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This article is about the mathematics of Student’s t-distribution. For its uses in statistics, see Student’s t-test. In this way, the t-distribution can be used to say how confident you are that any given range would contain the true mean. The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. In the English-language literature the distribution takes its name from William Sealy Gosset’s 1908 paper in Biometrika under the pseudonym “Student”. Gosset’s paper refers to the distribution as the “frequency distribution of standard deviations of samples drawn from a normal population”.

It became well-known through the work of Ronald Fisher, who called the distribution “Student’s distribution” and represented the test value with the letter t. Note that the numerator and the denominator in the preceding expression are independent random variables, which can be proven by induction. The probability density function is symmetric, and its overall shape resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. The normal distribution is shown as a blue line for comparison. The cumulative distribution function can be written in terms of I, the regularized incomplete beta function. Other values would be obtained by symmetry. 2F1 is a particular case of the hypergeometric function.

Let x1, , xn be the numbers observed in a sample from a continuously distributed population with expected value μ. I represents any other information that may have been used to create the model. The distribution is thus the compounding of the conditional distribution of μ given the data and σ2 with the marginal distribution of σ2 given the data. This is a form of the t-distribution with an explicit scaling and shifting that will be explored in more detail in a further section below.