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Who was student and why do we care so much about his t-test?1

1 Presented at the 37th Annual Meeting of the Association for Academic Surgery, November 13–15, 2003, Sacramento, CA.
https://doi.org/10.1016/j.jss.2004.02.003Get rights and content

Abstract

Statistics is the study of populations, how they relate to one another and what effect sampling has in terms of representing the original population. Use of statistical tools has been facilitated by modern computer technology; however, given the ease in which results are obtained, it is easy to overlook potentially incorrect use of the various statistical tests. Statistical tools are invaluable for investigators because they make it possible to determine if scientific results are important. When a test is used, it is important to know that the result of any statistical test is valid to avoid erroneous conclusions. To do so, the investigator must have a basic understanding of the test’s assumptions and limitations. Modern statistical packages often include a variety of results relating to the test’s applicability to the data analyzed. Not uncommonly, biologists are unfamiliar with these analyses. This review intends to improve the reader’s understanding of t-tests by providing a history of Student’s t-test, and its assumptions, applications, and limitations. Part 1 of the series (“The Mean and Standard Deviation: What Does It All Mean?”) reviewed basic aspects of distributions, measures of central tendency, and dispersion assessment. Small sample size effects on accurate estimation of a population mean and group comparisons for continuous data are presented in this review.

Section snippets

Detecting differences

The fundamental question asked in most scientific investigations is if some intervention has an effect on a measurable, biologically important parameter. Alternatively, various populations that are observed are tested to determine if they have characteristics that are the same or different. As an example, we have asked the question: when evaluating a cohort of individuals about to undergo weight loss surgery are the males heavier than the females? For continuous data such as body weight, the

Student

Student was the nom de plume of William Sealy Gossett who worked as the statistician for the Guinness brewery [1]. Neither well known, nor an academic, he was reluctant to publish under his actual name. Gossett felt insecure about his analysis of sample size effects on significance testing. He chose to remain anonymous until R.A. Fisher, one of the best-known statisticians of the day, publicly validated and refined Student’s analysis [2].

The Guinness brewery made an effort to hire highly

Central limit theorem

Sample size determines the probability that the sample mean is the same as the population mean. Figure 2 illustrates this concept. Figure 2A and C demonstrates the scattergram and frequency distribution of 100 repeated calculations of the mean using five randomly selected values from a data set of 1067 patients undergoing weight loss surgery. Figure 2B and D demonstrates the effect seen when 100 values are included in each mean. These figures demonstrate the effect of the central limit

t-values

Given the uncertainties present when sample sizes are small, when determining statistical significance one must account for the difference between mean values, the scatter inherent in the data, i.e., its standard deviation, and the sample sizes. Student combined these concepts into a single equation that calculated a t-value from which statistical significance could be determined [3]: t=μ1−μ2σ(1/n1)−(1/n2) t is the t-statistic whose numerical value is proportional to the probability that the

Central limit theorem

The means of random samples from any distribution will themselves have a normal distribution. Increasing the number of values increases the probability that the calculated mean value is the same as the real one for the population. Stated another way: The probability that any calculated mean value is the same as the actual total population mean value decreases as the sample size becomes smaller.

Gaussian distribution

The familiar bell-shaped curve.

Homoskedasticity

Variances between two groups to be compared are equal (similarly,

References (4)

  • L McMullen

    “Student” as a man

    Biometrika

    (1939)
  • Studies in the history of probability and statistics XXSome early correspondence between W. S. Gossett, R. A. Fisher and Karl Pearson, with notes and comments

    Biometrika

    (1968)
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