σ² Variance Calculator

Calculate population and sample variance (σ² and s²) with step-by-step squared-deviation working. See how variance relates to standard deviation for any numeric data set.

Need standard deviation and the empirical rule?

This page focuses on variance (σ² and s²). For SD-first analysis with 68-95-99.7 rule context, use the Standard Deviation Calculator →

Data Set (comma or space separated)

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Mean (μ)

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Variance (σ²)

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Std Dev (σ)

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Count (n)

Formula (with your data)

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Deviation Table

xᵢ	xᵢ − μ	(xᵢ − μ)²
Sum	—	—

What is Variance?

Variance is the average of squared deviations from the mean. Population variance (σ²) divides by N; sample variance (s²) divides by N−1 (Bessel’s correction) when data is a sample from a larger population. Variance is foundational in statistics, finance (portfolio risk), and quality control.

Because variance is in squared units, practitioners often report standard deviation (the square root of variance) for interpretability. This page focuses on variance computation, population vs sample choice, and when σ² vs s² applies.

For standard deviation as the primary output — with mean, median, empirical rule, and SD-focused examples — use the Standard Deviation Calculator. SD and variance are two views of the same spread; pick the page that matches how you need to report results.

Variance Formulas

Population Variance

σ² = Σ(xᵢ − μ)² / N

Sample Variance (Bessel's correction)

s² = Σ(xᵢ − x̄)² / (N − 1)

Standard Deviation

σ = √σ²

How to Calculate Variance

1

Choose Population or Sample

Use population variance (σ²) for complete data sets. Use sample variance (s²) when your data is a subset of a larger population.
2

Calculate the Mean

Add all values and divide by the count. This is the reference point for measuring how spread out the data is.
3

Find Squared Deviations

Subtract the mean from each value (deviation), then square each deviation. Squaring removes negatives and amplifies larger differences.
4

Average the Squared Deviations

Divide the sum of squared deviations by N (population) or N−1 (sample). This is your variance. Take the square root for standard deviation.

Worked Example

Data set: 2, 4, 4, 4, 5, 5, 7, 9

N = 8 | Sum = 40 | Mean = 40 ÷ 8 = 5

Σ(xᵢ − 5)² = 9+1+1+1+0+0+4+16 = 32

Population Variance = 32 ÷ 8 = 4

Population SD = √4 = 2

Sample Variance = 32 ÷ 7 ≈ 4.571

How the Variance Calculator Works

Formula, assumptions, and calculation steps for this statistics tool.

Formula Used

Sample Variance = sum((x - mean)^2) / (n - 1)

Methodology

Statistics calculators organize sample data, apply the selected descriptive or inferential formula, and report the statistic with interpretation.

Calculation Steps

Enter raw values or summary statistics.
Clean separators and count the sample size.
Apply the relevant statistic, probability, or confidence formula.
Display the result with context such as degrees of freedom, percentile, or strength.

Assumptions and Limits

Samples should be representative of the population being studied.
Normality or independence assumptions apply only where the selected method requires them.
Rounded results may differ slightly from spreadsheet software.

Frequently Asked Questions

What is variance?

Variance measures how far a set of numbers is spread out from their mean. It is the average of the squared differences from the mean. A variance of 0 means all values are identical; a high variance means the data is widely spread.

What is the difference between population and sample variance?

Population variance (σ²) divides by N and is used when you have data for every member of a group. Sample variance (s²) divides by N−1 (Bessel's correction) and is used when your data is a sample from a larger population. The correction prevents underestimation of the true variance.

Why do we square the deviations?

Squaring serves two purposes: it eliminates negative values (so positive and negative deviations don't cancel each other out), and it gives extra weight to larger deviations, making variance more sensitive to outliers than simpler measures like mean absolute deviation.

What is the relationship between variance and standard deviation?

Standard deviation is simply the square root of variance. Variance is in squared units of the original data, which can be hard to interpret. Standard deviation is in the same units as the data, making it easier to understand in context.

Real-World Applications

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Investment Portfolio Risk Analysis

In financial portfolio theory (Markowitz mean-variance optimisation), variance is the mathematical measure of investment return volatility — a higher variance means returns fluctuate more widely around the average, representing greater risk. Portfolio variance is not a simple sum of individual asset variances; it also incorporates the covariance between assets — the key insight of modern portfolio theory is that combining assets with low covariance reduces portfolio variance below the average of individual asset variances, quantifying the benefit of diversification.

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Quality Control & Manufacturing Process Analysis

Statistical Process Control (SPC) and Six Sigma use variance (and its square root, standard deviation) to measure manufacturing process consistency. A production process with low variance produces parts that are consistently close to the target specification; high variance indicates an unstable process that produces parts distributed widely around the target. Control charts (Shewhart charts) plot sample variance over time to detect when a process has shifted from its normal operating variance — the signal for investigation and corrective action.

