What does a day in the S&P 500 look like, statistically?
Four numbers describe a distribution's shape — where it sits, how wide it is, whether it leans, and how heavy its tails are — and for daily equity returns each of the four says something a normal distribution would not. This lesson measures them on SPY.US.
Measured, 2026-09-04
Daily returns on adjusted closes from 1 September 2006 to 3 September 2026: 5,031 observations.
| Statistic | Value | What it says |
|---|---|---|
| Mean | +0.050% a day | 12.6% a year at 252 days, before compounding; the equity premium, seen daily |
| Standard deviation | 1.224% a day | 19.4% a year annualised by √252 |
| Skewness | 0.00 | Symmetric on this window — the big up days of 2008 and 2020 balance the big down days |
| Excess kurtosis | 14.8 | A normal distribution has 0; this is a tail far heavier than the bell curve |
Over the ten years to 3 September 2026 alone: 2,514 days, mean +0.063%, standard deviation 1.132% (18.0% annualised), skewness −0.31, excess kurtosis 14.8 again.
Reading the four
The mean is tiny and the noise is huge. A twentieth of a percent a day against a spread of more than a percent: the signal-to-noise ratio of one day is about 1 to 25. That ratio is why nobody can tell from a week of returns whether a market is going up, and the sample-size unit is that arithmetic taken seriously.
The standard deviation is the number people quote as volatility, annualised by multiplying by the square root of 252 trading days. It is an average over regimes, and the volatility-clusters lesson shows how different the months inside it are.
Skewness near zero is a surprise to anyone who has read that markets "take the stairs up and the elevator down". Over twenty years the worst day was −10.9% (16 March 2020) and the best +14.5% (13 October 2008), and the largest moves in both directions came in the same crises. On the ten-year window the skew is −0.31, mildly negative. The asymmetry that matters is not in the daily moves but in the ordering — the elevator down is a cluster of bad days, not one bad day — and the clustering lesson has it.
Excess kurtosis of 14.8 is the finding. A normal distribution with the same standard deviation predicts that a move beyond three standard deviations happens on about 0.27% of days — 13.6 days in 5,031. The measured count is 83. The next lesson is that number and its consequences.
The normal distribution's role
It is the wrong model and the right yardstick. Every "how unusual was that?" question in this course is answered by comparing the data with what a normal distribution of the same mean and standard deviation would produce — and the answer is always that the data has more of the extremes. The bell curve is kept as the thing markets are not.
The shape changes with the horizon
The fat tails above are a daily fact, and they thin as returns are added up. Measured on 2026-09-04 on the same twenty years, SPY.US's 240 monthly returns had a standard deviation of 4.41%, an excess kurtosis of 0.99 and one month beyond three sigma against 0.65 predicted — close to a normal distribution where the daily series was nowhere near one. The skew moved the other way: −0.56 monthly and −1.18 on the nineteen calendar years, because the worst months and years are further from the mean than the best ones. A risk model built on daily data needs the tails; one built on annual data needs the lean. Neither can borrow the other's shape.
In the data
The endpoint is /eod/SPY.US?from=2006-09-01&to=2026-09-03&fmt=json, 5,032 rows, and the four numbers are one pass over the ratios of consecutive adjusted_close values. Nothing in the API states a distribution's shape; /technical/SPY.US?function=stddev&period=20 returns a standard deviation of the price, in price units, over a rolling window — 4.31 on 3 September 2026 against a close of 773 — which is a different object from the return volatility above, and the rolling-windows lesson explains the difference.
Try it now
- One month,
/eod/SPY.US?from=2026-08-01&to=2026-09-03&fmt=json, is 24 sessions and 23 daily returns. Computed on adjusted closes on 28 September 2026:
| Window | Daily returns | Mean | Standard deviation |
|---|---|---|---|
| 3 Aug to 3 Sep 2026 | 23 | +0.090% | 0.615% |
| 1 Sep 2006 to 3 Sep 2026 | 5,031 | +0.050% | 1.224% |
Compare the two rows. Then compute the standard error of the month's mean (its standard deviation over √23) and say why a month cannot estimate a mean and can estimate a standard deviation. 2. The window the table describes, so the two crises that supply its tails are visible:
Measure September 2008 to March 2009, then February to March 2020. Those two stretches contain most of the 83 days beyond three sigma. 3. In your own words: what does "annualised volatility of 19.4%" promise about next month, and what does it not?