Standard Deviation of Returns: Calculate and Interpret Volatility

Specify the series and denominator

Choose a consistent return interval, such as daily close-to-close returns after documented cash-flow and cost treatment. The sample variance sums squared deviations from the sample mean and divides by n − 1. Its square root is sample standard deviation. A population calculation instead divides by n, so the two outputs should not be compared without labels.

NIST defines variance and standard deviation as measures of scale. Squaring gives large deviations more influence. Taking the square root restores the original units, which matters when distinguishing decimal returns from percentages.

Calculate a four-observation example

Use hypothetical returns of −2%, 0%, +2% and +4%. Their arithmetic mean is +1%. Deviations from that mean are −3, −1, +1 and +3 percentage points. Squared deviations sum to 9 + 1 + 1 + 9 = 20 squared percentage points.

Sample variance is 20 ÷ 3, and sample standard deviation is the square root of that value, approximately 2.58199 percentage points per observation interval. Using the population denominator gives the square root of 20 ÷ 4, approximately 2.23607. If returns were entered as −0.02, 0, 0.02 and 0.04, the sample result would be 0.0258199; multiply by 100 to express it in percentage points.

Do not turn a convention into a guarantee

A common annualization convention multiplies daily standard deviation by the square root of an assumed number of observations per year. That scaling is connected to assumptions about stable variance and uncorrelated increments. Serial correlation and changing volatility can make it a poor description of longer-horizon behavior.

Keep the chosen annual observation count visible. A daily series that includes weekends is not automatically comparable with one containing only trading sessions. Missing days, stale valuations and mismatched close times can artificially lower or alter measured variability. Check the return construction before debating the annualization factor.

Inspect information the statistic leaves out

Standard deviation treats upside and downside deviations symmetrically. Two series can share the same value while differing in asymmetry, extreme losses and sequence. A strategy with rare severe outcomes may appear stable in a sample that missed those outcomes. Do not convert a standard deviation into a loss probability without an explicit, justified distributional model.

Report sample length, dates, mean, large observations and drawdown alongside the result. A single observation cannot support sample standard deviation with n − 1 in the denominator, and a constant series has zero observed dispersion without proving zero future risk. For source and follower accounts, align currencies, valuation timestamps and cash-flow treatment. Differences in measurement frequency can otherwise be mistaken for differences in trading quality.