Variance vs. Standard Deviation: Key Differences Explained

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Variance is the average of the squared differences between each value in a data set and the mean. Standard deviation is the square root of the variance. Both measure how widely values are spread around their average, and both are zero only when every value is identical. The practical difference is units. Variance is expressed in squared units (dollars squared, percent squared), while standard deviation is in the same units as the original data. That is why standard deviation is the figure most people read and quote, and variance is the figure most formulas use underneath.

Variance vs. standard deviation: key differences

How it works

The formulas

For a population of N values with mean μ:

Population variance: σ² = Σ(xᵢ − μ)² ÷ N

Population standard deviation: σ = √σ²

For a sample of n values with sample mean x̄:

Sample variance: s² = Σ(xᵢ − x̄)² ÷ (n − 1)

Sample standard deviation: s = √s²

The steps are the same in both cases:

1. Calculate the mean.
2. Subtract the mean from each value to get its deviation.
3. Square each deviation.
4. Add the squared deviations.
5. Divide by N (population) or n − 1 (sample) to get the variance.
6. Take the square root to get the standard deviation.

Why the deviations are squared

If you simply averaged the raw deviations, the values above the mean would cancel the values below it, and the result would always be zero. Squaring makes every deviation positive, so they accumulate instead of offsetting.

Squaring has a second effect: it weights large deviations more than small ones. A value 10 units from the mean contributes 100 to the sum, while a value 2 units away contributes only 4. Variance is therefore sensitive to outliers. That sensitivity is useful when large moves are what you care about, and a drawback when a single unusual data point distorts the picture.

An alternative, the mean absolute deviation, averages the absolute deviations instead of squaring them. It avoids the squared units and the outlier weighting but lacks the algebraic properties that make variance convenient, which is why variance and standard deviation dominate in statistics and finance.

Population vs. sample: why divide by n − 1

Use the population formula when the data covers every member of the group you are describing, such as the returns of a fund in every year since it launched, if those years are the whole question. Use the sample formula when the data is a subset used to estimate a larger, unseen group, such as five years of returns used to estimate a fund’s long-run volatility.

Dividing by n − 1 instead of n (known as Bessel’s correction) compensates for the fact that a sample’s values sit closer, on average, to their own sample mean than to the true population mean. Without the correction, sample variance would tend to understate the population variance. The adjustment matters most for small samples: dividing by 4 instead of 5 changes the result by 25%, while dividing by 999 instead of 1,000 barely changes it.

In spreadsheets, the functions follow this split: VAR.S and STDEV.S for samples, VAR.P and STDEV.P for populations.

When standard deviation is larger than variance

People often assume variance is always the bigger number. It is only when the variance is greater than 1. For values between 0 and 1, the square root is larger than the number itself, so standard deviation exceeds variance. This comes up constantly with returns written as decimals: a variance of 0.04 corresponds to a standard deviation of 0.20, or 20%. The comparison of their sizes says nothing about risk; it is a side effect of squaring.

Example

The following figures are hypothetical, chosen to show the mechanics.

Suppose Fund A posted these annual returns over five years: 10%, −5%, 15%, 8%, 2%.

Step 1: Mean. (10 − 5 + 15 + 8 + 2) ÷ 5 = 30 ÷ 5 = 6%.

Steps 2 and 3: Deviations and squares.

• 10 − 6 = 4, squared = 16
• −5 − 6 = −11, squared = 121
• 15 − 6 = 9, squared = 81
• 8 − 6 = 2, squared = 4
• 2 − 6 = −4, squared = 16

Step 4: Sum of squares. 16 + 121 + 81 + 4 + 16 = 238.

Step 5: Variance.

• Population variance: 238 ÷ 5 = 47.6 (percent squared)
• Sample variance: 238 ÷ 4 = 59.5 (percent squared)

Step 6: Standard deviation.

• Population SD: √47.6 ≈ 6.90%
• Sample SD: √59.5 ≈ 7.71%

Treating these five years as a sample of the fund’s possible returns, the sample figures apply: variance of 59.5 %² and standard deviation of about 7.7%.

Notice how much the single −5% year drives the result: its squared deviation of 121 is more than half the total sum of 238. The standard deviation of 7.7% can be read directly against the 6% average return, while “59.5 percent squared” has no intuitive meaning.

If you express the same returns as decimals (0.10, −0.05 and so on), the sample variance becomes 0.00595 and the standard deviation 0.0771. The standard deviation is now the larger number, as described above, even though nothing about the fund has changed.

