Student

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Student: An Anthology of Letters
By: Matthew Berry, 5986409
































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For one hundred percent.



























Preface
The characters and events in this anthology are entirely fictitious in nature and are
nothing more than the imaginations and machinations of the author. The places are real though,
let‟s be entirely clear about that. Dublin is a real place, Guinness breweries, Oxford they are all
real; I didn‟t make those up. At least, I‟m pretty sure I didn‟t. I must confess I have not visited
them, so I have no concrete concept of if they exist or not. Come to think of it, some of the
events may be real too and, hell, maybe even some characters. Wow, glad I caught myself before
we proceeded. Well let‟s just say that the characters, events, and places in this anthology are
more or less not entirely fictitious with some grounds of truth to them though creative liberties
were in all likelihood taken. This is a work of historical fiction
1
.

1
Hey that‟s a good term! Maybe I came up with that. Wonder if I could copywrite…
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Student: An Anthology of Letters is written about a student, by a student, and for a
professor
2
all of whom are linked by a single idea, statistics. The statistics that connect these
three people was developed and written by the first student mentioned, William Sealy Gosset
who wrote under the pseudonym „Student‟, hence the title of this work. Gosset/Student‟s work
titled The Probable Error of a Mean has had profound influence on the fields of statistics and
behavioural sciences as it allows scientists and statisticians to draw inferences about large
populations from small samples of data. The Probable Error of a Mean is not the centrepiece of
this work but it does lend to the importance and understanding of Gossets work in statistics and
his contribution to Student‟s t-statistic. I will be going over that soon but first, how about a little
back-story? This is a preface after all…
Gosset, born in 1876 entered the Royal Military Academy in Woolwich to become a
Royal Engineer but was rejected due to his poor eyesight. Instead he went to Winchester College
and then New College in Oxford where he graduated with a first class degree in chemistry. The
year was 1899. During the October of that same year he was hired by Arthur Guinness, Son &
Co. Ltd. as a brewer to work at Guinness Brewery in Dublin. In 1899 Guinness had started a
program of barley plot experiments that compared different varieties and cultivations and
fertilizers of different farms that fed the brewery. It was a unique idea that would hopefully make
Guinness breweries more successful. And so, after completing two years as a junior learning his
duties under senior brewers in every department, Gosset was put in charge of a section of the
brewery and research work in the new Guinness Research Laboratory that was opened in 1900.
The work Gosset conducted had to do with the analysis of how brewing parameters and
agriculture affected what he liked to call the „behaviour of beer‟. While less eccentric brewers
would call it „crop yields‟.
One can imagine with many more brewers other than Gosset conducting large-scale tests
things would become expensive for Guinness
3
and so brewers could not afford to gather the large
amounts of data that would normally be used by statisticians of the time. But in this era of
history there were no accurate inferential methods for working with small samples. This forced
Gosset to develop his own homemade method.
Using a textbook on the theory of errors he had borrowed from a friend, Gosset started
his development and by 1903 he could calculate standard errors. A year later he wrote a report on
the subject for the brewery, which, in turn, led him to consult Karl Pearson in July of 1905.
Gosset returned to Guinness for a brief stint as Acting Brewer-in-Charge of the Experimental
Brewery
4
in 1906 but then left to Pearson‟s lab to answer two questions he had come across in
his experiments: how much wider should the error limits be to make room for the error
introduced by using estimates (m instead of µ and s instead of

o) and what level of probability
should be deemed „significant‟? From 1906 to „07 Gosset worked on these problems and finally
worked out the answer to his questions through „inspired guesses‟. While more eccentric authors
would call it „total shots in the dark‟.
At the end of 1907 Gosset returned to Guinness to become Acting Brewer-in-Charge
5

again for the following seven years where he took over the entire statistical analysis of the
company. During that time Gosset prepared his papers: The Probable Error of a Mean and

