Variance Calculator
Compute sample variance, population variance, standard deviation, coefficient of variation, pooled variance, and the chi-square 95% confidence interval from raw data, summary statistics, or two independent samples. Reproducible R, Python, and APA output.
Separators accepted: space, comma, tab, newline, semicolon. Minimum n = 2.
Result (n = 15)
Interpretation
Very high variability: spread exceeds 30% of the mean; verify the mean is meaningful and inspect for outliers or subpopulations.
How Variance Works and When Each Flavor Matters
Variance is the average squared distance from the mean, the canonical measure of dispersion. Karl Pearson (1894) introduced the term "standard deviation" for the square root of this quantity, and Ronald Fisher (1918) formalised "variance" itself while building the foundations of ANOVA. The choice of divisor matters: dividing the sum of squared deviations by n yields the population variance σ², an unbiased estimator only when the data are the entire population. Dividing by n − 1 yields the sample variance s², which corrects for the lost degree of freedom absorbed by estimating the mean from the same sample. This Bessel correction (Bessel 1838) is the reason every introductory text and every journal expects the n − 1 divisor for inferential work.
The two derived quantities you actually report depend on the audience. Standard deviation (SD = √s²) sits in the original units of measurement, which makes it the natural choice for clinical results, descriptive tables, and forest-plot summaries. Variance is awkward to interpret directly (mmHg² has no intuitive meaning), but it is the natural quantity for additive decomposition: the variance of a sum of independent random variables equals the sum of their variances, which is why ANOVA, meta-analysis, mixed-effects models, and regression diagnostics all work in variance units. The coefficient of variation, CV = SD / mean × 100%, normalises variability to the scale of the mean and is the standard precision metric in laboratory medicine, assay validation, and quality control (Westgard rules).
For confidence intervals on a variance, the chi-square distribution is the workhorse: under normality, (n − 1)s² / σ² follows χ²(n − 1), so the 95% CI for σ² is [(n − 1)s² / χ²(0.975, n − 1), (n − 1)s² / χ²(0.025, n − 1)]. The interval is asymmetric because χ² is right-skewed, which means the upper bound is always farther from s² than the lower bound. This CI is sensitive to departures from normality; for heavy-tailed or skewed data, bootstrap intervals (Efron 1979) are more reliable and remain valid without distributional assumptions. The calculator reports the chi-square CI by default; if your data are clearly non-Gaussian, treat it as indicative and re-run with the boot package in R.
When comparing two groups, pooled variance is the weighted average of the within-group variances: s²_p = [(n1 − 1)s1² + (n2 − 1)s2²] / (n1 + n2 − 2). It is the denominator for Student's two-sample t-test, for Cohen's d standardised to the pooled within-group SD, and for ANOVA mean-square error. Pooling is only valid when the population variances are equal, the homoscedasticity assumption. The F-test for equal variances (F = larger / smaller variance with df1 = n1 − 1 and df2 = n2 − 1) is the classical check, though it is sensitive to non-normality; Levene's test or the Brown-Forsythe variant is more robust. If the variances differ, switch to Welch's t-test, which does not pool and adjusts the degrees of freedom using the Welch-Satterthwaite approximation (Welch 1947).
Variance feeds directly into related tools. To run the two-sample t-test from the pooled SD (or with Welch's correction), use the two-sample t-test calculator. To convert standard error to standard deviation (a common requirement when extracting data from published trials), use the SE to SD converter. To plan an a priori sample size given a target SD and effect size, use the sample size calculator. For full data extraction, robust variance estimation, and APA-formatted reporting from a PhD statistician, our statistical analysis service covers everything end-to-end.
How to Use This Calculator
Pick a mode
Raw data if you have the observations themselves; summary stats if a paper reports only n, mean, and SD; pooled for two independent samples.
Paste or enter the values
Raw-data accepts spaces, commas, tabs, semicolons, or newlines. Summary needs n, mean, SD. Pooled needs n and SD per group.
Read s², σ², SD, and CV
Sample variance is highlighted by default. The calculator also returns population variance, SD, coefficient of variation, SEM, and the chi-square 95% CI for σ².
