Over the ten trading days to 6 October 2026, SPY (the S&P 500 exchange-traded fund, or ETF) moved by a typical 0.55% a day, up or down. Scaled up to a year, that is 8.7%. That yearly figure is called volatility.
That typical move is a standard deviation. This guide works one out by hand, on those ten days of SPY prices. It then tests the bell curve, the shape that makes a standard deviation easy to read, against every SPY trading day since 1993.
What is standard deviation in investing? Standard deviation measures how far returns typically sit from their mean, or average. A bigger standard deviation means bigger swings, up or down. In investing, it is the maths behind volatility.
Over the ten trading days to 6 October 2026, SPY's daily returns had a standard deviation of 0.55%. A typical year has 252 trading days. Multiplied by the square root of 252, that is 8.7% a year.
Under a bell curve, 68% of days fall within one standard deviation of the mean, while 95% fall within two.
Real returns have more extreme days. Since 1993, SPY moved more than four standard deviations on 54 days. A bell curve expects about 0.5.
What standard deviation measures
A daily return is the % change in a price from one close to the next. Over any stretch, those returns have a mean, the sum divided by the count. Standard deviation measures how far the returns typically sit from that mean.
Volatility is the standard deviation of daily returns, scaled up to a year. What Is Volatility? explains it in plain words, with SPY's calm and stressed periods. This guide does the arithmetic.
How to calculate the standard deviation of stock returns, step by step
The table takes SPY's closing price on 11 trading days, from 22 September to 6 October 2026. That gives ten daily returns. SPY paid no dividend over those days, so the plain closes are enough.
Day | SPY close ($) | Daily return (%) | Return minus the mean (%) | Squared |
|---|---|---|---|---|
22 Sep | 773.38 | – | – | – |
23 Sep | 767.81 | −0.720 | −0.795 | 0.632 |
24 Sep | 767.18 | −0.082 | −0.157 | 0.025 |
25 Sep | 771.35 | 0.543 | 0.469 | 0.220 |
28 Sep | 765.61 | −0.744 | −0.819 | 0.671 |
29 Sep | 764.20 | −0.184 | −0.259 | 0.067 |
30 Sep | 762.63 | −0.205 | −0.280 | 0.079 |
1 Oct | 763.99 | 0.178 | 0.103 | 0.011 |
2 Oct | 769.64 | 0.740 | 0.665 | 0.442 |
5 Oct | 774.83 | 0.674 | 0.599 | 0.359 |
6 Oct | 779.09 | 0.550 | 0.475 | 0.225 |
Total | 0.749 | 0.000 | 2.73 |
Source: YX Insights
There are five steps:
Work out each daily return. On 23 September, SPY closed at $767.81, against $773.38 the day before. That is a fall of 0.720%.
Find the mean. The ten returns add up to 0.749%. Divided by ten, the mean is 0.075% a day.
Take each return minus the mean. These gaps are the deviations. They always add up to zero, so they cannot be averaged as they stand.
Square each deviation, then add them up. Squaring makes every gap positive. The squares add up to 2.73.
Divide by nine, then take the square root. 2.73 divided by nine is 0.303. This figure is called the variance. Its square root is 0.551%, the standard deviation.
Why nine, one less than the number of returns? The mean came from the same ten returns. Once the mean is fixed, the tenth return is set by the other nine. So only nine are free to vary.
Dividing by nine corrects for that. In Excel, the STDEV.S function divides by nine in the same way.
Source: YX Insights
Chart 1 shows the ten daily returns. The dashed line is the mean of 0.075%. The shaded band runs one standard deviation either side, from −0.476% to 0.626%.
Six of the ten days sat inside the band. Four fell outside it: two falls of more than 0.7% and two rises above 0.67%.
Why you multiply by the square root of 252
A daily standard deviation is small, so it is scaled up to a year. A typical year has 252 trading days.
If one day's return does not depend on the day before, the variances of the days add up. So a year's variance is 252 times a day's. The standard deviation is the square root of the variance, so it grows with the square root of 252, about 15.87.
For SPY's ten days, 0.551% times 15.87 gives 8.74% a year. Over 20 days to 6 October, SPY's volatility was 10.4%, as What Is Volatility? shows. The window changes the answer.
The 68–95 rule: one and two standard deviations
A normal distribution, or bell curve, is a symmetric shape where most values sit near the mean. Under a bell curve, 68.3% of values fall within one standard deviation of the mean. 95.4% fall within two. 99.7% fall within three.