2
Who, tautologically, used to be a student.
3
Just imagine what the world would have been like if Guinness was put out of business due to expenditures on large
experiments. I shudder at the thought.
4
I am not making this up; the fancy title is actually a thing.
5
Still a thing.
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Probable Error of a Correlation Coefficient, under the pseudonym „Student‟ and put the final
touches on his t-table. Finally Goesset was able to see how beer behaved in any analysis he could
think of using his newly developed statistical method. One interesting experiment he did was to
analyze barley yield and quality together, which he did in value per acre of farmland. He also
conducted non-business related statistics using his developments as often as he could for his own
personal enjoyment
6
.
If you recall I mentioned Gosset‟s Probable Error of a Mean as important as it lends to
an understanding of statistics. I shall get to that… now. As the title suggests the piece pertains to
determining the likelihood that a sample mean (m) sufficiently approximates the mean of a
population (µ) from which it is drawn. Like the mean‟s standard error (σ
m
), „probable‟ or
„estimated‟ error is a specific estimate of the dispersion of its sampling distribution (s
m
= √s
2
/n).
This dispersion is a crucial step in inferential statistics: in order to draw inference about a
population parameter from a sampled mean the sampling distribution of the sample means must
be specified. It is the same for null hypothesis testing: to infer the probability that a certain
population would yield a sampled mean as extreme as the obtained value a sampling distribution
of means must be used for comparison. The central limit theorem
7
does this but only in terms of
parameters and population variance. This is not effective as in most research, such as Gossets,
parameters are not known or expensive to reach. Sample data and sample means must be used
despite the limits that pertain to not having the complete parameter (underestimations,
overestimations, increasing error due to decreasing sample size, biased measures etc.).
Gosset used sample data and variance to estimate the sampling distribution of the mean
(making a sampling distribution of sample means) and used it to draw inferences about
population means. Remember the statisticians of the time? They would take the time to gather
large quantities of data and assume a population and were able to use this data and simply
calculate a standard z score and use a unit normal table to find the corresponding p value. Gosset
could not do this because the unit normal table does not take into account the estimation of
population variance or that the error in the estimate depends on sample size. Sound familiar? It‟s
the two problems Gosset was sent to Pearson‟s lab to work on.
To answer his questions Gosset derived a sampling distribution of a statistic he would
call “z”. “z”, as per Gosset, was the deviation of the mean of a sample (m) from the mean of a
population (µ) divided by the standard deviation of the sample or estimated standard deviation
(s). Or in other words: (m - µ) is the calculated difference between the sample mean and the
hypothesized population mean and s is the difference between the sample mean and the
hypothesized population mean that would be expected by chance alone.



z =
m÷ µ
( )
s


z =
x ÷µ ( )
s


t =
x ÷ µ ( )
s
m

where


6
Because working in a Guinness brewery wasn‟t fun enough apparently.
7
For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will
have a mean of μ and a standard deviation of the standard error σ
m
= σ/√n and will approach a normal distribution as
n approaches infinity
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s =
(x ÷ m)
2
¿
n
in Gosset‟s original calculation and

s
m
=
s
2
n
=
s
n


Originally Gosset calculated s with the denominator as n but this meant that the estimate
of population variance s
2
(sample variance) would be biased
8
. Whereas n – 1
9
is unbiased but
would only be incorporated to replace n in the formula after Gosset met R.A. Fischer in 1912.


s =
(x ÷ m)
2
¿
n


s =
(x ÷ m)
2
¿
n ÷1


s =
(x
i
÷ x )
2
¿
n ÷1


But to derive the sampling distribution of “z” Gosset had to first derive the sampling
distribution of s, which he did using the first four moments of s
2
and using some more inspired
guesses
10
. After the derivation of this sampling distribution Gosset was finally able to derive the
“z” sampling distribution. But Gosset still wasn‟t finished. After completing the “z” distribution
he tabulated the probability values for “z” from sample sizes 4-10
11
and even used p-values from
normal approximations for comparison purposes and to show the degree of error in using his
approximation.
Well now, you might be saying, this is all jim-dandy but what does any of this have to do
with the t-statistic that is attributed to Student? Didn‟t he invent it? Does “z” have something to
do with the formulation of t? As it turns out, it does. The actual basis of the t-statistic, the
backbone if you will, is the “z” statistic Gosset invented. However “z” was not turned into the
current definition of t until after he met R.A. Fisher as I sort of mentioned before. Though I will
not go into too much detail and back-story with R.A. Fisher
12
he essentially used mathematics to
give proof of the “z” statistic and ended up extending and unifying Gosset‟s work into a solid
foundation for testing the significance of means, mean differences, correlation coefficients and
regression coefficients in the process. Finally, as mentioned before, in his proof of the “z”
statistic Fisher multiplied “z” by √n-1 to make the statistic unbiased and more accurate. This
finalized the transformation making “z” mathematically into the famous Student‟s t-statistic that
is so readily known around the world. All it needed was a proper letter.
Though R.A. Fisher had taken hold of his work, Gosset still had a hand in the finalization
of the t-statistic; he kept in constant correspondence with Fisher and was the one to coin the
name „t-statistic‟ in addition to calculating the new set of probability tables the new distribution
13

required.

