During the month of November or December, graduate students in my multivariate analysis class traditionally pay special homage to the celebrated CayleyHamilton theorem. It is accorded this high honor by Professor Maurice Tatsuoka in the chapter on linear transformations, axis rotation, and eigenvalues in his excellent textbook. The role of this theorem in the textbook is rather obscure. It is not, to my knowledge, applied or used in any multivariate technique or employed in the proof of any other theorem or formula in the entire text! It is presented as a standalone pillar of mathematical splendor. It is shocking to many students that a theorem can have absolutely no practical application what so ever but yet be intrinsically elegant in and of itself. However, one can reflect on innumerable things in real life that are of this very same nature.
In my graduatelevel statistics classes I have three, four and five star handouts graded according to importance. The onepage handout on CayleyHamilton is a SIXSTAR handout and is the only one I have ever given this highest distinction. This handout is printed in limited quantities and serially numbered to insure its value as a collector's item. My students are admonished not to discard or in any way, bend, fold, or mutilate this work of art. Remember, I tell them..."ASK NOT WHAT THE CAYLEYHAMILTON THEOREM CAN DO FOR YOU BUT WHAT YOU CAN DO FOR THE CAYLEYHAMILTON THEOREM."
Here is a spectacular demonstration of how like its eigenvalues a matrix behaves. I am proud to present in gif format the celebrated CayleyHamilton theorem. ENJOY!!!
The Above Was Archived on 21 September 1999
This month I would like to focus on perhaps one of the most significant news event that has occurred in my lifetime. It makes the discovery of the incandescent light bulb or man's landing on the moon very inconsequential. I am of course referring to Mark McGwire of the St. Louis Cardinals and his shattering of the single season home run record on Tuesday, September 8, 1998 at 9:18 PM EDT. This record is the most revered one in all of sport both in the USA and many foreign countries. All eyes were fixed on Busch Stadium that evening when Mr. McGwire in the wink of an eye rifled his 62nd home run over the left field wall.
What does this feat have to do with the field of statistics you ask? I maintain it has everything to do with statistics. Baseball is a game whose very objective and rich heritage is vitally dependent on the art of record keeping and the meaningful manipulation of these records. There is no other sport in the world that breeds the thousands upon thousands of numbers and summaries that baseball does year after year. Indeed, each season there are many new records contrived to fit the particular accomplishments and combinations of skills of certain ballplayers and or the teams that employ them. Observe the emergence in recent years of the 3030 or the 4040 player or the manager's detailed charting of pitches thrown.
My purpose here is not to discusss all these newfangled indices. I will leave that task to the writers and news media who scramble to produce these tidbits to justify their existence. I simply want to capture the wonder of that magical September night and relate to you my observations of what were the important coincidences and facts about that historic day. Here they are:
You can see how a statistician can easily become obsessed with facts and figures like the above set particularly when that statistician happens to be a Cardinal fan. However, there is much more to this story. I would truly like to thank both Sammy Sosa of the Cubs and Mark McGwire for the great show that they have put on this year ( and the home run derby is not over with this writing). I think these two fine role models have shown baseball and this country that their close friendship and encouragement of one another is what life is all about! All other athletes should take special note of this relationship.
This month we will present another neat probability experiment that can easily be conducted by students as a short assignment or an inclass project. If you have been a regular reader of my home page you may recall the pennyspinning problem whose disscussion now resides in my Archives of Statistics Fun. That experiment and the current one represent excellent opportunities for an instructor to highlight the value of the Monte Carlo method in estimating probabilities for unusual variations in experiments that don't lend themselves to the usual formulas.
Let's state the current question in simple language:
If a penny is flipped until a head first appears, what is the probability that this first head occurs on an oddnumbered trial (i.e.,first, third, fifth, etc.)?At first blush, a typical student would reason that since the first head is just as likely to occur on an odd trial as it is on an even trial (second, fourth, sixth, etc.), the probability is obviously .5. But wait! Another student mentions that maybe the probaility should be somewhat greater than .5 since the first opportunity for a head to pop up is on the very first trial, and an odd trial continues to preceed an even trial after the first two. At this point the bandwagon effect sets in and students begin to incrementally up their estimates slightly from .5. But after many values are offered, a hush settles over the room and students begin to look at one another and shrug their shoulders. No one is really sure!
Enter Captain Sigma (the instructor)! With a flourish of his cape and a wink of his eye, he quietly suggests that this is a problem that just begs for empirical data. He urges each student to take about five minutes at home and repeat the experiment 10 times, tally how many times the first head appears on an odd trial, and bring the data to the next meeting. The students happily concent to this simple task ( Someone in the back of the room asks, "How many points is it worth?") and they all eagerly await the pooling of their data at the next meeting.
