The Student's t Distribution

Here is a problem that shows up on every trading desk. A strategy has run for two weeks and made money on average. Is that real, or is it luck? The textbook move is to count how many standard deviations the average sits above zero. But you do not know the strategy's true standard deviation — you have to estimate it from the same fourteen numbers. You are dividing an uncertain thing by another uncertain thing, and the usual bell-curve answer quietly assumes the denominator is known exactly. It isn't. Ignore that and you will call flukes significant.

The analogy: a ruler you also had to measure

Imagine measuring a table with a tape measure, and you are confident the table is 210 centimetres. Now imagine the tape itself was cut by hand and might be a few percent short or long — and the only way to check it is to measure the same table again. Every measurement now carries two uncertainties: the reading, and the ruler.

Sometimes the wobbly ruler runs short, which makes the table look longer than it is. Because you divide by the ruler's length, a small ruler produces a big answer, and small rulers happen often enough to matter. That is precisely why the resulting distribution has fatter tails than the bell curve: extreme results are manufactured not just by extreme data, but by accidentally small estimates of the spread.

William Gosset worked this out in 1908 while testing barley at Guinness, and published under the pen name "Student" because his employer would not let him use his own.

Writing it down

Take $n$ observations $x_1, \dots, x_n$ from a normal population. Write $\bar{x}$ for the sample average, $s$ for the sample standard deviation (the usual formula, dividing by $n-1$), and $\mu$ for the unknown true mean you are testing against. Form

$$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}.$$

In words: how far the sample average sits from the hypothesised mean, measured in units of its own estimated standard error. If you replaced $s$ with the true standard deviation $\sigma$, this would be a standard normal. Because $s$ is itself random, it follows the t distribution instead.

The t distribution has exactly one shape parameter, written $\nu$ (Greek "nu") and called the degrees of freedom. Here $\nu = n - 1$: one observation's worth of information was spent estimating the mean, so only $n-1$ independent pieces are left to estimate the spread. Its density is

$$f(t) \;\propto\; \left(1 + \frac{t^2}{\nu}\right)^{-(\nu+1)/2}.$$

In words: the height of the curve falls off as a power of $t$, not as $e^{-t^2/2}$. That single difference is everything. The normal's tail dies exponentially; the t's tail dies polynomially, which is far slower, so genuinely extreme values keep real probability.

Worked example 1: does this desk actually make money?

Ten trading days of P&L, in thousands of dollars:

$$12,\; -5,\; 8,\; 20,\; -3,\; 15,\; 4,\; 9,\; -8,\; 18.$$

Step 1 — the average. They sum to 70, so $\bar{x} = 70/10 = 7.0$.

Step 2 — the deviations and their squares. Subtracting 7 from each gives $5, -12, 1, 13, -10, 8, -3, 2, -15, 11$. Squaring: $25, 144, 1, 169, 100, 64, 9, 4, 225, 121$, which sum to 862.

Step 3 — the sample standard deviation. Divide by $n - 1 = 9$: $s^2 = 862/9 = 95.78$, so $s = 9.79$.

Step 4 — the standard error. $s/\sqrt{n} = 9.79 / \sqrt{10} = 9.79/3.162 = 3.09$.

Step 5 — the interval. With $\nu = 9$ degrees of freedom, the 95 percent two-sided t multiplier is 2.262 (from a t table). The interval is

$$7.0 \pm 2.262 \times 3.09 = 7.0 \pm 7.00 = (0.00,\ 14.00).$$

The lower end lands exactly on zero. You cannot claim the desk is profitable.

Now do it wrong. Use the normal multiplier 1.96 instead: $7.0 \pm 1.96 \times 3.09 = 7.0 \pm 6.07 = (0.93, 13.07)$. That interval excludes zero, and you would have reported a statistically significant edge. Same data, opposite conclusion, and the only difference was pretending you knew the volatility.

Worked example 2: how much fatter is "fat"?

Suppose daily returns have a standard deviation of 1 percent, and you want the odds of a single 6 percent down-day.

Under a normal. Six percent is 6 standard deviations. The normal probability of a 6-sigma move in one direction is $9.9 \times 10^{-10}$. Invert it: one day in about one billion, or once every four million years.

Under a t with $\nu = 4$. A t distribution has variance $\nu/(\nu-2)$, which here is $4/2 = 2$, so its standard deviation is $\sqrt{2} = 1.414$. To make it match a 1 percent daily volatility you scale it down by that factor, which means a 6 percent move corresponds to $6 \times 1.414 = 8.49$ on the raw t scale. The tail probability past 8.49 for four degrees of freedom is $5.3 \times 10^{-4}$.

Invert that: one day in about 1,890, which at 252 trading days a year is once every seven and a half years.

One billion years versus seven years, from the same volatility. That gap is why risk desks that model returns as normal are repeatedly surprised, and why the t is a standard drop-in replacement for the normal in Value at Risk (VaR) work.

Use the explorer below to see the curve the t is converging toward. Drag the coverage band out to 2 and 3 standard deviations and read off the percentages — those are the normal-tail numbers the t always beats.

What this means in practice

• Any small-sample claim. Backtest Sharpe over one quarter, a pilot A/B test, an execution-cost study on forty fills — use the t multiplier, not 1.96. The correction is largest exactly where people are most tempted to over-claim.
• A better return model. Fitting a t to daily equity returns typically lands somewhere around $\nu = 3$ to $5$. Those tails are what let you size a stop or a VaR limit that survives contact with reality.
• A warning about $\nu \le 2$. With two or fewer degrees of freedom the variance is infinite; with one (the Cauchy) even the mean does not exist, and sample averages never settle down. If your fitted $\nu$ comes out near 2, standard risk arithmetic based on variance stops meaning anything.
• Regression output. Every coefficient t-statistic your regression package prints is this exact construction, with $\nu$ equal to observations minus fitted parameters. See Degrees of Freedom.