8
s
2
would not represent σ
2
though Gosset thought it would.
9
The addition of n-1 introduces the concept of degrees of freedom, which requires knowledge of n-dimesnional
geometry, one of the main elements that cause students grief in any statistical class, and therefore will not, by any
stretch of the imagination, be covered here.
10
Shots. Dark. In the. Just hammering it home.
11
He had to stop at 10 because the exhausting nature of his calculations was very deterring in extending the table.
12
This is only a preface after all…
13
The t-distribution is a sampling distribution of sample means and unlike that of the unit normal table it must take
into account degrees of freedom (n-1), as Fisher demonstrated. This gives an unbiased s
2
value. This means that each
and every new degree of freedom produces a new s
2
value and a new t-distribution, making t sensitive to degrees of
freedom. What is curious is that as t approaches thirty or more the distribution resembles that of the normal
distribution. This will not come as a complete surprise as the larger the degree of freedom the closer n-1 in the
denominator will approach n, which as we saw in the calculation of σ
m
will approach a normal distribution as n
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Finally, thanks to Gosset, the world had a means for using small samples to draw
inferences about large populations. In using the t-statistic scientists and statisticians simply
substitute any reasonable sample value as m for µ and use s

instead of σ. Provided they have a set
critical region and rejection criterion, the t-table will act as a unit normal table in that two or one
tailed measures can be taken. All that is required is for the estimated sampling error s
m
be
calculated from any data obtained using n-1 as an unbiased estimator of sample size, m as the
sample mean, and s as the estimated standard deviation. From this a t-value can be calculated and
then compared against the t-distribution that has already been tabulated
14
. If the obtained t-value
is more extreme than that of the tabulated t-value (or t
crit
) then the result is considered significant.
The general method is similar to that of z-score calculation and is used readily in null hypothesis
testing, a significant value allowing for the rejection of the null, and a non-significant value
allowing for the retaining of the null hypothesis. And I will get into a lengthy and detailed
discussion of null hypothesis testing… in a later installment. I can‟t spend too much time talking
about null hypothesis statistics and statistical practices here; this is only a preface after all…
Just a few more things before I do launch into the nitty-gritty. I think by now it is
apparent how the subject of Gosset, Student, and t-statistics is relevant to statistics
15
. It is a
beautiful and timeless tale, though to some slightly boring, of how a simple brewer came up with
one of the most influential methods of inferential statistics in the world. And though it wasn‟t
apparent at the time, as not even Gosset thought his discovery would be useful to anyone other
than Fisher, over 100 years later the Student t is constantly used in statistics either as its own test
for significance in sample data or as a post-hoc test for ANOVA in conjunction with the
Bonferroni correction
16
and effectively underlies the most frequently used statistical tests in
behavioral science! The t has ties to most things in statistics.
Gosset gave so much to the field of statistics, he paved the way to the age of small sample
research; I felt it necessary to discuss him as both himself and Student in my own special
(preface elongating) way. Student: An Anthology of Letters is my historical representation of the
events that lead Gosset to develop his t-statistic as Student. Carefully tip-toeing around some of
the more complex mathematics my goal is to provide context to the relationships he held with
various people, showing a representation of Gosset‟s character
17
with statistics and Guinness as a