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Experimental Science & Research Analysis

Researchers use sample variance to quantify measurement uncertainty, experimental error, and biological variability in their data. Analysis of Variance (ANOVA) decomposes total data variance into variance attributed to experimental treatments (between-group variance) and variance attributed to random error (within-group variance) — the F-statistic is the ratio of these two variances, and is used to test whether treatment effects are statistically significant relative to random variation.

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Machine Learning Model Evaluation

In machine learning, the bias-variance trade-off describes how model complexity affects predictive performance. Variance in this context refers to how much the model's predictions change across different training sets — high variance models (overfitted) produce very different predictions when trained on slightly different data, indicating overfitting. Mean Squared Error (MSE) — the standard regression loss function — is itself a variance-related metric: MSE = variance of prediction errors + bias² of prediction errors (the bias-variance decomposition theorem).

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Survey Research & Social Science

Social scientists use variance to summarise the spread of survey responses — a question about life satisfaction rated 1–10 might have a mean of 7.2 with a variance of 2.1 in one country and a mean of 7.0 with a variance of 8.5 in another, revealing that the second country has similar average satisfaction but far greater inequality in satisfaction levels. Factor analysis, principal component analysis (PCA), and structural equation modelling all use variance and covariance matrices as their mathematical input.

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Options Pricing & Financial Derivatives

The Black-Scholes options pricing model uses the variance of the underlying asset's log-returns (expressed as volatility, σ, the annualised standard deviation) as the primary driver of option value. Higher variance (volatility) means greater probability of large price movements, which increases the value of out-of-the-money options. Implied volatility — backed out from observed option prices — represents the market's consensus estimate of future price variance and is one of the most closely watched indicators in financial markets.

Common Mistakes

Using population variance (divide by N) when sample variance (divide by N−1) is required

The most common variance calculation error is using the wrong denominator. When the data is a sample drawn from a larger population (virtually all practical statistical analysis), sample variance (÷N−1) should be used — it produces an unbiased estimate of the true population variance. Population variance (÷N) is only appropriate when the data represents the complete population with no unobserved members. Excel's VAR() and VAR.S() functions use N−1 (sample); VARP() and VAR.P() use N (population). Spreadsheet users who use VARP() when they should use VAR() systematically under-estimate variance.

Interpreting variance without considering its squared-units problem

Variance is expressed in the square of the data's units — variance of weights in kilograms is in kg², variance of heights in centimetres is in cm². This makes variance impossible to interpret directly against the data scale: a variance of 25 kg² in a weight dataset doesn't directly tell you the typical deviation is 5 kg without taking the square root (standard deviation). For human-interpretable spread measures, report standard deviation alongside or instead of variance — variance is the statistically fundamental quantity, but standard deviation is the interpretable one.

Computing variance on ordinal or categorical data

Variance is only mathematically meaningful for continuous or interval-scale data — data where differences between values are meaningful and equal. Applying a variance calculation to ordinal survey responses ("rate 1–5 where 1 = strongly disagree") treats the response categories as having equal spacing and a meaningful arithmetic mean, which is a contested assumption in social science methodology. For truly ordinal data, non-parametric spread measures (interquartile range, median absolute deviation) are more appropriate than variance.

Not checking for outliers before computing variance

Variance is highly sensitive to outliers because deviations are squared — an outlier 10 units from the mean contributes 100 to the sum of squared deviations, while a typical data point 2 units from the mean contributes only 4. A single extreme outlier can dominate the variance calculation, producing a misleadingly high value that misrepresents the spread of the majority of the data. Always inspect data for outliers (box plots, dot plots) before calculating and reporting variance, and consider robust spread measures (median absolute deviation) if outliers are present.

Confusing variance with range or interquartile range

Range (max − min) and interquartile range (Q3 − Q1) are alternative dispersion measures that describe spread using specific data points rather than aggregating across all observations. Variance uses all data points and gives more weight to extreme values. For symmetric, roughly normal distributions, variance (and standard deviation) is the most informative spread measure. For skewed distributions or datasets with outliers, IQR is more robust. None of these measures is universally superior — the correct choice depends on the data distribution and the analytical purpose.

Statistical Dispersion Measures Comparison

Measure	Formula	Outlier Sensitivity
Range	Max − Min	Extreme (uses only 2 points)
IQR	Q3 − Q1	Resistant (central 50%)
Variance	Σ(x−μ)² / (n−1)	High (deviations squared)
Std Deviation	√Variance	High (same data as variance)
Mean Abs Deviation	Σ\|x−μ\| / n	Moderate

References

Freedman, D., Pisani, R. and Purves, R. Statistics, 4th edition. W.W. Norton, 2007.
Field, A. Discovering Statistics Using IBM SPSS Statistics. SAGE, 2018.
Hogg, R.V., McKean, J. and Craig, A. Introduction to Mathematical Statistics. Pearson, 2018.
NIST/SEMATECH. e-Handbook of Statistical Methods — Measures of Scale. itl.nist.gov, 2024.
Markowitz, H. "Portfolio Selection." The Journal of Finance, 1952.