Student's t-distribution

student t pdf

student t cdf

William Sealy Gosset

Student's t-distribution, often simply referred to as the t-distribution, is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population standard deviation is unknown. It plays a crucial role in many areas of statistics, especially in the development of t-tests for hypothesis testing and in constructing confidence intervals.

Overview

The t-distribution was first introduced by William Sealy Gosset under the pseudonym "Student" in 1908. Gosset was employed by the Guinness Brewery in Dublin, Ireland, and his work in small-sample statistics was motivated by problems in quality control and experimentation in brewing. The distribution is defined by a parameter known as the degrees of freedom, which is related to the sample size.

Definition

The probability density function (pdf) of the Student's t-distribution for a given number of degrees of freedom \(v\) is given by the formula:

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

where \(t\) is the variable, \(v\) is the degrees of freedom, and \(\Gamma\) is the Gamma function. The distribution is symmetric and bell-shaped, similar to the normal distribution, but with heavier tails, meaning it is more prone to producing values that fall far from its mean.

Properties

The Student's t-distribution has several important properties:

• It is symmetric about zero.
• As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
• It has heavier tails than the normal distribution, which allows for the accommodation of outliers and provides a more reliable estimate when dealing with small sample sizes.

Applications

The t-distribution is widely used in statistics, particularly in the context of t-tests, including:

• The one-sample t-test, for determining whether the mean of a single group differs from a specified value.
• The two-sample t-test, for comparing the means of two independent groups.
• The paired t-test, for comparing the means of two related groups.

It is also used in constructing confidence intervals for the mean of a normally distributed population when the population standard deviation is unknown and the sample size is small.

See also

• Normal distribution
• Chi-squared distribution
• F-distribution
• Z-test
• Confidence interval
• Hypothesis testing