approaches infinity. This means that the larger the denominator the more representative the sample is becoming to
its respective population and the closer it is to becoming a population itself. The mean of the t-distribution is the
hypothesized mean of the population. It‟s standard error is s
m
= √s
2
/n which shows the average dispersion of the
sample means from the hypothesized mean.
14
Thank you Gosset.
15
One can only hope though.
16
Run multiple t-tests with lowered alpha levels so that the experiment wise alpha level is not inflated – mostly used
with only three groups because the calculations and comparisons of more groups would be strenuous, vigorous,
arduous, laborious, onerous, gruelling, and tough at best.
17
On a personal note, I feel a connection with Gosset. He was described as someone who was very bright with high
ideals and an impish sense of humor, a most appealing character, unaffectedly friendly, helpful, patient and loyal,
someone who everyone can like and trust and while I do not deem myself as quiet as he did I believe I hold the rest
of these qualities
18
. Perhaps this is why I was drawn to his writings and story, it wasn‟t an interest in beer and
brewery that drove me as I originally thought but the following in the footsteps of a likeminded soul as he made a
world-changing discovery. I too find it fascinating that one can derive knowledge about a large subject or subjects
from the analysis of something small and specific. To draw inference unto something (and be correct in doing so)
has so many benefits. It reduces time (and resources) having to collect data and run tests, which equates to a faster
turn around time in understanding. As a student, I love to learn and understanding and knowledge is like currency to
me: to be collected and given out for the maximum gain of everyone
19
. If we as people can know things faster and
react upon them faster I feel this holds limitless possibilities. Numerous fields can benefit not just behavioural
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back drop. I fully acknowledge there may be some creative and humorous liberties that I might
have taken but I am sure they do not detract from the story. I hope you enjoy reading it as much
as I enjoyed writing it.
But I digress. I believe I have prefaced enough for the time being. Now, without further
ado I present my work: Student: An Anthology of Letters.


science but medical and health science (drug and treatment tests), ecology and environment (reduction in carbon
footprint and renewable energy tests), business and industry (marketing campaigns and consumer tests), and also
right at home or at school (evaluation of teaching practices). If sample testing isn‟t already incorporated in the
furthering of understanding of these fields I suspect it will not be long until it is
20
.
18
My vision however is 20:20.
19
I am the Robin Hood of education.
20
I know you‟re thinking it, Footnoteception.

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12
th
October 1899

Canterbury.
England

Dear Mum,
I knew I had to tell you the moment I found out. Please excuse any hasty mistakes as my
fingers are trembling as I type this note. It was no more than a few days ago that a tall man with
an air of importance approached me. I had just finished my morning cycle when he came down
from the dormitory steps and asked for me by name. At first I thought he was looking for
someone else and was mistaken but I did not voice these concerns as he looked keen on talking
to me, and I was the only one in the courtyard at that hour. I knew it was important. As it turns
out the man was none other than Arthur Guinness, the very same man who owns the brewery in
Dublin, Ireland. He and his right hand man La Touche had travelled to England in search of
bright young graduates to help him revolutionize Guinness‟s brewery and conduct experiments.
Having only finished my studies a few months back I have, to your chagrin I might add, had yet
to acquire a decent paying job and here this man was offering room, board, pay and food. It was
too good of an opportunity to pass up. So I accepted on the spot.
I know many at home wished me the best in going abroad and studying chemistry.
Though I still wish I could have followed in father‟s footsteps I do not now, nor ever will regret
the disappointment my eyesight caused me in the Royal Military Academy. I might add that we
are never to speak of that event ever again. I am beyond myself with giddiness at this new
opportunity.
You can hold your head up high now mum. I have been hired, straight out of college, by
Arthur Guinness, Son & Co., Ltd. to work as a brewer. Tell the family, as I have not the ability to
send any more letters at present. I must pack for I will soon be off for Dublin. Wish me luck and
love and I will hopefully see you around Christmas.


Your son,
William S. Gosset







20
th
January 1900

Canterbury,
England
Dear Mum,
I hope you are doing well and that nothing has changed since we last spoke at Christmas. I would
like to inform you, now that I am safely out of the city, that I had it all planned from the beginning that I
would come home earlier than expected to surprise you. It was good to see you and father singing hymns
at church caught completely unawares by my surprise and the one your children had made for you. It was
so grand hearing you cheer as the five of us walked up the aisle without a word and sat beside you during
the Lords Prayer. Albeit, we lit the candles on the cake with malice of forethought and I hope you forward
the apology letter and monies enclosed to Mrs. Higgins for a new hat. Though you must agree it will be a
Christmas we shall never forget.
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In addition to my apology I would like to inform you that there has been another addition to the
ranks of junior brewers here in Dublin. You may have already heard that Mrs. Phillpotts‟ son, Geoffrey,
from up the road has also joined on and did so earlier this month. He is an excitable lad and I‟ve taken it
upon myself to show him around, as, even though we are all new, I feel right at home here. Guinness has
us living at St. James‟ Gate which is where, as he says, all unmarried brewers will be housed together. It
isn‟t all that bad; it takes me back to my time in college, though the beds are a lot more comfortable and I
like the people I am living with. I have shown Geoffrey to his quarters and he has had lunch with all of
us: myself, Arthur, Alan, Peake and Case all of whom I told you about in my last letter. I am getting a
little tired of the dining room where all of us brewers must eat together, and the library where we all must
read together but the money we are making is more than enough to keep me from getting too agitated.
Though the weather is getting poor, which is hard to believe considering the dreary weather
Dublin gets year round. I still have time to get a few more cycle runs in before the snows come. And, as I
have been told by some of the senior brewers I am working under, the skiing here is fantastic. I must give
it a try. But I cannot wait for the summer. I hear we get to go fishing or sailing or golfing. This position
has simply been an extension of my time at college. It is wonderful. But I must not let pleasure take
precedence over work. Soon Guinness will open a new Research Laboratory and our work will really take
off. I cannot say more than this as Guinness is keeping every thing hushed to stave off nosey competitors.
With business back after the holidays things are going to get busy and so I do not know when
next I will write. Until then, take care mum.