Comparison with Fund B. Now take a second hypothetical fund with returns of 7%, 5%, 6%, 8%, 4%. Its mean is also 6%. The deviations are 1, −1, 0, 2, −2; the squares sum to 10; the sample variance is 2.5 and the sample standard deviation is about 1.58%. Both funds averaged the same return, but Fund A’s yearly results were spread almost five times as widely, a difference the mean alone would hide.

Why it matters in investing

Standard deviation as a volatility measure

When analysts, advisors and fund documents refer to an investment’s volatility, they usually mean the standard deviation of its returns. A higher standard deviation means returns have historically landed farther from their average in both directions, so the outcome over any given period has been less predictable. A lower one means returns have clustered more tightly.

Higher volatility is not automatically bad. Assets with wider swings, such as stocks compared with short-term government bonds, have historically been associated with higher long-run return potential, which is the trade-off investors accept. Standard deviation is the denominator in the Sharpe ratio, which measures excess return per unit of volatility so that funds with different risk levels can be compared on one scale.

Reading standard deviation with the normal distribution

If returns were normally distributed, about 68% of outcomes would fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. Applied to the hypothetical Fund A, that would suggest roughly two years in three landing between −1.7% and 13.7% (6% ± 7.7%). This rule of thumb is widely used, but it only holds to the extent that returns actually follow a normal distribution, which financial returns often do not (see Limitations).

Variance in portfolio math

Variance becomes the more useful number once you combine assets. For a two-asset portfolio with weights w₁ and w₂, standard deviations σ₁ and σ₂, and correlation ρ:

σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂

Hypothetical example: a 50/50 portfolio of an asset with 20% standard deviation and one with 10%, with a correlation of 0.2.

• 0.25 × 0.04 = 0.0100
• 0.25 × 0.01 = 0.0025
• 2 × 0.5 × 0.5 × 0.2 × 0.20 × 0.10 = 0.0020
• Portfolio variance = 0.0145; portfolio standard deviation = √0.0145 ≈ 12.0%

The weighted average of the two standard deviations is 15%, but the portfolio’s is about 12%. That gap is the diversification effect, and it can only be computed through variance and correlation. Averaging standard deviations directly would overstate the risk whenever correlation is below 1. This is the core of modern portfolio theory: what matters is how assets move together, not just how volatile each one is alone.

Scaling over time

The same additive property governs how volatility scales with time. If returns in each period are independent, variance grows in proportion to the number of periods, so standard deviation grows with the square root of time. A monthly standard deviation of 4% annualizes to roughly 4% × √12 ≈ 13.9%, not 4% × 12 = 48%. The square-root rule depends on the independence assumption; if returns trend or mean-revert, it misstates annual volatility.

Volatility also affects compounding. For a given arithmetic average return, higher variance lowers the compound growth rate, which is one reason a fund’s average annual return and its CAGR can differ noticeably.

When to use SD and when to use variance

• Reporting or comparing spread: use standard deviation. It is in the data’s units and can be set against the mean.
• Combining risks, building portfolios, or running statistical models: use variance. It adds across independent variables and appears in regression, analysis of variance (ANOVA) and optimization formulas.

Limitations

Upside and downside count equally. A return 10 points above the mean raises variance exactly as much as a return 10 points below it. Most investors only mind the second kind. Semivariance and downside deviation address this by measuring only returns below the mean or below a target.

Outliers dominate. Because deviations are squared, one extreme observation can drive most of the result, as the −5% year did in the Fund An example. A single crash in the measurement window can make an asset look far riskier than in a window that excludes it, and vice versa.

Real returns are not perfectly normal. Financial returns often show fat tails (extreme moves occur more often than a normal distribution predicts) and skew. Standard deviation summarizes spread but says nothing about the shape of the distribution, so the 68/95/99.7 rule can understate the chance of large losses. Tail measures such as value at risk and conditional value at risk exist partly for this reason.

Results depend on the window and frequency. The standard deviation of daily, weekly or monthly returns over one, three or ten years can differ materially. Two figures are only comparable when they use the same period, frequency and annualization method.

History is not a forecast. Both measures describe past dispersion. Volatility changes over time, and a calm stretch can be followed by a turbulent one that the historical figure did not anticipate.

FAQs

The Bottom Line

Investors comparing funds should read standard deviation, because it sits in the same units as returns and can be set against the average. Anyone combining assets, annualizing volatility or running statistical models needs variance, since it adds across independent variables. Treat both as descriptions of past spread that ignore direction and distribution shape, and compare figures only when they share the same period and frequency.