Standard deviation measures how widely a set of returns has spread around its own average. In investing it is the usual yardstick for volatility: a fund whose monthly returns cluster tightly has a low standard deviation, and one that lurches between big gains and big losses has a high one. Gains count exactly as much as losses, because the measure looks only at the distance from the average, not the direction. When fund documents, research reports or traders quote volatility, they almost always mean the standard deviation of returns scaled to a year. The stocks hub covers the investments it is most often applied to.
How do you calculate standard deviation?
Find the average return, square each return's distance from it, add the squares, divide by one less than the number of returns and take the square root.
Squaring does two jobs. It stops positive and negative distances from canceling out, and it gives large misses more weight than small ones. The figure before the square root is the variance, which comes out in squared percentage points and is hard to read; taking the root brings the answer back to the same units as the returns. Dividing by one less than the count, the sample version that spreadsheets call STDEV.S, allows for the fact that the average was taken from the same data. With a few years of daily returns, the difference from dividing by the full count is negligible.
How do you turn daily returns into annual volatility?
Calculate the standard deviation of daily returns, then multiply it by the square root of 252, about 15.87, the usual number of US trading days in a year.
Worked example: S&P 500 volatility from 10 trading days, October 15 to 28, 2026
Closing levels come from FRED, the St. Louis Fed's database. Each daily return is the close divided by the previous close, minus 1. The 10 returns average 0.035% a day, and deviations are in percentage points.
| Date, 2026 | Close | Daily return | Deviation from the average | Deviation squared |
|---|---|---|---|---|
| October 14 | 7,718.94 | |||
| October 15 | 7,744.06 | 0.33% | 0.29 | 0.085 |
| October 16 | 7,659.44 | −1.09% | −1.13 | 1.271 |
| October 19 | 7,690.73 | 0.41% | 0.37 | 0.140 |
| October 20 | 7,712.57 | 0.28% | 0.25 | 0.062 |
| October 21 | 7,757.17 | 0.58% | 0.54 | 0.296 |
| October 22 | 7,663.43 | −1.21% | −1.24 | 1.545 |
| October 23 | 7,623.68 | −0.52% | −0.55 | 0.306 |
| October 26 | 7,676.77 | 0.70% | 0.66 | 0.438 |
| October 27 | 7,722.60 | 0.60% | 0.56 | 0.316 |
| October 28 | 7,743.98 | 0.28% | 0.24 | 0.059 |
| Sum | 4.518 |
- Variance: 4.518 ÷ 9 = 0.502
- Daily standard deviation: √0.502 = 0.709%
- Annualized: 0.709% × 15.87 = 11.25%
Ten days keep the arithmetic visible but give a noisy estimate. The same steps on the 251 daily returns in the year to October 28, 2026 give 0.80% a day and about 12.7% a year. The FRED series leaves out dividends, which add a tiny, steady amount each day and barely change the spread.
The square-root rule assumes each day's return is independent of the last, so that variances add up: 252 independent days carry 252 times the daily variance, and the standard deviation grows by √252. Monthly figures are scaled by √12, about 3.46, so the 12.43% three-year standard deviation that Vanguard reports for its S&P 500 ETF, built from 36 monthly returns, corresponds to about 3.59% a month. The NYSE's holiday calendar leaves 251 trading days in 2026, but 252 remains the convention. Crypto trades every day and is annualized with 365, a choice the bitcoin vs S&P 500 study shows matters.
What do one and two standard deviations mean?
If returns follow a bell curve, about 68% of them fall within one standard deviation of the average and about 95% within two.
NIST's statistics handbook gives the exact shares for a normal distribution as 68.27%, 95.45% and 99.73% for one, two and three standard deviations. Applied to the US stock market's calendar-year returns from Kenneth French's data library, which include dividends, the bands turn out wide.
| Band around the 12.19% average | Annual returns in the band, 1927 to 2025 | Bell curve's share inside | Years inside, of 99 |
|---|---|---|---|
| ±1 standard deviation of 19.84% | −7.6% to 32.0% | 68.27% | 68 |
| ±2 standard deviations | −27.5% to 51.9% | 95.45% | 93 |
| ±3 standard deviations | −47.3% to 71.7% | 99.73% | 99 |
For whole years the bell curve held up: 68 years inside one standard deviation is exactly what it predicts. The misses were lopsided, though. Five of the six years beyond two standard deviations were losses, in 1930, 1931, 1937, 1974 and 2008, and the only gain among them was 1933's 56.8%. A one-standard-deviation band nearly 40 points wide also shows how little a single year reveals, since a bell curve expects one year in three to land outside it.
The same figures explain why an average year overstates long-run growth. Money left in the market over those 99 years compounded at 10.27% a year, 1.92 points below the 12.19% average and close to half the variance: 19.84% squared and halved is 1.97 points. CAGR covers compound growth rates.
Why do fat tails matter?
Real markets produce extreme days far more often than a bell curve allows, so standard deviation understates how large the worst moves can be.
Kenneth French's daily data for the US stock market run from July 1, 1926 to September 30, 2026: 26,338 trading days, with an average move of 0.04% and a standard deviation of 1.08%.
| Daily move larger than | A bell curve expects | Days it happened |
|---|---|---|
| 3 standard deviations, about 3.2% | 71 | 450 |
| 4 standard deviations, about 4.3% | 1.7 | 194 |
| 5 standard deviations, about 5.4% | 0.015 | 101 |
On a bell curve a five-standard-deviation day should come about once in 6,900 years of trading; the market has had 101 in a century. On October 19, 1987 it fell 17.41%, a move of 16 standard deviations. The other side of fat tails is a crowded middle: only 19% of days moved more than one standard deviation, against 32% for a bell curve, and the distribution's kurtosis, a measure of tail weight, is about 19 where a bell curve's is 3. Quiet stretches are more common than the standard deviation suggests, and crashes far more common.
Much of this comes from volatility arriving in bursts. Annualized, the market's daily swings measured 5.0% in 1964 and 7.2% in 2017, but 40.0% in 2008 and 43.6% in 1932. A three-year standard deviation describes those three years rather than a fixed trait of the fund, which is why the study of the biggest stocks' volatility also counts large daily moves directly. How deep the losses went is a separate question, covered by drawdown and bear markets and corrections.
How is standard deviation different from beta, downside deviation and drawdown?
Standard deviation measures total spread around an investment's own average, while the measures people confuse it with track the market, only the losses or the deepest fall.
| Measure | What it captures | Counts gains as risk? |
|---|---|---|
| Standard deviation | Spread of returns around their own average | Yes |
| Variance | Standard deviation squared, the form used inside portfolio math | Yes |
| Implied volatility, such as the VIX | Volatility the S&P 500 options market expects over the next 30 days | Yes |
| Beta | Sensitivity of returns to a benchmark's moves | Yes |
| Downside deviation | Spread of returns below a chosen target only | No |
| Maximum drawdown | Largest fall from a peak to a later low | No |
Standard deviation looks back at what happened, while Cboe designs the VIX to capture the volatility the options market expects over the next 30 days. Beta measures how much of an investment's movement follows the market, so a fund can have a high standard deviation and a low beta if its swings are its own. Downside deviation feeds the Sortino variant of the Sharpe ratio, which itself divides excess return by standard deviation. A portfolio's volatility is not the average of its holdings' figures either, because it depends on how they move together; how much crypto to hold in a portfolio works through that calculation with variances.