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Student's t‐distribution is any member of a family of continuous probability distributions that arises when estimating the mean of a Normally distributed variable (in the population) in situations where the sample size is small and the population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym Student.

The t‐distribution plays an important role in a number of widely used statistical analyses, including Student's t‐test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.

The t‐distribution is symmetric and bell‐shaped, like the Normal distribution, but has heavier tails, meaning that it is more prone than a Standard Normal distribution to producing values that fall far from its mean (Figure 4.14a). The exact shape of the t‐distribution is determined by the mean and variance plus what are known as the degrees of freedom, df. These are derived from the sample size. As the df increases, the shape of the t‐distribution becomes closer to the Normal distribution; and when the sample size (and degrees of freedom) are greater than 30, the t‐distribution is very similar to the Standard Normal distribution.

Figure 4.14 Examples of probability density/distribution functions for the t‐, chi‐squared, F‐ and Uniform distributions. (a) t‐distribution. (b) chi‐squared distribution. (c) F‐distribution. (d) Uniform distribution.