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Explanations, Calculations, Codes and Tables
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Introduction
Calculations
Codes
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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.
One and two tails When calculating t, a one tail or two tail model needs to be specify. A one tailed t is conceptually similar to the z, and assumes all the excluded values are on one side (tail) of the normal distribution, as shown in the following diagram.
A two tailed t however, assumes the area excluded are on both sides (tails) of the t distribution, so that each side contains only half of the excluded area, as shown in the following diagram. In calculations involving the confidence interval, the two tailed t is usually used. For examples :
https://en.wikipedia.org/wiki/Student%27s_t-distributionWikipedia on t Javascript algorithm adapted from Press WH, Flannery BP, Teukolsky SA, and Vetterling WT. (1994) Numerical recipes in Pascal. Cambridge University Press ISBN 0-521-37516-9. p.189 |
