Statistical notation
Sigma is the eighteenth letter of the Greek alphabet. In statistics it appears in two forms with separate roles: capital sigma, Σ, commonly denotes a summation; lowercase sigma, σ, denotes standard deviation, a measure of dispersion around a center.
The relationship is structural. A standard deviation calculation includes a Σ because it adds a series of squared distances from the mean. Summation is therefore one step in measuring spread.
This page uses the notation as it is used in modern statistics and finance; it does not rely on a historical attribution claim for the symbols. Read the variance study
Two funds, one average
Sigma resolves a problem that average return cannot. Consider five years of results from two funds.
| Year | Fund A | Fund B |
|---|---|---|
| 1 | +10% | +30% |
| 2 | +6% | −14% |
| 3 | +12% | +25% |
| 4 | +4% | −10% |
| 5 | +8% | +9% |
| Mean | 8% | 8% |
Each column totals 40 percentage points, so each arithmetic average is 8%. On a fact sheet showing only mean return, the funds appear identical. The lived paths are not: Fund A stays within four points of its average and never posts a loss; Fund B ranges from −14% to +30%.
Sigma separates them. It asks how far from the 8% average each year actually lands.
Why signed deviations cancel
The instinctive approach is to measure each year's distance from the mean and average those distances. Fund A's deviations are +2, −2, +4, −4, and 0. Fund B's are +22, −22, +17, −18, and +1.
Both sets add to zero. This is not a coincidence: the mean is the balance point of a data set, so positive and negative deviations cancel by construction. A measure that produces the same answer for every collection of returns has not captured dispersion.
The solution is to square the deviations before adding them. Squaring makes every contribution positive, so a −22-point deviation becomes 484 and cannot cancel a positive miss elsewhere. It also weights a large miss more heavily: doubling a deviation quadruples its contribution.
That choice makes variance algebraically useful in portfolio work, because the combined dispersion of holdings can be calculated with their variances and covariances. Simply removing minus signs does not have that property.
Constructing the number
Root mean square deviation describes the process in reverse: calculate deviations, square them, average those squares, then take the square root.
- Deviation
- Subtract the mean from each return to find its distance from center.
- Square
- Square every distance. Fund B's −14% year is 22 points below its 8% mean, contributing 484.
- Mean
- Average the squared distances. This is variance, written σ².
- Root
- Take the square root of variance. The result is sigma, expressed back in percentage points.
| Fund | Sum of squared deviations | Variance | Sigma |
|---|---|---|---|
| A | 40 | 8 | 2.8% |
| B | 1,582 | 316.4 | 17.8% |
Fund B's squared deviations are 484, 484, 289, 324, and 1, for a total of 1,582. Dividing by five produces a population variance of 316.4; its square root is 17.8%. Fund A's population sigma is 2.8%. The two funds share a mean, while Fund B carries more than six times the dispersion.
Show the math
σ = √[ Σ (rᵢ − μ)² / n ]
For each return rᵢ, subtract the mean μ, square the result, sum those squares, divide by the number of observed returns n, then take the square root. The resulting sigma is in the same units as the returns.
If the data are a sample used to estimate a broader population, the conventional sample standard deviation divides by n − 1 rather than n. The example above treats the displayed five returns as the complete population for the example. More on sigma squared / variance.
Why sigma is the natural unit of risk
Under a normal-distribution model, two inputs describe the curve: μ, its center, and σ, its spread. Raising sigma flattens and widens the curve while holding its total area constant. The points where the curve changes curvature sit one sigma to either side of the mean.
The familiar 68/95/99.7 proportions are conditional on that normal-distribution assumption: about 68% of outcomes fall within one sigma, 95% within two, and 99.7% within three. Markets are not obligated to follow that model; these bands are a way to understand the model, not a schedule for future returns.
| Band | Illustrative return range | Normal-model share |
|---|---|---|
| 1σ | −7% to +23% | ~68% |
| 2σ | −22% to +38% | ~95% |
| 3σ | −37% to +53% | ~99.7% |
A planning conversation can use an illustrative range to test whether a person's time horizon, cash needs, and capacity to withstand losses fit the risk being considered. It cannot convert modeled bands into guarantees.