# T Distribution

Reviewed by Annapoorna | Updated on Aug 27, 2020

Catalogue

## What is T Distribution?

The T distribution also called the Student's T distribution, is a type of probability distribution that has heavier tails but is similar to a normal distribution with its bell shape. T distributions have a greater chance for extreme values than normal distributions. Hence when putting on a graph, it depicts fatter tails.

## How to interpret the T Distribution?

Tail heaviness is calculated by a T distribution parameter called degrees of freedom, with smaller values giving heavier tails, and with higher values having the T distribution resembling a regular standard distribution having "0" as its mean, and "1" as a standard deviation.

When a sample of n observations is picked from a normally distributed population with mean- M and standard deviation- D, the mean sample- m and standard sample deviation- d will differ from M and D due to the sample randomity.

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

Normal distributions are used when the distribution of the population is presumed normal. The T distribution resembles the normal distribution, with only fatter tails. Both assume a population which is normally distributed. T distributions have kurtosis, which is higher than normal distributions. With a T distribution, the probability of getting values that are very far from the mean is higher than with a normal distribution.

## Strength and Limitations of the T Distribution

Since the T distribution has fatter tails than the normal distribution, it can be used as a model for financial returns exhibiting excessive kurtosis, enabling a more realistic calculation of the Value at Risk (VaR) in such cases.

The T distribution can skew the accuracy concerning the normal distribution. Its shortcoming only arises when perfect normality is required. However, there's a relatively minor difference between using a normal distribution and T distribution.