Your son,
William S. Gosset


13
th
February 1903

Canterbury,
England
Dear Mum,
I am writing to you now to apologize for my lack of correspondence over the past two months
and limited correspondence over the past year. Events at the brewery have really escalated and while
Guinness has not placed us under any vows of silence they may as well have, seeing as how busy they are
keeping us. That is not to say I do not enjoy my work. To the contrary, I am finding it more enjoyable and
self-fulfilling everyday in its difficulty. What has made me particularly busy, to the point that I have not
been able to keep up our once lucrative correspondence, is that due to my mathematical abilities (of which
I must confess I did not even know I had), I find myself performing many of the maths of my co-workers
and fellow brewers. Personally I think that I am simply less afraid of the tasks of calculation as I have
seen similar maths at Oxford. It may seem strange, so many intelligent minds working here, that
reasoning of this mathematical nature has not been more widely made use of, but this is due to, and I
repeat myself, the popular dread of mathematics. And though I am quick to grasp their concerts and try to
aid them I find myself potentially taking on too much. I am burning the candle at both ends, as you would
say. But I know I cannot rest, even if I am reduced to simply using first principles and working from there
I will argue out a solution for my sake more for my fellow brewers. I have a reputation now to keep up.
A friend of mine, Airy, actually came to me the other day as I worked in the library. I feel he
sensed my distress in the burdens I had taken on and he offered to loan me a textbook on the theory of
errors, which may end up helping. I have since asked for it and he will be delivering it shortly as I type
this note. With any luck it may hold the key so that I no longer have to continue doing the work of others.
Maybe it has a formula I can use and teach to others to use in calculating and analyzing their data. As you
know I have many a time complained about the laborious process of collecting such large scales of data to
appease Guinness in analyzing qualitative measures such as resin consistency and which fertilizer gives
the best malting quality of barley. They would have us collect on thousands of farms to avoid large errors.
It cannot be done. It is far to daunting a task and to ask each brewer to do so must be taxing on their
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coffers. And so we deal with small samples, and I must blindly stumble through maths to find errors that I
know exist but cannot yet see.
Perhaps this new text will allow me to see them and perhaps I can once again return to my regular
correspondence with you mum. Give my best to father and my siblings. I will see you all as soon as I can.

Your son,
William S. Gosset.




7
th
July 1905


Woodlands, Monkstown
Co. Dublin
Dear Pearson,
As you are a man who is well aware of many things, I am sure that you are aware that I
will be cycling the twenty miles from my parents home in Canterbury to your summer home in a
few days. This meeting was arranged, again as you may be aware, by the company I am serving
Guinness, Sons & Co., Ltd. due to my discovery on how to calculate standard errors two years
ago and my report to Guinness the subsequent year, of which I am aware you have read. You are
an expert in the field of mathematics and may be able to aid me in understanding some
difficulties with what I would like to call my „homemade‟ measure of correlation. I am
examining the difference of ∑(A + B)
2
and ∑(A – B)
2
and would like to know more about the
product moment correlation coefficient. To say the least I am excited at the possibility to further
my understanding of this, my pet project.
In addition we at the brewery are having great difficulty in interpreting experimental data
with small samples. For example the barley experiments we started three years ago started with
four farms each growing one plot of each variety. The estimate m of the mean based on a sample
of four is obviously not as exact as using a larger parameter and the error in the estimate s of the
standard deviation simply cannot be ignored. I recognize this but have yet to find a sufficient
answer. It is my hopes that in this meeting you can provide an answer as to how much wider
should the error limits be to make allowance for the error introduced buy using estimates of m
and s instead of parameters µ and σ. And maybe you can teach me in the way of learning the
practise of all the statistical methods currently use. I am an excellent student and estimate this
should take no longer than a half hour.
I look forward to meeting you and having a splendid conversation. Please have the tea
ready when I arrive. I take it black with two sugars.





