Student's t-Distribution

The Student's t distribution looks like a normal distribution but with fatter tails, meaning extreme values are more likely. It was published in 1908 by William Sealy Gosset, a statistician at the Guinness brewery in Dublin, under the pen name "Student". Traders meet it in two places: in statistical tests with small samples, where it gives more honest confidence intervals, and in risk models, where it captures the fat tails of market returns better than the normal distribution.

Shape and degrees of freedom#

The t distribution has one key parameter, the degrees of freedom (ν):

For ν > 2, the variance is ν / (ν minus 2), so a t distribution must be rescaled to match a target standard deviation.

Use 1: small sample tests#

When estimating a mean from a small sample, using the sample standard deviation adds uncertainty. The t distribution accounts for this, giving wider intervals than the normal distribution.

t = (sample mean - hypothesised mean) / (s / √n), with n - 1 degrees of freedom

Use 2: modelling fat tailed returns#

Daily returns of stocks, indices, currencies and crypto have more extreme days than the normal distribution predicts. Fitting a t distribution often gives degrees of freedom between about 3 and 6 for daily returns, capturing those fat tails.

Where traders use the t distribution#

Limits#

• Symmetric: the standard t distribution has no skew, while equity returns often have negative skew. Skewed t versions exist.
• Static: it does not capture volatility clustering on its own; combining it with GARCH helps.
• Extreme tails: even t distributions may understate the most extreme events; extreme value theory focuses on the very far tail.

Frequently asked questions#

What is the Student's t distribution?#

A bell shaped distribution similar to the normal but with fatter tails, controlled by its degrees of freedom.

Why use the t distribution instead of the normal?#

For small sample tests, because it accounts for uncertainty in the estimated standard deviation, and for returns, because it captures fat tails.

What degrees of freedom fit stock returns?#

Daily returns are often fitted with about 3 to 6 degrees of freedom, reflecting fat tails.

Next, learn the distribution of win counts in Binomial and Bernoulli Distributions.