Please review the following article on Student t-Test. Your review should discuss the discovery of the t-test and its role in Statistics. Article: Studentttest. Your review should be between 100 and 150 words. Please post your review as a reply to this post. Your deadline to reply to this post is April 2nd, 2015. This is an optional assignment. It is our fifth blog assignment.

## 16 thoughts on “Let’s discover the Student t-test”

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Overall, the article itself was hard to understand and follow. The format of the article was a narrative style, making it confusing to follow because the important information about the t table was hidden between the storyline. The t table was a combined effort of Fisher and Student; their relationship blossomed over the years with their work on the t table. The t table was born from a modified z table; it is able to do different applications that the z table could not do. z was switched to t, with t equal to z multiplied by n-1 to the ½ exponent. The t table switched to become based off of degrees of freedom instead of observations. The t table tests the significance of two means, and of linear, curvilinear, and multiple regressions.

This article discussing Gosset, Fisher, and the T-distribution was pretty badly written in my opinion. The fact that it was written from more of a narrative standpoint made it extremely difficult to follow and obtain information from. The relationship between Fisher and Gosset started when Fisher would write to Gosset regarding changes to his papers. The t test tables came into being when the two of them worked on modifying the z table so that as the number of observations grew, the table would not become extremely difficult. To do this, the degrees of freedom were used, instead of the number of observations, and the t score was set equal to z(n-1)1/2. This table is used to determine the significance of two means, linear, curvilinear, and multiple regressions.

This article told the story of two mathematicians and how their collaboration led to the discovery of the t-test. Gosset and Fisher first began their correspondence in the early 1900’s. Gosset had laid much of the groundwork in this area of study and Fisher drew inspiration from his tabulations. Two important changes to former t tables were that the number of observations in a table was changed to the number of the degrees of freedom, and the formula t = z(n – 1)1/2 was employed instead of z values. This created a more standardized table and allowed statisticians to account for a larger n. The role of the t-test in Statistics is to allow for one to test the significance of two means as well as to carry out linear, curvilinear, and multiple regression.

The progression of Student’s original rudimentary equation, stemming from the theory of distribution, to the widely accepted and applicable t tabulation was an extended and slow process. Years of correspondence between W.S. Gosset and R.A. Fisher documented the trials and errors that eventually lead to the development of the statistical t test. Interestingly enough, the statistical tests originated from practical purposes related to Guinness brewery conditions, as opposed to an abstract theory translated for practical purposes. Gosset provided Fisher with his original statistical table that utilized z values. The table, although useful, was fundamentally flawed: as n grew increasingly large, the math became increasingly complicated. By substituting the z values for t values, statistical calculations became simpler with increasing n. Following finalizations of the statistical formula, the t table was eventually published. As more and more applications were discovered, the significance of the t tables became obvious.

In my opinion, this is a poorly written article. It is written as a type of narrative describing the story of two mathematicians, Gosset and Fisher. Fisher eventually began to use the value of t rather than z. He found that probabilities could be found by an “expansion inverse powers of n”. Student’s cos formula became more complicated with increasing n, whereas Fisher’s method became easier and easier. Essentially it took years of collaboration as well as trial and error between the two mathematicians to develop the statistical t test. Though the paper is somewhat difficult to follow as a story, I feel as though it was a successful way to convey the story of how the t test was developed.

I found this article very interesting. It amazes me that the t-table, such a widely used and pretty much a requirement for most data to be published, was determined by two men writing letters back and fourth. The process was tedious though and often hard to follow in this paper. I did not understand a lot of what they were talking about. Alway each man would check and prove each other. I have to wonder though, before this, was their any way to determine the if data was significantly different? Or I guess that wasn’t even an option before this was determined. I did understand the part about degrees of freedom and the difference in using n and n-1.

This article surprised me in a number of ways. It was fascinating to learn that Gossett, who contributed so much to statistics, was not first and foremost a theoretical mathematician, but rather a man looking to perform his brewery analyses as practically and efficiently as possible. Additionally, I was surprised that the t-table was primarily the work of one man and his friends. I would have assumed that such integral part of statistics would have been the work of a large research team. The t-table is intrinsic to nearly all of the major tests we’ve learned this past month in Biostatistics, be they confidence intervals or hypothesis testing. And these tests are necessary to evaluate the results of experiments. Gossett’s collaboration with Fisher and Pearson has revolutionized statistics.

Although this article was somewhat difficult to understand due to it’s narrative style, it still provided interesting information regarding the background of the T-test and was probably the most effective way to introduce the Student t-test background. Something I found entertaining was that Gossett actually used “Student” as a pseudonym, and his groundwork studies in tabulation and the z table interested Fisher. Throughout the 1900s, the pair corresponded through mail and examined the concept of the degrees of freedom and the proof of Student’s distribution, using t rather than z. By the summer of 1924, Gossett and Fisher were able to publish their t-test after modifying the formula so that the number of observations in a study would not be a burden, and instead used the concept of degrees of freedom. The t score was then set equal to z(n-1)/(1/2) and helped to test the significance of not only one mean, but two, and analyze multiple types of regressions.

The article was written in the style of a narrative. It tells the story of how Fisher and Gosset came up with the t table. Over the years the two corresponded by letters. Gosset had already been working in this area of study when fisher first contacted him. Together they came up with the t table. The t table was created by modifying a z table. The t table could do things the z table could not such as handle a higher number of observations. This is because it use the degrees of freedom instead of number of observations. The t table could determine the significance of two means, linear, curvilinear, and multiple regressions.

