Standard Deviation

Variance measures spread, but it does so in squared units, dollars squared, percent squared, numbers you can't lay next to the outcomes themselves. Standard deviation fixes that by taking the square root, landing you back in the original units. It answers the plain question "how far from the average does a typical outcome fall?" in a number you can actually use.

If $X$ has variance $\mathrm{Var}(X)$, its standard deviation is

$$\sigma = \sqrt{\mathrm{Var}(X)} = \sqrt{E\big[(X-\mu)^2\big]},$$

where $\mu = E[X]$ is the mean and $\sigma$ (the Greek letter "sigma") is the symbol everyone uses. Because it shares the data's units, $\sigma$ compares directly to $\mu$: a stock with a $0.1\%$ average daily return and a $1\%$ standard deviation moves about ten times more than it drifts, on a typical day.

For the common bell shape, $\sigma$ comes with rules of thumb worth memorizing: roughly $68\%$ of outcomes land within $\pm 1\sigma$ of the mean, and roughly $95\%$ within $\pm 2\sigma$. So a "two-sigma move" is one that happens only about $5\%$ of the time, split across both tails. This is the everyday language of the normal distribution.

Worked example: five days of returns

Suppose a stock's daily returns over a week are $+2\%,\ -1\%,\ 0\%,\ +1\%,\ -2\%$. First the mean:

$$\mu = \frac{2 - 1 + 0 + 1 - 2}{5} = 0\% .$$

Now the squared distances from that mean are $4,\ 1,\ 0,\ 1,\ 4$ (in percent-squared). Average them to get the variance:

$$\mathrm{Var} = \frac{4 + 1 + 0 + 1 + 4}{5} = \frac{10}{5} = 2 .$$

Finally take the square root to return to percent:

$$\sigma = \sqrt{2} \approx 1.41\% .$$

So on a typical day this stock moves about $1.4\%$ away from its average of zero. That single number, its daily volatility, is what a desk quotes, scales, and compares across names.

Common pitfalls

• Population vs. sample. Dividing the summed squared distances by $n$ gives the population $\sigma$; dividing by $n-1$ gives the sample estimate used when you only have a data sample. For small datasets the difference matters.
• The 68–95 rule assumes a bell shape. For skewed or fat-tailed data those percentages are wrong, sometimes badly. Real market returns have more extreme moves than a normal curve predicts.
• Standard deviation is not additive. You cannot add the $\sigma$ of two things to get the $\sigma$ of their sum. Variances add (when independent); standard deviations do not.
• It treats up and down the same. A big gain and a big loss both enlarge $\sigma$ equally, even though only one hurts. That's why some risk measures look only at the downside.

Standard deviation is the ruler you'll reach for in almost every risk calculation, and it's the bridge from the abstract variance to the concrete bell curve where those tidy percentages come from.