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Variance and standard deviation both describe how spread out data is around its mean. Variance is the average squared deviation from the mean; standard deviation is the square root of variance. Standard deviation is usually easier to interpret because it is expressed in the data’s original units. Variance is often more useful in statistical calculations, where squared deviations are mathematically convenient.
Variance and standard deviation at a glance
| Feature | Variance | Standard deviation |
|---|---|---|
| Relationship | The average squared deviation from the mean | The positive square root of variance |
| Units | Squared units, such as dollars squared or cm² | Same units as the original data |
| Typical use | Statistical models, ANOVA, and calculations involving squared variation | Reporting and explaining the spread of observations |
| Symbols | σ² for a population; s² for a sample | σ for a population; s for a sample |
Neither measure is inherently better. They describe the same underlying spread, but serve different purposes.
What variance measures
Variance is the mean of the squared differences between each observation and the mean. To calculate it, find the mean, subtract it from each value, square each difference, and average those squared differences. Squaring prevents positive and negative deviations from canceling out. It also gives larger deviations more influence: a deviation of 10 contributes 100, while a deviation of 2 contributes 4. NIST describes variance as a measure based on squared deviations.
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1Fix the driver behind crashes, sound loss and screen glitches2Repair Windows errors before they cause bigger problems3Scan for outdated or missing drivers - takes under a minuteWhat standard deviation measures
Standard deviation is the positive square root of variance. Taking the square root returns the measure to the original scale: if data is in dollars, standard deviation is in dollars, while variance is in dollars squared. The same applies to measurements in inches, milliseconds, or other units.
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It is common to call standard deviation a “typical distance” from the mean, but that is an intuition rather than its exact definition. It is the root-mean-square deviation—not the arithmetic mean of the absolute distances.
Population and sample formulas
The correct denominator depends on what the data represents. Use the population formula if the dataset includes every member of the population of interest. Use the conventional sample formula when the observations are a sample used to estimate a larger population.
Population
For a population of N values with population mean μ:
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σ² = Σ(xᵢ − μ)² / N
σ = √[Σ(xᵢ − μ)² / N]
Sample
For a sample of n values with sample mean x̄:
s² = Σ(xᵢ − x̄)² / (n − 1)
s = √[Σ(xᵢ − x̄)² / (n − 1)]
The sample variance formula’s n − 1 denominator is called Bessel’s correction. Estimating the mean from the sample uses one degree of freedom, and deviations from the sample mean tend to be smaller than deviations from the unknown population mean. Dividing by n − 1 makes s² an unbiased estimator of population variance under the usual assumptions. That unbiasedness applies to the variance estimator, not generally to the sample standard deviation itself. Penn State explains the conventional sample formula and its rationale.
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This is not a universal rule for every estimation goal: some procedures, including maximum-likelihood estimation in common settings, divide by n. Choose the denominator that matches the quantity and estimator you intend to report.
Worked example: the same values, two assumptions
Take the values 2, 4, 4, 4, 5, 5, 7, 9. Their mean is 5. The deviations from the mean are −3, −1, −1, −1, 0, 0, 2, 4; squaring them gives 9, 1, 1, 1, 0, 0, 4, 16. Their sum is 32.
- If these eight values are the whole population: variance = 32 ÷ 8 = 4; standard deviation = √4 = 2.
- If these eight values are a sample: sample variance = 32 ÷ 7 ≈ 4.571; sample standard deviation = √(32 ÷ 7) ≈ 2.138.
The observations have not changed. The answer changes because the population-versus-sample assumption changes.
When to use variance and when to use standard deviation
Use standard deviation when you want to describe how much individual observations vary in the same units as the measurements—for example, the spread of test scores, delivery times, or manufacturing dimensions. It is generally the clearer choice for communicating variability to readers.
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Use variance when a calculation or model is built around squared deviations. Variance is central to ANOVA, variance-component analysis, mean squared error, covariance matrices, and many regression and uncertainty calculations. In uncertainty work, variance may be used in calculations while its square root is reported as a standard uncertainty. NIST’s uncertainty guidance illustrates the relationship between variance and standard uncertainty.
When combining independent sources of uncertainty, variances can be combined under the relevant assumptions; standard deviations generally cannot simply be added. The model’s assumptions matter, so don’t apply a combination rule without checking them.
Outliers, shape, and what these measures leave out
Both measures are sensitive to outliers because variance is based on squared deviations and standard deviation is its square root. A distant value can raise both substantially. That sensitivity can be useful when large errors should count heavily, but it can make the measures less representative of a skewed or contaminated dataset. Variance and standard deviation rank datasets identically when calculated from the same values, since one is a monotonic transformation of the other.
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Neither measure describes the complete shape of a distribution. Two datasets can have the same mean and standard deviation yet differ in skewness, tail behavior, or clusters. Use a histogram or box plot, and consider the interquartile range or median absolute deviation for a more outlier-resistant summary. The range is simple but depends strongly on the two most extreme observations.
Standard deviation does not prove that data is normally distributed. For an approximately normal, bell-shaped distribution, the empirical rule says about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Those are approximations for normal data, not universal percentages. NIST discusses standard deviation in the context of normal-distribution and process behavior.
Standard deviation is not standard error
Standard deviation describes spread among individual observations. Standard error describes the estimated spread of a statistic, often the sample mean. For independent observations under the usual conditions, the standard error of the sample mean is:
SE(x̄) = s / √n
A larger sample can have the same underlying standard deviation but a smaller standard error for its mean. Use standard deviation to describe individual-level variability; use standard error when discussing the precision of an estimated mean.
How changes of scale affect the measures
For a transformation Y = aX + b:
Var(aX + b) = a² Var(X)SD(aX + b) = |a| SD(X)
Adding a constant shifts all values but does not change either measure. Multiplying values by a factor changes standard deviation by the factor’s absolute value and variance by its square. For example, converting meters to centimeters multiplies standard deviation by 100 and variance by 10,000.
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A coefficient of variation, often written CV = s / x̄ and expressed as a percentage, describes spread relative to the mean. It is most meaningful for ratio-scale data with a meaningful zero and a positive, nonzero mean; it can mislead when the mean is near zero or values can be negative.
Calculate variance and standard deviation in a spreadsheet
First decide whether your values are a sample or the complete population; a spreadsheet cannot make that modeling decision for you. Then choose the matching function:
| Assumption | Variance | Standard deviation |
|---|---|---|
| Sample | Excel VAR.S(range)Google Sheets VAR(range) |
Excel STDEV.S(range)Google Sheets STDEV(range) |
| Population | Excel VAR.P(range)Google Sheets VARP(range) |
Excel STDEV.P(range)Google Sheets STDEV.P(range) or STDEVP(range) |
For new Excel work, the explicit .S and .P function names make the assumption visible; older names may remain for compatibility. See Microsoft’s documentation for VAR.S, VAR.P, and STDEV.P. Google Sheets documents VAR, VARP, and STDEV.
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Numerical calculations: avoid an unstable shortcut
For large values that are close together, calculating variance as the difference between two large quantities—such as [Σxᵢ² − n(x̄)²] / (n − 1)—can lose numerical precision. Use software that applies a numerically stable algorithm rather than implementing the raw-sums shortcut yourself. NIST warns about instability in raw sums of squares.
Quick Recap
A quick decision rule
- Decide whether your dataset is the complete population or a sample from a larger population.
- Use standard deviation to explain spread in the original units.
- Use variance when the model, formula, or decomposition depends on squared variation.
- If outliers or skew are substantial, supplement these measures with a plot and a robust summary such as the median and interquartile range.
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