Standard deviation

Standard deviation is the statistical measure that quantifies how spread out a set of numbers is around their average. In finance, it is what sits underneath volatility — when you see a stock has 15% annualized volatility, you are seeing its annualized standard deviation of returns.

How it's calculated

The steps:

1. Compute the average of the data points (the mean)
2. For each data point, find the difference from the mean
3. Square those differences (so positives and negatives don't cancel)
4. Average the squared differences (this is the variance)
5. Take the square root of the variance — that is the standard deviation

stdev = sqrt(sum((x_i - mean)^2) / n)

What the number means

For data that follows a normal distribution (the bell curve), standard deviation has clean interpretive properties:

• ~68% of values fall within one standard deviation of the mean
• ~95% of values fall within two standard deviations
• ~99.7% of values fall within three standard deviations

For a stock with 20% annualized volatility and a 10% expected return, a normal-distribution model would say there is roughly a 68% chance next year's return falls between -10% and +30%, and roughly a 95% chance it falls between -30% and +50%.

Why finance uses it

Standard deviation has three useful properties for portfolio math: it is symmetric, it is well-understood statistically, and it combines cleanly when you mix assets together. The risk of a portfolio is a function of the standard deviations of its holdings and the correlations between them. The Sharpe ratio uses standard deviation as its denominator.

The big caveat: returns are not normal

Real-world stock returns have fat tails. Extreme moves — both crashes and rallies — happen far more often than a normal distribution would predict. The 1987 crash, the 2008 financial crisis, and the 2020 COVID drawdown were all multi-standard- deviation events that, under a normal model, should have happened roughly once in the history of the universe.

Standard deviation is still the most useful single number for risk, but using it alone systematically understates how often the worst happens. Pair it with drawdown to get a more honest picture.