StatTools : Probability of Student's t : Explanations and Tables

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Introduction Tails (1&2) Examples Tables References
William Gosset was a brewer, but had an interest in statistics. He found the estimation of the probability for the standard deviate z unreliable if the observations were few. He derived a correction of the probability estimate according to sample size and called it t.  Gosset published his papers under the pseudonym of Student and this became known as Student's t.

Student's t allows the use of a small number of measurements to estimate what may be true of the whole population.  This forms the basis of modern inferential statistics, where a small number of observations are made, and the results are generalized to the wider population.

The t distribution curve is wider than the normal one.  Therefore, a larger area (or higher probability) of being greater than a particular deviate is obtained compared to the normal distribution. This difference varies with sample size (degrees of freedom), such that the probability of t approaches that of z when the sample size increases towards infinity. Conceptually, this is represented by the diagram to the left.

With infinite degrees of freedom (i.e., a large sample size), the one tailed t and z have the same value for a particular probability, but with fewer cases, t will be larger than z in obtaining the same probability.