Student's t Distribution | Sahithyan's Notes

A continuous sampling distribution. Used to estimate population mean of a population with unknown population standard deviation using a small sample. First described by William Sealy Gosset under the pseudonym “Student” in 1908.

Suppose a sample of size $n$ is taken from a population $N(\mu, \sigma^2)$ . It’s mean is $\bar{x}$ and standard deviation is $s$ . The $t$ -statistic is given by:

t_{n-1} = \frac{\bar{x} - \mu}{s / \sqrt{n}}

Here, $t_{n-1}$ represents the t-statistic with $n-1$ degrees of freedom.

Properties

Symmetric about 0
Bell-shaped
Heavier Tails
Compared to the normal distribution. This means it is more prone to producing values that fall far from its mean.

Degrees of Freedom

Usually $n-1$ , where $n$ is the sample size. Denoted by $\nu$ .

As degrees of freedom increase:

Curve becomes narrower
Approaches the standard normal distribution.

Mean

For $\nu > 1$ , the mean is $0$ .

Variance

For $\nu > 2$ :

\frac{\nu}{\nu - 2}

Test statistic

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

Definition

Mathematically, the t distribution is defined by the following probability density function (PDF):

f(t) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}

where:

$t$ is the value of the random variable,
$\nu$ is the degrees of freedom (typically, $\nu = n - 1$ for a sample of size $n$ ),
$\Gamma$ is the gamma function.

Relationship to the Normal Distribution

For large degrees of freedom (typically $\nu > 30$ ), the t distribution and normal distributions becomes indistinguishable.

Uses

Used to make inferences about means when sample sizes are small and the population standard deviation is unknown.

Most commonly used in the following scenarios:

Estimating the Mean: When you want to estimate the mean of a population based on a small sample and the population standard deviation is unknown.
Hypothesis Testing: In t-tests (such as one-sample, two-sample, and paired t-tests) to determine if there is a significant difference between means.
Confidence Intervals: To construct confidence intervals for the mean when the sample size is small.