THAT DEAR MR. GOSSET
That ancient and honorable firm, the Guinness Brewing Company of Dublin, Ireland, began the twentieth century with an investment in science. Young Lord Guinness had recently inherited the enterprise, and he decided to introduce modern scientific techniques into the business by hiring the leading graduates in chemistry from Oxford and Cambridge Universities. In 1899, he recruited William Sealy Gosset, who had just graduated from Oxford at age twenty-three with a combined degree in chemistry and mathematics. Gosset’s mathematical background was a traditional one of the time, including calculus, differential equations, astronomy, and other aspects of the clockwork universe view of science. The innovations of Karl Pearson and the first glimmerings of what was to become quantum mechanics had not yet made their way into the university curriculum. Gosset had been hired tor his expertise in chemistry. What use would a brewery have for a mathematician?
Gosset turned out to be a good investment for Guinness. He showed himself to be a very able administrator, and he eventually rose in the company to be in charge of its entire Greater London operations. It was, in fact, as a mathematician that he made his first major contribution to the art of brewing beer. A few years before, the Danish telephone company was one of the first industrial companies to hire a mathematician, but they had a clearly mathematical problem: how big to make the switchboard of a telephone exchange. Where in the making of beer and ale would there be a mathematical problem to be solved?
Gosset’s first published paper, in 1904, deals with such a problem. When the mash was prepared for fermentation, a carefully measured amount of yeast was used. Yeast are living organisms, and cultures of yeast were kept alive and multiplying in jars of fluid before being put into the mash. The workers had to measure how much yeast was in a given jar in order to determine how much fluid to use. They drew a sample of the fluid and examined it under a microscope, counting the number of yeast cells they saw. How accurate was that measure? It was important to know, because the amount of yeast used in the mash had to be carefully controlled. Too little led to incomplete fermentation. Too much led to bitter beer.
Note how this matches Pearson’s approach to science. The measurement was the count of yeast cells in the sample, but the real “thing” that was sought was the concentration of yeast cells in the entire jar. Since the yeast was alive, and cells were constantly multiplying and dividing, this “thing” did not really exist. What did exist in some sense was the probability distribution of yeast cells per unit volume. Gosset examined the data and determined that the counts of yeast cells could be modeled with a probability distribution known as the “Poisson distribution.”3 This was not one of Pearson’s family of skew distributions. In fact, it is a peculiar distribution that has only one parameter (instead of four).
Having determined that the number of live yeast cells in a sample follows a Poisson distribution, Gosset was able to devise rules and methods of measurement that led to much more accurate assessments of the concentration of yeast cells. Using Gosset’s methods, Guinness was able to produce a much more consistent product.
Gosset wanted to publish this result in an appropriate journal. The Poisson distribution (or the formula for it) had been known for over 100 years, and attempts had been made in the past to find examples of it in real life. One such attempt involved counting the number of soldiers in the Prussian Army who died from horse kicks. In his yeast cell counts, Gosset had a clear example, along with an important application of the new idea of statistical distributions. However, it was against company policy to allow publications by its employees. A few years before, a master brewer from Guinness had written an article in which he revealed the secret components of one of their brewing processes. To avoid the further loss of such valuable company property, Guinness had forbidden its employees from publishing.
Gosset had become friendly with Karl Pearson, one of the editors of Biometrika at the time, and Pearson was impressed with Gosset’s great mathematical ability. In 1906, Gosset convinced his employers that the new mathematical ideas were useful for a beer company and took a one-year leave of absence to study under Pearson at the Galton Biometrical Laboratory. Two years before this, when Gosset described his results dealing with yeast, Pearson was eager to print it in his journal. They decided to publish the article using a pseudonym. This first discovery of Gosset’s was published by an author identified only as “Student.”
Over the next thirty years, “Student” wrote a series of extremely important papers, almost all of them printed in Biometrika. At some point, the Guinness family discovered that their “dear Mr. Gosset”
had been secretly writing and publishing scientific papers, contrary to company policy. Most of “Student”’s mathematical activity took place at home, after normal working hours, and his rise in the company to ever more responsible positions shows that the Guinness company was not ill-served by Gosset’s extracurricular work. There is an apocryphal story that the first time the Guinness family heard of this work occurred when Gosset died suddenly of a heart attack in 1937 and his mathematical friends approached the Guinness company to help pay the costs of printing his collected papers in a single volume. Whether this is true or not, it is clear from the memoirs of the American statistician Harold Hotelling, who in the late 1930s wanted to talk with “Student,” that arrangements were made to meet him secretly, with all the aspects of a spy mystery. This suggests that the true identity of “Student” was still a secret from the Guinness company. The papers by “Student” that were printed in Biometrika lay on the cusp of theory and application, as Gosset moved from highly practical problems into difficult mathematical formulations and back out into practical reality with solutions that others would follow.
In spite of his great achievements, Gosset was an unassuming man. In his letters, one frequently finds expressions like “my own investigations [provide] only a rough idea of the thing,” or protestations that he was given too much credit for some discovery when “Fisher really worked out the complete mathematics … .” Gosset, the man, is remembered as a kind and thoughtful colleague who was sensitive to the emotional problems of others. When he died at the age of sixty-one, he left his wife, Marjory (who was a vigorous athlete, having been captain of the English Ladies Hockey Team), one son, two daughters, and a grandson. His parents were still alive at the time of his death.
If nothing else, all scientists are indebted to Gosset for a short paper entitled “The Probable Error of the Mean,” which appeared in
Biometrika in 1908. It was Ronald Aylmer Fisher who pointed out the general implications of this remarkable paper. To Gosset, there was a specific problem to be solved, which he attacked in the evenings in his home with his usual patience and carefulness. Having discovered a solution, he checked it against other data, reexamined his results, tried to determine if he had missed any subtle differences, considered what assumptions he had to make, and calculated and recalculated his findings. He anticipated modern computer Monte Carlo techniques, where a mathematical model is simulated over and over again to determine the probability distributions associated with it. However, he had no computer. He painstakingly added up numbers, taking averages from hundreds of examples and plotting the resulting frequencies—all by hand.
