independent variables (Var(X+Y) = Var(X) + Var(Y)), used in ANOVA, regression, and mathematical statistics. Standard deviation: same units as data, interpretable with the empirical rule (68-95-99.7 rule), used in z-scores and confidence intervals. Together they provide a complete picture of data spread for both practical interpretation and mathematical analysis.