T distribution

t-distribution or Student's t-distribution is a type of probability distribution that is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning it is more prone to producing values that fall far from its mean. It was first described by William Sealy Gosset under the pseudonym "Student".

History

The t-distribution was first introduced by William Sealy Gosset in 1908. Gosset was a chemist working for the Guinness brewery in Dublin, Ireland. He was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as low as 3. To account for the uncertainty inherent in dealing with small sample sizes, Gosset developed the t-distribution.

Definition

The t-distribution is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. It is defined by the degrees of freedom, which essentially is the number of independent ways by which a dynamic system can move, without violating any constraint imposed on it.

Properties

The t-distribution has the following properties:

• It is symmetric and bell-shaped, like the normal distribution.
• It is a unimodal distribution, meaning it has one peak.
• The exact shape of the t-distribution depends on the degrees of freedom. As the degrees of freedom increases, the t-distribution approaches the normal distribution.
• The t-distribution has more probability in its tails than the normal distribution because it has more spread.

Applications

The t-distribution is used in a variety of statistical applications, including:

• Hypothesis testing: The t-distribution is used in the Student's t-test for testing the mean of one or two normally distributed populations.
• Confidence intervals: The t-distribution is used in the construction of confidence intervals for small sample sizes.
• Regression analysis: The t-distribution is used in regression analysis to test the significance of parameters.

See also

• Normal distribution
• Chi-squared distribution
• F-distribution