On This Page
Key Takeaways
- Standard deviation is the square root of variance and measures how far individual returns or prices sit from their average.
- In finance, volatility is the annualized standard deviation of returns, so a higher value means a wider spread of outcomes.
- Under a normal distribution, 68% of outcomes fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
- Price times implied volatility times the square root of time gives the one-standard-deviation move, the expected range options traders compare with their strikes.
- Standard deviation measures spread in any data set. The odds inside a sigma band come from the distribution, and fat-tailed returns put more moves past three sigma than the normal curve does.
What is standard deviation?
Standard deviation is the square root of variance. It measures the spread of observations around their mean in the same units as those observations, so a price series gives a price amount and a return series gives a return amount. In finance, volatility means the annualized standard deviation of returns, and a higher value means a wider spread of outcomes.
The symbol sigma commonly denotes population standard deviation, while s denotes the sample statistic.
How is standard deviation calculated?
Find the mean, subtract it from each observation, square the deviations and add them. Divide by N when the data is the whole set, or by N − 1 when it is a sample, such as a recent window standing in for the stock's longer history. Take the square root.
Measuring spread around the sample’s own mean understates it. Dividing by N − 1 corrects that, and the correction matters most on short windows.
Both calculations work on any data, skewed or symmetric. A distribution enters only when you turn a sigma band into odds.
Formula
σ = √( (1 / N) × sum of (xi − mean)² )- xi is each observation, mean is the average of the observations, and N is the count of observations.
- Divide by N to describe the observed set. Divide by N − 1 when the data is a sample of a longer history, which corrects for measuring spread around the sample mean.
- Multiply a per-period standard deviation by √k to scale it to k periods when period returns are uncorrelated and equally volatile. Count k in the same days as the inputs: trading days for historical returns, calendar days for the option example.
Standard deviation example: a one-sigma stock move
A $100 stock carries 20% annualized implied volatility over 30 calendar days, on a 365-day year. Price × volatility × √time gives the one-sigma move, and a normal curve centered at $100 turns it into odds for the day-30 price.
- One-standard-deviation move = $100 × 0.20 × √(30 / 365) = $5.73
- One-sigma range: $100 plus or minus $5.73 = $94.27 to $105.73
- Two-sigma move = 2 × $100 × 0.20 × √(30 / 365) = $11.47, a range of $88.53 to $111.47. A normal curve puts 95% of outcomes inside ±2 standard deviations.
The one-sigma move is $5.73, so the range runs $94.27 to $105.73 around $100. On the normal curve, 68% of day-30 prices land inside that range and 95% land inside two sigma.
Stock price × annualized volatility × √time gives the one-sigma move. At $100, 20% implied volatility and 30/365 of a year, the move is $5.73, which puts the one-sigma range at $94.27 to $105.73.
The two-sigma move, 2 × $100 × 0.20 × √(30/365) = $11.47, widens the range to $88.53 to $111.47.
The chart draws the normal curve for the day-30 price, centered at $100. One standard deviation on each side holds 68% of outcomes, two hold 95%, and three hold 99.7%.
Those odds cover where price finishes on day 30. Price touches a band during the 30 days more frequently than it finishes outside it, so a stop or short strike at one sigma gets tested more than the closing odds imply. Options traders compare the range with strike locations, spread width and expiry, and the Options Profit Calculator shows how a position responds across it.
Price standard deviation vs return volatility
Price standard deviation measures dispersion around an average price. Return volatility measures dispersion in price changes expressed as returns. A steadily rising price path carries substantial price dispersion even when its percentage returns never change and its return volatility is zero.
Bollinger Bands put price standard deviation around a moving average. Realized volatility applies return standard deviation to historical returns. Implied volatility is the volatility priced into the options: the input that makes a pricing model return the option’s market price.
Annualizing and scaling standard deviation over time
Standard deviation scales with the square root of time when period returns are uncorrelated and equally volatile, so a daily figure times √252 gives the annual one. Work in log returns, which add across periods; simple returns compound and need converting first.
Match the day count to the inputs. The worked option range uses 365 calendar days; historical daily returns annualize over 252 trading days.
Limitations of standard deviation
- Upside and downside moves count the same, so a sharp rally lifts the reading as much as an equal selloff. Pair it with a downside measure, such as the Sortino ratio, before you size for loss.
- The 68%, 95% and 99.7% odds belong to the normal curve. When returns run fat-tailed, moves beyond three standard deviations come more frequently than the normal 99.7% implies. Size short premium for that tail.
- Check the window for one-off gaps. Squaring the deviations lets a few large moves dominate, and one earnings gap lifts the reading until it rolls out of the window.
- Historical standard deviation measures the window behind you. Compare it with implied volatility, which prices the period ahead, before you buy or sell premium.
- Square-root-of-time scaling holds when period returns are uncorrelated and equally volatile. In a trending or mean-reverting stretch, measure the longer horizon directly instead of scaling the daily figure.
Bottom line
Standard deviation puts dispersion in the same units as the data, which makes it the common language for volatility, sigma bands and expected moves. Match the units, horizon and day count to the question, and choose a distribution that fits the asset before turning a sigma band into a probability.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Fidelity: Standard Deviation Indicator
- CFA Institute: Understanding Investment Risk
- Fidelity: Historical Volatility
How entries are sourced, calculated and updated: editorial methodology.