Run the F-test (pooled mode)
The pooled tab returns the F-statistic and two-tailed p-value for equality of variances so you can decide between Student's and Welch's t-test.
Copy R, Python, or APA
Reproducible code snippets paste straight into RStudio or Jupyter, and the APA sentence uses italicized statistic letters and p < .001 truncation.
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Variance Formulas at a Glance
Sample variance (s²)
Σ(xi − x̄)² / (n − 1)
Unbiased estimator of σ². Use for any inferential analysis (CIs, t-tests, ANOVA, regression). The default in R's var(), Python's np.var(ddof=1), and Excel's VAR.S.
Population variance (σ²)
Σ(xi − μ)² / n
Use only when the data are the entire population. The default in np.var() without ddof and in Excel's VAR.P. Convert from s² via σ² = s² × (n − 1) / n.
Standard deviation
s = √s² or σ = √σ²
Square root of variance, in the original units. Reported alongside the mean in most descriptive tables. Used as the denominator of Cohen's d, z-scores, and standardised effect sizes.
Coefficient of variation
CV = (s / |x̄|) × 100%
Unit-free relative dispersion. Only meaningful on a ratio scale with a positive mean. Common precision metric in laboratory medicine and assay validation.
Pooled variance
s²_p = [(n1−1)s1² + (n2−1)s2²] / (n1 + n2 − 2)
Weighted average of two within-group variances. Used in Student's t-test, ANOVA mean-square error, and Cohen's d standardised to the pooled SD. Valid only under homoscedasticity.
Chi-square CI for σ²
[(n−1)s² / χ²(1−α/2, n−1), (n−1)s² / χ²(α/2, n−1)]
Asymmetric because χ² is skewed. Assumes normality; for non-Gaussian data, use bootstrap CIs instead. APA and Cochrane Handbook accept the chi-square interval for well-behaved data.
Frequently Asked Questions
What is variance in statistics?
Variance measures how far a set of numbers spreads out from their mean. The sample variance, s², is the average squared deviation of each observation from the sample mean using a divisor of n − 1 (Bessel's correction), and it is the unbiased estimator of the population variance σ². The standard deviation is the square root of the variance, expressed in the original units of measurement. Variance is foundational to inferential statistics: confidence intervals, t-tests, F-tests, ANOVA, and regression all depend on accurate variance estimates.
What is the difference between sample variance and population variance?
Sample variance divides the sum of squared deviations by n − 1 and corrects for the fact that the sample mean itself absorbs one degree of freedom, making s² an unbiased estimator of the true population variance σ². Population variance divides by n and is only appropriate when the data are the entire population, not a sample drawn from it. Spreadsheets and calculators often offer both; for any inferential analysis (CIs, p-values, ANOVA), use the sample variance with the n − 1 divisor. Bessel's correction is named after Friedrich Bessel, who derived it in the early 19th century.
How do I calculate variance from raw data?
Compute the mean of the values, subtract the mean from each observation, square each deviation, sum the squared deviations, then divide by n − 1 for sample variance or by n for population variance. In symbols: s² = Σ(xi − x̄)² / (n − 1). The sample standard deviation is s = √s². The calculator's raw-data tab performs this computation along with skewness, kurtosis, range, IQR, the standard error of the mean, and a chi-square confidence interval for σ².
How do I calculate variance from a mean and standard deviation?
Square the standard deviation: variance = SD². If the source reports a sample SD (the journal convention), then s² is the sample variance. If you also need the population variance from a reported sample SD, multiply by (n − 1)/n to remove Bessel's correction. The calculator's summary-stats tab handles both directions and additionally returns the coefficient of variation, the standard error of the mean (SD / √n), and a chi-square confidence interval for σ².
What is the coefficient of variation?
The coefficient of variation (CV) is the standard deviation expressed as a percentage of the mean: CV = (s / |x̄|) × 100%. It is a unit-free measure of relative variability that lets you compare dispersion across variables on different scales, for example, comparing the variability of blood pressure (mmHg) to that of body weight (kg). CV is only meaningful when the data are on a ratio scale with a true zero and a positive mean. In laboratory medicine a CV under 10% is usually considered acceptable for assay precision; in field epidemiology, 30% is more typical.
