What Is Variance? Definition, Formula and How to Interpret It

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Variance is a statistical measure of how widely the values in a data set are spread around their mean: the average of the squared differences between each value and the mean. A small variance means the values cluster tightly around the average; a large variance means they are scattered far from it. In finance, variance applied to a series of returns describes how much those returns fluctuate, which makes it a basic measure of volatility and risk.

How Variance Works

Variance answers one question: if you pick a value from the data set, how far from the average should you expect it to be? Simply averaging the raw deviations from the mean does not work, because positive and negative deviations always cancel out and sum to zero. Squaring each deviation solves this. Every squared term is zero or positive, so the result reflects the full amount of spread regardless of direction.

The Variance Formula

For an entire population:

σ² = Σ (xᵢ − μ)² / N

For a sample drawn from a larger population:

s² = Σ (xᵢ − x̄)² / (n − 1)

Where:

• xᵢ = each value in the data set
• μ or x̄ = the population mean or the sample mean
• N or n = the number of values
• Σ = the sum across all values

The n − 1 denominator is known as Bessel’s correction. Sample values sit slightly closer to their own mean than to the true population mean, so dividing by n − 1 instead of n makes the sample variance an unbiased estimate of the population variance. Historical returns are usually treated as a sample of possible outcomes, so finance typically uses the n − 1 version.

Steps to Calculate Variance

1. Calculate the mean of the data.
2. Subtract the mean from each value to get its deviation.
3. Square each deviation.
4. Add up the squared deviations (the “sum of squares”).
5. Divide by N for a population or by n − 1 for a sample.

In Excel and Google Sheets, VAR.P returns the population variance and VAR.S returns the sample variance.

Interpreting Variance

A few properties shape how to read a variance value:

• Variance is never negative. It is zero only when every value in the set is identical; a negative result signals a calculation error.
• It is in squared units. Returns measured in percent give a variance in “percent squared,” which has no intuitive meaning, so the raw number is hard to read on its own.
• It is relative. A variance of 0.007 is neither high nor low in isolation. It becomes meaningful only when compared with another asset, another period or a benchmark measured on the same basis (same return frequency, same units).
• Higher variance means wider outcomes. A higher variance produces a flatter, wider probability distribution; a lower variance produces a narrow, peaked one.

Example of Variance

In a hypothetical example, two funds each return an average of 8% a year over four years.

Fund A: 8%, 12%, −4%, 16% Fund B: 7%, 9%, 8%, 8%

Working in decimals for Fund A, the mean is (0.08 + 0.12 − 0.04 + 0.16) / 4 = 0.08.

The sum of squares is 0.0224. Treating the four years as a sample, the variance is 0.0224 / 3 ≈ 0.00747. The standard deviation is √0.00747 ≈ 0.0864, or about 8.6%. (As a population, the variance would be 0.0224 / 4 = 0.0056, with a standard deviation of about 7.5%.)

For Fund B, the deviations are −0.01, 0.01, 0, and 0. The sum of squares is 0.0002, the sample variance is about 0.000067, and the standard deviation is roughly 0.8%.

Both funds have the same average return, yet Fund A’s variance is more than 100 times larger: an investor in Fund A experienced a losing year and a much wider range of results.

Why Variance Matters in Finance

Variance is the starting point for most quantitative risk measures:

• Volatility. Annualized volatility, the figure most often quoted for stocks and funds, is a standard deviation, which is derived directly from variance.
• Portfolio construction. Harry Markowitz’s modern portfolio theory (1952) defines portfolio risk through variance. For two assets with weights w₁ and w₂, portfolio variance is w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(1,2). Because the covariance term can be small or negative, combining assets can produce a portfolio variance lower than the weighted average of the individual variances. This is the mathematical basis of diversification and the efficient frontier.
• Risk-adjusted returns. The Sharpe ratio divides excess return by standard deviation, so any change in variance changes the ratio.
• Comparing time periods. If returns are assumed to be independent from period to period, variance scales with time: monthly variance multiplied by 12 approximates annual variance, while standard deviation scales by the square root of 12. That assumption does not always hold, so annualized figures are approximations.

Variance also says nothing about the level of returns. A fund can have low variance and a poor average return. Pair it with a return measure such as CAGR or ROI to see both sides.

Limitations of Variance

• Outliers carry extra weight. Squaring magnifies large deviations. In the Fund An example, the single −4% year contributes 0.0144 of the 0.0224 total, nearly two-thirds of the variance.
• Upside and downside count equally. A surprise 30% gain raises variance just as much as a 30% loss. Investors who care mainly about losses may prefer downside measures such as semivariance.
• It is backward-looking. Historical variance describes what happened over the chosen window and frequency. A different period or a switch from daily to monthly returns can produce a different picture, and past dispersion does not guarantee future dispersion.
• It does not describe shape. Two return series can share a variance but differ in skew or in how often extreme results occur (kurtosis). Variance alone does not reveal fat tails.

Variance vs. Related Terms

Variance vs. standard deviation. Standard deviation is the square root of variance. It carries the same information but in the original units, so a standard deviation of 8.6% can be read directly against an average return of 8%. Variance is often used in calculations (variances of independent variables add, standard deviations do not), while standard deviation is used for reporting.

Population vs. sample variance. The gap between dividing by N and by n − 1 matters most for small data sets and shrinks as the number of observations grows.

Variance vs. covariance. Variance measures how one variable moves around its own mean. Covariance measures how two variables move together. The variance of a variable is its covariance with itself.

Statistical variance vs. budget variance. In accounting and corporate budgeting, “variance” means the difference between an actual figure and a planned or standard figure, such as spending $10,000 over budget. That usage involves no squaring or averaging and is unrelated to the statistical measure.

FAQs

The Bottom Line

Variance matters to investors comparing how consistently funds or assets deliver their returns, and to anyone building portfolios with modern portfolio theory. Because it sits in squared units and treats gains and losses alike, read it through its square root, the standard deviation, and compare it only on the same basis. Pair it with a return measure, since low variance says nothing about whether average returns are good.