What is t distribution?



T distribution is a continuous type of probability distribution, which is symmetric about zero (0). This is like the normal distribution, which has a bell-shaped curve. Unlike the normal distribution, it has fatter tails. Consider the following figure.

Density curves of Z and t distributions


Definition of the Student’s t distribution

Suppose {X1, X2, X3… Xn} be the set of n standard normal variables. Let T be the continuous random variable defined as below.

Student's T random variable



Here,

Y = X1² + X2² + X3² + … + Xn²

The random variable T defined above has Student’s t distribution with n degrees of freedom. The number of standard normal variables, n, required to define variable Y are degrees of freedom. The shape of the T distribution (broadness in the tails) is determined by its degrees of freedom.


Properties of T distribution

  • T distribution is symmetric about 0.
  • T distribution has fatter tails than the normal distribution.
  • T distribution behaves like normal, as degrees of freedom gets increase.


Use of T distribution

While studying the average height of a group of people, studying the average income, etc. you may come across a situation where you do not know the population standard deviation (σ). In such situations, you use T distribution to study and test the claim about population mean (µ).

We use T distribution to study the population mean when small samples are given. Therefore, T-tests are known as a small sample tests.


How to use the t distribution table?

While constructing the confidence intervals or making decisions in hypothesis testing, critical values are essential. Follow the procedure below to find critical values using the T distribution table.

Before going to find critical values using the T distribution table, let us understand some facts about the t distribution table.

T distribution curve is symmetric about 0. This gives, the right-tailed area to t is same as that of the left-tailed area –t. Therefore, the t distribution has only positive values (right-tailed critical values).

To get a left-tailed critical value, first, follow the following guidelines to find a positive critical value. Then just multiply this value by (-1). Consider the following figures.

Right tailed curve - t distribution



 

Left tailed curve - t distribution


The given significance level (α) is equally distributed at both the tails. Consider the following figure.

Two tailed curve - t distribution


There are three-row headings in the T table, namely, one-tailed area, two-tailed area, and C (Confidence level). While df (degrees of freedom) is the row headings in the T table.

Heading - t distribution table



T table is restricted to some of the confidence levels that we often use.

How to find the critical values for the T-test

  1. Choose the tail of the test.
  2. Choose the significance level (α).
  3. Choose the degrees of freedom (df).
  4. Then search for the value corresponding to the column headed (α) and row headed (df) in the body of the table to get the critical value.
  5. To get a left-tailed critical value, just multiply the obtained value by (-1). For instant, say t0 be the right-tailed critical value then left-tailed critical value is, (-1) *t0 = -t0.


Illustration

Find the left-tailed critical value for a t-test with degrees of freedom, df = 10 and significance level, α=0.025.


Procedure

  1. The test is a left-tailed test, so choose a one-tailed area.
  2. The significance level is Î±=0.025.
  3. The degrees of freedom, df=10.
  4. Then search for the value corresponding to the column headed 0.025 and row headed 10 in the body of the table.
  5. From the table we get, the right-tailed critical value as, t0=2.228. This gives, left-tailed critical value as, (-1)*t0 = -2.228.

T distribution table - illustration 1


Apply the same procedure to find the critical value for a right-tailed test as well as the two-tailed test.


How to find the critical values to construct confidence interval (T interval)

  1. Choose the confidence level (C).
  2. Choose the degrees of freedom (df).
  3. Then search for the value corresponding to the column headed (C) and row headed (df) in the body of the table to get the critical value (tc).


Illustration

Find the critical value (tc) for t interval with degrees of freedom, df = 15, and confidence level, C=0.90.


Procedure

  1. The confidence level, C = 0.90.
  2. The degrees of freedom, df=15.
  3. Then search for the value corresponding to the column headed 0.900 and row headed 15 in the body of the table to get critical value, tc = 1.753.
T distribution table - illustration 2