Every Statistician should know how to use the T distribution table to find critical values?

Student’s T distribution

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

Definition

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

Where

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 T distribution (broadness in the tails), is determined by its degrees of freedom.

Properties of T distribution:

• T distribution is symmetric about 0.
• It has fatter tails than the normal distribution.
• It behaves like normal, as degrees of freedom get an 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).

For small samples, we use T distribution to study the population mean. Therefore, T-tests are known as 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 table has only positive values (right-tailed critical values).

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

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

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

T table is restricted to few confidence levels, which are often used.

Let us see how to find the critical values for the T-test using the T distribution table.

1. Choose the tail of the test.
2. Then choose the significance level (α).
3. Then choose the degrees of freedom (df).
4. Based on this, search for the value corresponding to the column (α) and the row (df) in the body of the table to get a 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.

Now let us see how to find the critical values to construct confidence interval (T interval) using the T distribution table.

1. Choose the confidence level ©.
2. Then choose the degrees of freedom (df).
3. Based on this, search for the value corresponding to column © and the row (df) in the body of the table to get the critical value (tc).