The origin of W.S. Gosset’s work stemmed from his employment as a brewer at Guiness Brewery and his need to determine values such as the Laboratory Analysis of malts and the length of time the beer remained potable. Gosset had already published two papers (under the pseudonym Student) regarding the “Probable Error of the Mean” and “Probable Error of a Correlation Coefficient” when R.A. Fisher introduced the maximum likelihood estimate for variance of a normal sample, which used n as the divisor instead of Gosset’s n – 1. Fisher provided a mathematical proof based on Gosset’s distribution using a geometric representation of the sample in n-dimensional space. Fisher then used this geometric representation to derive the general sampling distribution of the correlation coefficient. Fisher and Gosset struck up a friendship, and together were able to modify Gosset’s z-table so that as the number of observations grew (and the standard error decreased), the calculations would not become impossibly complex. When Gosset’s z-values were substituted with Fisher’s t-values (t = z(n – 1)^(1/2)), which took into account the degrees of freedom and not the number of observations, the calculations became increasingly simple as the number of observations grew. From Fisher and Gosset’s work, it then become possible to determine the significance of two means, linear, curvilinear, and multiple regressions.

The article “Gosset, Fisher, and the T-distribution” was an interesting narrative on the development of the Student t-test, which was formulated from the collaborative work of W.S. Gosset and R.A. Fisher. Gosset (who used the pseudonym Student) and Fisher began a correspondence in the 1900’s in an effort to expand upon the z table distribution. Their goal was to expand the z table in a way that accommodated larger numbers of observations in a sample. In order to do this, they used degrees of freedom to modify the formula, preventing the difficulties of calculations because of increases in observations that led to decreases in standard error. By using degrees of freedom, the z score was replaced by a t score, defined as z(n-1)^(1/2). By altering the table in this way, the Student t-test could be used to determine the significance of two means, and had the capability to carry out linear, curvilinear, and multiple regressions.

It is interesting that this article was written as a narrative (I would not expect that in a statistics or any other STEM journal article). Gosset and Fischer were indeed interesting individuals and their names aren’t directly applied to the t-test i.e. Gosset’s and Fischer’s t-test instead of Student t-test (but I do understand that they wanted to expand the z score table). This article gave me more insight into the lives of mathematical/statistical researchers and how they communicate abstract concepts. They understood the constraint of z-score values: the inability to account for larger sample sizes. They solved this problem with a fairly simple formula: t=z (n-1)^1/2, which incorporates the degrees of freedom. It is phenomenal that such a simple equation would expand the applications of a test i.e. various regressions and the the determination of the significant difference between two means.

The article, “Gosset, Fisher, and the t Distribution,” discusses how two statisticians, Gosset and Fisher, came into contact through a series of letters, which would lead to their collaboration in creating the tabulation of student’s t test. From 1899, Gosset had already been conducting statistical research due to the need of statistics at his brewery job. In 1970, Fischer wrote a letter to Gosset that suggested the calculation of new tables of student’s distribution and a new method of calculation. Together, Gosset and Fischer changed the tabulation of student’s t test by changing the number of observations to the number of degrees of freedom. In addition, they no longer used z values, but instead t values, which were defined as t=z(n-1)^1/2. The student’s t test is important to statistics because it is used to determine if two sets of data are significantly different from each other. In biostatistics, data must be proven statistically significant in order to make further conclusions.

I thought the article, although hard to read, was written well. The style of writing from the 1920’s and 1930’s is different than it is now, causing the difficulty while reading it. However, I felt as though the author did a good job of portraying what the letters between Fisher and Gosset were trying to convey. Aside from the writing style, it was very interesting to see how something so commonly used in math (t-table) came about. Seeing how something like a brewery can cause the necessity to delve into math further really opens my eyes to how widely statistics is used. The t test tables came into being after modifying z tables to incorporate larger data sets. This is also how degrees of freedom came about and is used to this day.

In summary, the article, while interesting was difficult for me to understand the overall message of the creation of the t table. Because it was written like a story meant for entertainment, I had issues noticing the critical information about Student’s and Fisher’s development of the t table. From the two mathemetician’s professional relationship- which was initiated through peer revision of each others articles-they began working together to alter the already existing z table. Their efforts were meant to make a table that was still easy to utilize and understand regardless of how large the number of observations was. Their changes to the z table beside the letter included t change in the value of t which was the same as z multiplied by n-1 to the 1/2 power: t=z(n-1)^1/2. With these calculations, the table was no longer based on observations but instead on degrees of freedom. The final table was then found useful in determining the significance of 2 means and linear, curvilinear, and multiple type regressions.

Overall, I thought that the article was dense and hard to follow at times. As well as the choice to write it in narrative format was interesting as a STEM related article. It focused primarily on the relationship between Gosset and Fisher throughout the years and how their collaboration ultimately produced what is known as the student t-table. It was interesting to discover that the need for statistics was heavily utilized in a brewery and that this can be noted as influencing a statistical discovery. Gosset and Fischer acknowledged the constraints of the z-score system and were able to expand upon it by utilizing the formula t=z(n-1)^1/2, incorporating the idea of degrees of freedom. The concept of using degrees of freedom rather than number of observations, they were able to conduct calculations for increasingly larger sample sizes.