When do I need a pooled variance?
Pooled variance is the weighted average of two independent sample variances, used in Student's two-sample t-test, in ANOVA mean-square error, and as the denominator for Cohen's d standardized to the pooled within-group SD. The formula is s²_p = [(n1 − 1)s1² + (n2 − 1)s2²] / (n1 + n2 − 2). Pooling is only valid when the population variances are equal (the homoscedasticity assumption). If an F-test rejects equal variances, switch to Welch's t-test, which does not pool. The calculator's two-sample tab reports the pooled variance, pooled SD, and the F-test for equal variances so you can choose.
How do I test whether two variances are equal?
The F-test for equal variances computes F = larger variance / smaller variance with df1 = n1 − 1 (numerator) and df2 = n2 − 1 (denominator). A large F (above the critical value at α = 0.05) suggests the variances differ. The F-test is sensitive to departures from normality, so for non-Gaussian data prefer Levene's test or the Brown-Forsythe variant, both implemented in R (car::leveneTest) and Python (scipy.stats.levene). The calculator returns the two-tailed F-test p-value, which you can use to decide between Student's and Welch's t-test.
How do I build a confidence interval for a variance?
Use the chi-square distribution. For a sample of size n with sample variance s², a 100(1 − α)% CI for the population variance σ² is [(n − 1)s² / χ²(1 − α/2, n − 1), (n − 1)s² / χ²(α/2, n − 1)]. The interval is asymmetric because the chi-square distribution is skewed, and it assumes the data are approximately normal. For non-normal data, bootstrap intervals are more reliable. The calculator's raw-data and summary-stats tabs both return the 95% CI for σ² automatically.
What is the relationship between variance and standard deviation?
Variance and standard deviation are two views of the same spread: SD = √variance, and variance = SD². Variance is in squared units of the original variable (squared kilograms, squared mmHg), so it is harder to interpret directly; standard deviation is in the original units. Variance is the natural quantity for additive decomposition (the variance of a sum of independent variables equals the sum of their variances), which is why ANOVA, regression, and meta-analysis use variance components rather than SDs.
What is the variance of common probability distributions?
Normal(μ, σ²) has variance σ². Binomial(n, p) has variance np(1 − p). Poisson(λ) has variance λ (equal to its mean, the basis of the Poisson assumption). Uniform(a, b) has variance (b − a)² / 12. Exponential(λ) has variance 1/λ². The variance is a property of the distribution itself, independent of sample size. To check whether your data plausibly came from one of these distributions, plot the histogram and compare the sample variance to the theoretical variance implied by the fitted parameters.
Related Research Tools
To run a two-sample t-test from the pooled SD (or with Welch's correction), see the two-sample t-test calculator. To convert standard error to standard deviation (a common requirement when extracting data from published trials), see the SE to SD converter. For the median-and-IQR to mean-and-SD transformation used in meta-analysis, see the median & IQR to mean & SD estimator. To plan an a priori sample size given a target SD, see the sample size calculator. For confidence intervals on means, proportions, and ratio measures, see the confidence interval calculator. And to convert a test statistic into a p-value across z, t, chi-square, F, and r distributions, see the p-value calculator.
Reviewed by
Dr. Sarah Mitchell
Dr. Sarah Mitchell holds a PhD in Biostatistics from Johns Hopkins Bloomberg School of Public Health and has over 15 years of experience in systematic review methodology and meta-analysis. She has authored or co-authored 40+ peer-reviewed publications in journals including the Journal of Clinical Epidemiology, BMC Medical Research Methodology, and Research Synthesis Methods. A former Cochrane Review Group statistician and current editorial board member of Systematic Reviews, Dr. Mitchell has supervised 200+ evidence synthesis projects across clinical medicine, public health, and social sciences. She reviews all Research Gold tools to ensure statistical accuracy and compliance with Cochrane Handbook and PRISMA 2020 standards.
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