# t-Distribution

**Definition: **The **t-Distribution**, also known as **Student’s t-Distribution** is the probability distribution that estimates the population parameters when the sample size is small and the population standard deviation is unknown.

It resembles the normal distribution and as the sample size increases the t-distribution looks more normally distributed with the values of means and standard deviation of 0 and 1 respectively.

## Properties of t-Distribution

- Like, standard normal distribution the shape of the student distribution is also bell-shaped and symmetrical with mean zero.
- The student distribution ranges from
**–****∞**to**∞**(infinity). - The shape of the t-distribution changes with the change in the degrees of freedom.
- The variance is always greater than one and can be defined only when the degrees of freedom
**ν ≥ 3**and is given as:**Var (t) = [****ν****/****ν -2]** - It is less peaked at the center and higher in tails, thus it assumes platykurtic shape.
- The t-distribution has a greater dispersion than the standard normal distribution. And as the sample size ‘n’ increases, it assumes the normal distribution. Here the sample size is said to be large when
**n ≥ 30.**

The following are the important **Applications of the t-distribution**:

- Test of the Hypothesis of the population mean.
- Test of Hypothesis of the difference between the two means.
- Test of Hypothesis of the difference between two means with dependent samples.
- Test of Hypothesis about the coefficient of correlation.

Thus, student distribution is the statistical measure that compares the observed data with the expected data obtained with a specific hypothesis. It complies with the central limit theorem which says that the distribution approaches the standard normal distribution as long as the sample size is large.