Sigma²: variance | Sigma² education

Same average

Two funds. One mean.

These five-year examples both average 8%. Sum either column and the total is 40 percentage points; divide by five and the arithmetic mean is 8%. Yet their paths are visibly different.

Five illustrative annual returns
Year	Fund A	Fund B
1	+10%	+30%
2	+6%	−14%
3	+12%	+25%
4	+4%	−10%
5	+8%	+9%
Mean	8%	8%

Fund A stays close to its mean. Fund B moves far above and below it. The difference is dispersion: how far individual data points scatter around the mean.

Variance comes first

To calculate standard deviation, a statistical measure of how much an investment's returns fluctuate around their average over time, we first calculate variance: the average squared distance of returns from that average.

Population variance

σ² = Σ(xᵢ − μ)² / N

Population-variance equation key
σ²	Sigma squared; population variance
Σ	Sigma, the symbol for summation: add all values that follow
xᵢ	An individual return or observation
μ	Mu; the average (mean) return
N	Total number of observations or returns

Once variance is calculated, standard deviation is obtained by taking its square root. This returns the measure to the same percentage-point units as the original returns.

Standard deviation

σ = √σ²

Why signed deviations cancel

Because the arithmetic mean acts as a dataset's natural center of gravity, the sum of all distances above the average will always cancel out the sum of the distances below it. To fix this issue, we must square the individual deviations before summing them. Squaring these numbers is mathematically essential: it eliminates the negative signs to prevent this mutual cancellation, while simultaneously ensuring that extreme market swings are penalized proportionally to the true risk they introduce.

Population-variance calculation using percentage-point deviations
Measure	Fund A	Fund B
Deviations from 8%	+2, −2, +4, −4, 0	+22, −22, +17, −18, +1
Squared deviations	4, 4, 16, 16, 0	484, 484, 289, 324, 1
Total	40	1,582
Variance (total ÷ 5)	8.0	316.4

This uses the population convention because all five displayed years are being summarized. When a sample estimates a broader population, analysts often divide by n − 1 instead. Either convention must be stated.

The number and its readable twin

Variance is expressed in squared return units: percentage points squared in this example. It is useful in calculations but awkward to read beside returns. Its square root is standard deviation, or sigma (σ), which returns the same information to percentage-point units.

Variance compared with standard deviation
Measure	Fund A	Fund B
Variance (σ²)	8.0	316.4
Standard deviation (σ)	2.8%	17.8%

Standard deviation (σ) and variance (σ²) represent the exact same underlying risk data, but they serve different analytical purposes. While standard deviation is the preferred metric for clear, everyday communication, variance is the mathematically essential component used to aggregate risk in portfolio optimization.

Build a portfolio

See how five variances combine

Everything so far has measured the variance of a single return stream. A real portfolio holds many at once, and its variance is not simply the average of the pieces. Whether the holdings' swings pile up or partly cancel depends on how much they move together. Set five positions below, give each a weight, an expected return, and a volatility, then use the single dial to change how closely they track one another and watch the portfolio's variance respond.

The dial sets one shared correlation, ρ, for how tightly the five holdings move together, from nearly independent to near lockstep. The related concept, covariance, has its own page. Here the shared correlation dial is simply a lever, so you can see what co-movement does to variance.

σₚ² = Σ Σ wᵢ wⱼ σᵢ σⱼ ρᵢⱼ

Portfolio-variance equation key
σₚ²	Variance of the whole portfolio
wᵢ, wⱼ	The weights of positions i and j
σᵢ, σⱼ	The volatilities of positions i and j (σ² is a variance)
ρᵢⱼ	The correlation between positions i and j; this calculator uses the shared correlation dial for every pair

When i and j are the same position the term is just wᵢ²σᵢ², that holding's own variance. The remaining terms, where i and j differ, are where co-movement enters and where a portfolio's risk parts ways with the average of its holdings.

All inputs and outputs are illustrative educational examples, not recommendations, forecasts, or guidance.

225.00

289.00

484.00

36.00

400.00

Shared correlation (ρ): how closely holdings move together 0.30

move independentlymove together

At this shared correlation setting, the five holdings combine to 9.9% volatility. If they all moved together they would carry 14.1%, so lower correlation removes 4.2 points of risk for the same expected return. That is why variance is measured for the portfolio as a whole, not one holding at a time.

Portfolio metrics