Yours very sincerely,
W.S. Gosset.



Berry 12
31
st
April 1907



Woodlands, Monkstown
Co. Dublin
Dear Pearson.
Can you believe it has been a little less than two years since our very first meeting at your
summer home? My goodness how time does fly from one‟s mind as he concentrates. I will have
you know I returned to the brewery in Dublin safe as can be earlier this week, no thanks to the
cab you arranged. The man was a lunatic. I would have much preferred to cycle across the ocean
than get into the back of that mans car again. Not much has changed here in the year since I was
asked by Guinness to take up the study of the law of error (to which I assume the workings of
which they had found a great service to the brewery) in your laboratory at the University College
in London. Upon my return they have granted me the title of Acting Brewer-In-Charge of the
Experimental Brewery again, as they did when I returned from that summer vacation in 1905.
Some things I suppose do not change as time flies. I feel that our time together was well used and
I have found many answers to the questions I had originally asked, and received little help from
you dear friend, in that first meeting. All of that has been cleared up now. And to answer the
question I know you are going to ask, no I cannot return to your lab as my place is here at the
brewery. I am sorry but it must be so. Fret not for we shall always keep in constant
correspondence.
What I wanted to tell you is that I have finally worked out an answer to my question
about the probable error of the mean. You were so keen on rushing me off into that death-cab
you never let me tell you before I left. So I am resigned to doing so over paper and ink. I have
even tabulated the probability values of my criterion of z=(m-µ)/s for samples n = 2, 3…10. I
may find the time to tabulate larger sample sizes but it is a considerable investment of my time.
Truth be told I would not have gotten this far without your help and some well placed inspired
guesses. Thank you dear friend for what you have allowed me to do.
Attached you will find what proof I can provide for my answers. I plan to use these
calculations and conceptual arguments in two papers I will be preparing publishing hopefully by
next year. Please proof read them and make sure I have not made any embarrassing mistakes.
Now I say next year for two reasons. First, in addition to my duties over the Experimental
Brewery I am also to take over the entire statistical analysis of the company. This includes barley
yields, measurements, and assessments for different varieties, different farms, and districts and
different seasons, malting assessments and brewing results for the same barley lots and many
other fanciful things that I am sure would bore the moustache right off your face. This, of course
will increase my workload and limit the time I have in perfecting my papers. Pushing a
publishing date a year may end up being a bit to ambitious. Second, it has come to my attention
that Guinness will not allow me to publish under my real name. As per the Board and that
infernal man La Touche who heads it, the only way they will permit publication is if I do so
under a pseudonym. There is other bureaucratic nonsense that will surely delay publication but
this is by far the most frustrating. I cannot even take credit for my own work or mention anything
about brewing, brewer‟s names, or the company I have worked for nearly a decade. This is
almost as frustrating as being a student back in Oxford when submitting chemistry labs and
being called out for plagiarism that one didn‟t commit. Once again I must deal with bureaucratic
nonsense that is put in place by higher powers in hopes of protecting themselves. But as much as
I do not like it I must abide by it. In putting my thoughts to paper here I have actually just
Berry 13
thought of a suitable pseudonym that I do plan on using for my publication. I tell you because I
want at least one other person to know of my work in case it ever becomes useful.
Thank you so much for all your past and future help in my endeavours in the world of
statistics and mathematics. You must come by for a cup of tea some time. Two milks one sugar,
just the way you like. Until then my friend.

Yours very sincerely,
W.S. Gosset a.k.a. Student



















12
th
September 1912



Woodlands, Monkstown
Co. Dublin
Dear Pearson,
Please excuse, as I will forgo any pleasantries dear friend, as this is a matter of, well, pride I
suppose. Enclosed in this letter is another letter which gives a proof of my formulae for the frequency
distribution of z ( =x/s), where x is the distance of the mean of n observations from the general mean and s
is the standard deviation of the n observations. Would you mind looking at it for me? I do not feel at
home in more than three dimensions even if I could understand it otherwise.
The question arose because this man, Fisher, his tutor is a Caius man, Stratton, whom I have met
on occasion when I visit my agricultural friends at Cambridge. Stratton is an astronomer and has applied
what you may call Airy to their statistics. Well, this student of his Fisher produced a paper giving 'A new
criterion of probability' or something along those lines. A neat but, as far as I could understand it, quite
unpractical and unserviceable way of looking at things. (I may have understood it when I read it but it‟s
gone out of my head as well as my hands as you shall hear that I have lost it). By means of this he thought
he proved that the proper formula for the standard deviation of a sample is:


vice


Berry 14
Stratton, the tutor, made him send me and with some exertion I mastered it, spotted the fallacy,
as I believe, and wrote him a letter showing, I hope, an intelligent interest in the matter and incidentally
making an embarrassing blunder. To this he replied with two foolscap pages covered with mathematics in
which he proved, by using n dimensions that the formula was, after all the second I have supplied above
and of course exposed my mistake. I couldn't understand his stuff and wrote saying 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, such as you. In any case I should be glad of your opinion of it dear
friend.
Thank you for your help. Frankly I simply lack the ability to deal with stiff mathematics as this
Fisher is supplying. If the original letter turns up I will be sure to send it to you post haste. You cannot
imagine my embarrassment upon loosing it.


Yours very sincerely,
W.S. Gosset.



15
th
September 1915


Holly House,
Blackrock, Co. Dublin

Dear Mr. Fisher

I would like to give you my very many thanks for the copy of your paper in Biometrika.
When I first saw it I nearly wrote to thank you for the kind way in which you referred to my
unscientific efforts but my natural indolence, joined to the fact that our last correspondence
petered out owing to my lack of courtesy and mathematics, allowed me to let the opportunity
slip. I am very glad that my problem is a step nearer solution.
I don't know if it would interest you to hear how things came to be of importance to me
but it happened that I was mixed up with a lot of large scale experiments somewhat agricultural
but mainly in an Experimental Brewery, whose name I am not at liberty to mention. The
agricultural (and indeed almost any) experiments naturally required a solution of the
mean/standard deviation problem and the Experimental Brewery which concerns such things as
the connection between analysis of malt or hops, and what I like to call the behaviour of the beer,
and which takes a day to each unit of the experiment, thus limiting the numbers, demanded an
answer to such questions as “If with a small number of cases I get a value r, what is the
probability that there is really a positive correlation of greater than (say) *25?” After my own
investigations I got of course a rough idea of the thing which has been very useful to me but I
should still like a full solution.
I feel that, like Pearson of whom you may know little of but I am sure will meet in due
time, our correspondence, friendship and partnership in terms of mathematics will flourish. I
look forward to creating numerous formulae with you and possibly t tests and distributions.

Yours truly,
W.S. Gosset.


Berry 15







16
th
September 1915


Holly House,
Blackrock, Co. Dublin

Dear Mr. Fisher
I apologize for my outlandish typo at the end of my last letter. I was meaning to say “I
look forward to creating numerous formulae with you and possibly tests and distributions.” I
hope I did not cause any confusion.

Best regards,
W.S. Gosset.





























Berry 16
Appendix:
If you think I didn‟t do a good enough job recreating the events of Gosset‟s life, perhaps
an error in dates may be upsetting you, or you would like to find out more about Gosset, Student,
the creation of the t-test, Guinness breweries and farming, or you really, really, really just like
reading References, please check out these References:

Box, J.F. (1987). Guinness, Gosset, Fisher, and Small Samples. Statistical Science 2 45 –
52.

Pearson, E.S. (1968). Studies in the history of probability and statistics. XX. Some early
correspondence between W.S. Gosset, R.A. Fisher, and K. Pearson, with notes and
commentary. Biometrika 55 445 – 457.

Student (1908a). The probable error of a mean. Biometrika 6 1-25



Formulae Used for the t-test:
n = sample size xbar = mean of the sample
n-1 = degrees of freedom µ = estimated population mean or given
s = standard deviation population mean
s
2
= squared deviation x
i
= a score in the distribution.
s
m
= estimated standard error




Acknowledgements:
I would like to thank Alex Willms of the University of Waterloo, Cambridge for his
cover design, Willian S. Gosset for some of his writings and letters to base my work off of and
Professor Nick Watier for the opportunity to write this anthology.
( )
m
s
x
t
µ ÷
=
n
s
n
s
s
m
= =
2

s =
(x
i
÷ x )
2
¿
n ÷1