SPY's daily returns from February 1993 to 6 October 2026 had a standard deviation of 1.17%. That is 18.5% a year. So one standard deviation was a move of 1.17% in a day, while two was 2.33%.
Source: YX Insights
Chart 2 sorts all 8,478 of SPY's daily returns by size, measured in standard deviations from the mean. The orange line is what a bell curve would expect.
SPY's shape is taller in the middle than the bell curve. Moves within 1/4 of a standard deviation either side of the mean came on 31.6% of days. A bell curve expects 19.7%.
Overall, 79.2% of SPY's days fell within one standard deviation, against 68.3% for the bell curve. Within two, the match was close: 95.2% against 95.4%.
Fat tails: how often SPY moved 3 and 4 standard deviations
The bell curve misses most at the edges, called the tails. A distribution with more extreme values than a bell curve has fat tails.
Source: YX Insights
Chart 3 counts SPY's extreme days since February 1993. A day more than three standard deviations from the mean, or 3.50%, came 125 times. A bell curve expects about 23.
A day more than four standard deviations away, or 4.67%, came 54 times. A bell curve expects about 0.5, or one such day every 63 years. Beyond five standard deviations, SPY had 25 days, against an expected count under 0.01.
The extreme days bunched together. Of the 54 four-standard-deviation days, 18 came in 2008 and 12 in 2020. The largest was a rise of 14.5% on 13 October 2008, 12.4 standard deviations from the mean. The largest fall was 10.9% on 16 March 2020, 9.4 standard deviations.
One reason is that volatility itself changes. A single standard deviation for 33.7 years mixes calm spells with stressed ones. What Is Volatility? shows how those spells come and go.
What standard deviation misses
It treats rises and falls alike. A fund that jumps is as volatile as one that drops.
It says nothing about how far a price has fallen. A drawdown, the fall from a previous high, measures loss directly. What Is a Drawdown? covers it.
It understates extreme days. Reading SPY's history through a bell curve would have expected about 0.5 days beyond four standard deviations. There were 54.
How to read a standard deviation figure
Check the period. A daily figure and a yearly figure differ by a factor of 15.87.
Check the window. Ten days, 20 days and 33.7 years gave SPY 8.7%, 10.4% and 18.5% a year.
Compare like with like. Use the same window for each asset. Measure returns the same way too, for example daily returns with dividends counted for both.
Know where else it appears. Variance also sits inside beta, a measure of how much a share moves with the market. What Is Stock Beta? works through it.
Standard deviation is the typical distance of returns from their mean. Scaled by the square root of 252, it becomes volatility. Within two standard deviations, SPY's history matched a bell curve closely. Beyond that, SPY had far more extreme days than a bell curve allows.
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Common questions about standard deviation in investing
What is a good standard deviation for a stock?
There is no single good standard deviation. It depends on how much swing an investor can accept. For scale, SPY, an S&P 500 fund, had a standard deviation of 18.5% a year from 1993 to October 2026. Compare any figure with others over the same dates.
How do you calculate the standard deviation of stock returns in Excel?
Put the daily closing prices in a column. Work out each day's return as today's close divided by yesterday's, minus 1. Apply STDEV.S to those returns to get the daily standard deviation. Multiply by SQRT(252), about 15.87, to turn it into a yearly figure, which is the volatility.
Is standard deviation the same as volatility?
Volatility is a standard deviation. It is the standard deviation of daily returns, scaled up to a year by the square root of 252. So a daily standard deviation of 1% equals a volatility of about 15.87% a year. What Is Volatility? explains how to read it.
What does a 3 standard deviation move mean?
A three-standard-deviation move is a daily return more than three standard deviations from the mean. For SPY since 1993, that was a move of more than 3.50% in a day. A bell curve expects 0.27% of days to be that large. SPY had 125 such days, or 1.47% of all trading days.
What does the 68–95–99.7 rule mean?
The 68–95–99.7 rule describes a bell curve. 68.3% of values fall within one standard deviation of the mean, 95.4% within two and 99.7% within three. SPY's daily returns since 1993 fit the two-standard-deviation part closely, at 95.2%. They had far more days beyond three standard deviations than the rule implies.
DISCLAIMER: This article is strictly educational. Any information or analysis in this note is not an offer to sell or the solicitation of an offer to buy any securities. Nothing in this note is intended to be investment advice and nor should it be relied upon to make investment decisions. Any opinions, analyses, or probabilities expressed in this note are those of the author as of the note's date of publication and are subject to change without notice.