The t‐distribution and its relationship to the normal distribution

The t-distribution is used as an alternative to the normal distribution when sample sizes are small in order to estimate confidence or determine critical values that an observation is a given distance from the mean. It is a consequence of the sample standard deviation being a biased or underestimate (usually) of the population standard deviation. Many early statistical studies were performed using limited sample sizes so it was necessary to develop approaches to deduce more information about the underlying population: even modern applications such as process control and metabolomics often suffer from modest sample sizes. Previously 1, we distinguished between samples and populations and looked at how the statistics obtained from a sample can be used to estimate parameters in a population. In chemometrics, we are also concerned with distributions. We have already discussed the normal and chi-squared distributions 2, 3. These, however, are usually employed to describe populations. When sample sizes are relatively small, we find that these ideal distributions can be distorted. Many early statistical studies were performed using limited sample sizes so it was necessary to develop approaches, not just for estimating means and standard deviations, for using these datasets to deduce more information about the underlying population. Even in modern studies, applications such as process control and metabolomics often suffer from modest sample sizes. The t-distribution was developed for such situations. The first recognisable description in the literature was credited to Frederick R. Helmert in the late 19th century 4. However, its current and more widely accepted description is due to William Gossett 5 in 1908. He published under the pseudonym 'Student' because he was working for the Guinness brewery and they did not, according to legend, allow him to use his own name. An alternative explanation is that they did not want rivals to know they were using this method. Although modern statistical ideas had many of their origins in the 19th century, our modern way of thinking was formulated primarily in the first few decades of the 20th century, especially due to the advocacy of R. A. Fisher and collaborators; hence, the terminology in common use is based primarily on Gossett's work, which was taken up by Fisher 6. The t-distribution is also sometimes called Student's distribution or Student's t-distribution to reflect the pseudonymous authorship of this original paper. The probability distribution function of the t-distributions for varying degrees of freedom is illustrated in Figure 1. Although the t-distributions do superficially appear quite similar to the normal distribution once ν is greater than 5, actually the proportion of measurements within 1.96 standard deviations on either side of the mean, which should be 95% for a normal distribution, differs noticeably even with there are 10 degrees of freedom. Although it is quite easy to calculate the probability of a sample being at any specified number of standard deviations from the mean in Excel or Matlab or most computational packages, it is still common to use statistical tables. Always check whether a table represents one-tailed or two-tailed probabilities. Note that as the number of degrees of freedom increases, the critical values start to resemble those for the normal distribution. There are several more comprehensive tables available on the Web of which we reference one 7. Whereas the distinction between t-statistic and normal statistics may not appear very great at first, relatively modest samples sizes could make a substantial difference in decision-making, for example, in areas such as process control with decision limits (critical limits) at the 95% or 99% level. This is especially true in cases where there may be many other factors at play, such as whether samples do indeed fall into an underlying normal distribution or whether sampling is representative or random (both different concepts, which we will explore in later articles). Finally, it is important to note that there is a multivariate t-distribution in analogy to the multivariate normal distribution. We will not dwell further on this as it is rarely encountered in chemometrics, with the more widespread F-distribution being described in the next article.