The specific problem Gosset attacked dealt with small samples. Karl Pearson had computed the four parameters of a distribution by accumulating thousands of measurements from a single distribution. He assumed that the resulting estimates of the parameters were correct because of the large samples he used. Fisher was to prove him wrong. In Gosset’s experience, the scientist seldom had the luxury of such large samples. More typical was an experiment yielding ten to twenty observations. Furthermore, he recognized this as very common in all of science. In one of his letters to Pearson, he writes: “If I am the only person that you’ve come across that works with too small samples, you are very singular. It was on this subject that I came to have dealings with Stratton [a fellow at Cambridge University, where] … he had taken as an illustration a sample of 4!”
All of Pearson’s work assumed that the sample of data was so large that the parameters could be determined without error. Gosset asked, what happens if the sample were small? How can we deal with the random error that is bound to find its way into our calculations?
Gosset sat at his kitchen table at night, taking small sets of numbers, finding the average and the estimated standard deviation, dividing one by the other, and plotting the results on graph paper.
He found the four parameters associated with this ratio and matched it to one of Pearson’s family of skew distributions. His great discovery was that you did not have to know the exact values of all four parameters of the original distribution. The ratios of the estimated values of the first two parameters had a probability distribution that could be tabulated. It did not matter where the data came from or what the true value of the standard deviation was. Taking the ratio of these two sample estimates brought you into a known distribution.
As Frederick Mosteller and John Tukey point out, without this discovery, statistical analysis was doomed to use an infinite regression of procedures. Without “Student”’s t,4 as the discovery has come to be called, the analyst would have to estimate the four parameters of the observed data, then estimate the four parameters of the estimates of the four parameters, then estimate the four parameters of each of those, and so on, with no chance of ever reaching a final calculation. Gosset showed that the analyst could stop at the first step.
There was a fundamental assumption in Gosset’s work. He assumed that the initial set of measurements had a normal distribution. Over the years, as scientists used “Student”’s t, many came to believe that this assumption was not needed. They often found that “Student”’s t had the same distribution regardless of whether the initial measurements were normally distributed or not. In 1967, Bradley Efron of Stanford University proved that this was true. To be more exact, he found the general conditions under which the assumption was not needed.
With the development of “Student”’s t, we are sliding into a use of statistical distribution theory that is widespread in the sciences but with which there are deep philosophical problems. This is the
use of what are called “hypothesis tests,” or “significance tests.” We shall explore this in a later chapter. For the moment, we only note that “Student” provided a scientific tool that almost everyone uses—even if few really understand it.
In the meantime, “dear Mr. Gosset” became the middleman between two towering and feuding geniuses, Karl Pearson and Ronald Aylmer Fisher. He maintained a close friendship with both, although he often complained to Pearson that he did not understand what Fisher had written him. His friendship with Fisher began while the latter was still an undergraduate at Cambridge University. Fisher’s tutor5 in astronomy introduced them in 1912, when Fisher had just become a wrangler (the highest mathematical honor) at Cambridge. He was looking at a problem in astronomy and wrote a paper in which he rediscovered “Student”’s 1908 results—young Fisher was obviously unaware of Gosset’s previous work.
In this paper, which Fisher showed Gosset, there was a small error, which Gosset caught. When he returned home, he found waiting for him two pages of detailed mathematics from Fisher. The young man had redone the original work, expanded upon it, and identified an error that Gosset had made. Gosset wrote to Pearson: “I am enclosing a letter which gives a proof of my formulae for the frequency distribution of [”Student“’s t] … . Would you mind looking at it for me; I don’t feel at home in more than three dimensions, even if I could understand it otherwise … .” Fisher had proved Gosset’s results using multidimensional geometry.
In the letter, Gosset explains how he went to Cambridge to meet with a friend who was also Fisher’s tutor at Gonville and Caius College and how he was introduced to the 22-year-old student. He goes on: “This chap Fisher produced a paper giving ’a
new criterion of probability’ or something of the sort. A neat but as far as I could understand it, quite unpractical and unserviceable way of looking at things.”
After describing his discussion with Fisher at Cambridge, Gosset writes:
To this he replied with two foolscap pages covered with mathematics of the deepest dye in which he proved [this is followed by a group of mathematical formulas] … . I couldn’t understand the stuff and wrote and said I was going to study it when I had time. I actually took it up to the Lakes with me—and lost it!
Now he sends this to me. It seemed to me that if it’s all right perhaps you might like to put the proof in a note. It’s so nice and mathematical that it might appeal to some people … .
Thus did one of the great geniuses of the twentieth century burst upon the scene. Pearson published the young man’s note in Biometrika. Three years later, after a sequence of very condescending letters, Pearson published a second paper by Fisher, but only after making sure that it would be seen as a minor addition to work done by one of Pearson’s coworkers. Pearson never allowed another paper by Fisher into his journal. Fisher went on to find errors in many of Pearson’s proudest accomplishments, while Pearson’s editorials in later issues of Biometrika often refer to errors made by “Mr. Fisher” or by “a student of Mr. Fisher” in papers in other journals. All of this is grist for the next chapter. Gosset will appear again in some of the succeeding chapters. As a genial mentor, he was instrumental in introducing younger men and women to the new world of statistical distribution, and many of his students and coworkers were responsible for major contributions to the new mathematics. Gosset, in spite of his modest protestations, made many long-lasting contributions of his own.