### What Is a T Distribution?

A T distribution is a type of probability distribution that is similar to the normal distribution with its bell shape, but has heavier tails (i.e., greater chance for extreme values).

Tail heaviness is determined by a parameter of the T distribution called degrees of freedom, with smaller values giving heavier tails, and with higher values making the T distribution resemble a standard normal distribution with a mean of 0, and a standard deviation of 1. The T distribution is also known as "Student's T Distribution."

### The Basics of T Distributions

When a sample of n observations is taken from a normally distributed population having mean M and standard deviation D, the sample mean, m, and the sample standard deviation, d, will differ from M and D because of the randomness of the sample.

A z-score can be calculated with the population standard deviation as Z = (m – M)/{D/sqrt(n)}, and this value has the normal distribution with mean 0 and standard deviation 1. But when this z-score is calculated using the estimated standard deviation, giving T = (m – M)/{d/sqrt(n)}, the difference between d and D makes the distribution a T distribution with (n-1) degrees of freedom rather than the normal distribution with mean 0 and standard deviation 1.

### *Fast Facts*

*The T distribution is a continuous probability distribution of the z-score when the estimated standard deviation is used in the denominator rather than the true standard deviation.**The t-distribution, like the normal distribution, is bell-shaped and symmetric, but it has heavier tails, which means it tends to produce values that fall far from its mean.*

### Real World Example of a T Distribution Application

- A confidence interval for the mean is a range of values, calculated from the data, meant to capture a “population” mean. This interval is m +- t*d/sqrt(n), where t is a critical value from the T distribution. For example, a 95% confidence interval for the mean return of the Dow Jones Industrial average in the 27 trading days prior to 9/11/2001, is -.33% +- 2.055)*1.07/sqrt(27), giving a (persistent) mean return as some number between -.75% and +0.09%. The number 2.055 is found from the T distribution.
- Because the T distribution has fatter tails, it can be used as a model for financial returns that exhibit excess kurtosis, rather than the normal distribution, allowing more realistic calculations of Value at Risk in such cases.