Student's t-Distribution

The backbone of hypothesis testing with small sample sizes.

The "Small Sample" Distribution

The t-distribution is a type of probability distribution that is similar to the normal distribution but has heavier tails. This means it assigns more probability to extreme values. It is used in place of the normal distribution when you have small sample sizes (typically n < 30) and the population standard deviation is unknown.

In finance, this is incredibly common. You rarely know the true volatility of an asset and often work with limited historical data. The t-distribution provides a more cautious and robust framework for constructing confidence intervals and performing hypothesis tests (like the t-test) in these real-world scenarios.

The Formula
The probability density function (PDF) is defined by its single parameter: the degrees of freedom (ν\nu or `df`).
f(t)=Γ(ν+12)νπΓ(ν2)(1+t2ν)−ν+12f(t) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})} \left(1 + \frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}
  • ν\nu (nu) represents the degrees of freedom, which is typically the sample size minus one (n - 1).
  • Γ\Gamma is the Gamma function.
Interactive t-Distribution
Adjust the degrees of freedom to see how the t-distribution compares to the standard normal distribution. Notice how it converges to the normal distribution as `df` increases.