Two days later the instructor rushes into the classroom and puts all the results from 35 students on the board. The students sit on the edge of their seats in awe as the numbers accumulate. The final tally results in 245 out of 350 replications ending on an odd trial. Zowie! THAT IS 70%! Something is wrong. The pennies must have been seriously flawed.
The instuctor showing no emotion on his face allows the buzzing and chattering to go on for several minutes. Finally he cracks a grin and informs the students that this result is a very good estimate although it is a tad too high. He proudly states that the actual answer is P=2/3 or 67%. The students are dumbfounded and become quite excitable. They actually all cheer for the instructor and demand a formal proof (Did I say "cheer" in a stat class? I must be delirious from a high fever!).
Here is what the instructor wrote on the board:
The solution involves the sum of the first n terms of a geometric series expressed as:
S = a + ar + ar^{2} + ... + ar^{n1}
Where
a = first term of the series
n = number of terms
r = the common ratio
S = the sum of the first n terms calculated by
S = a (1  r^{n}) / (1  r)
In our case, a =1/2 = .5 and r = (1/2)(1/2) = .5^{2} or .25 and using the first expression for S we have:
S = .5 + .5^{3} + .5^{5} + ...
In words, the above is stating that the probability of getting the first head on an odd trial is the probability of getting a head on the first trial (.5) plus the probability of geting a head on the third trial (.5)(.5)(.5) plus the probability of getting a head on the fifth trial (.5)(.5)(.5)(.5)(.5) plus etc.,etc. for n trials.
Now to compute what this sum would be for n trials we calculate using the second formula:
S = .5 (1  .25^{n}) / (1  .25)
Finally taking the limit of this calculation as n approachs infinite, we arrive at
S = (.5) / (1  .25) = (1/2) / (3/4) = 2/3 or .67
Truly Remarkable! The students all give the instructor a standing ovation and shout "QED" "QED" "QED".... The instructor smiles sheepishly while taking a bow and thinks to himself how rewarding it is to be a statistics professor.
The Above Was Archived on 13 September 1998
Here are the answers to last month's crossword puzzle. As warned previously, some of these statististicians are not exactly household words. Use the following scale to grade your perfomance as a Statistician Trivialist:
CROSSWORD
PUZZLE SOLUTION
Statisticians
Across
4. Likes a good match  Down
1. Inventor of cotton gin 

The Above Was Archived on 10 July 1998
One of the most intriguing and frequently mentioned probability questions of all time is the socalled "Birthday Problem." I really do not remember when I first encountered this question but I know it has been around for decades and pops up in many treatments of probability in basic statistics textbooks. Let us revisit this interesting problem and hopefully shed some new light on this timehonored topic.
First, for those readers who are not familiar with this problem, let us pose the question in its simplest form:
Given a room with a random collection of N people, what is the minimum N needed for an observer to state there is greater than a 5050 chance of at least two people having identical birthdays?Responses to this question are many and varied depending on a person's exposure to probability topics. However, the three most frequently offered answers are (a) 183 (b) 20 and (c) 23. The correct answer,of course, is (c) 23 which may shock some of you and prompt you to immediately head out and bet some of your buddies on a coincidence of birthdays in rooms with this few people present. Before you make this rash decision read the remainder of this discussion.Note:We shall assume in our discussion that a year has 365 days rather than the 366 in a leap year. We shall also assume that by "identical birthday" or "birthday coincidence" or "duplicate birthday" we mean the same month and day disregarding the year of birth.
The (a) response of 183 has much intuitive appeal for the ordinary person on the street. He or she would reason that in order to be absolutely certain that two birthdays coincide, 366 people would be needed in the room. Now since a probability just greater than .50 of a duplicate is all that is wanted, simply take 1/2 of 366 and arrive at 183. This seems logical but the laws of probability tell us the correct N is dramatically smaller than 183! Just how much smaller?
Many people who have studied a little probability would give the (b) response of 20.
Wow! This intuitively seems way to small to give us even a slight chance of a
coincidence of birthdays let alone a better than even chance. But the reasoning merits
close examination and goes something like this:
Check the birthdays in the room one by one. After the first person has given his or her
birthday, the second person will have one chance in 365 of having the same birthday as
the first. The third person could have the same birthday as the first or second person,
so the third person has 2 chances in 365. Added to the chance from the second person,
there are a total of 3 chances in 365. By the same logic, the fourth person has 3
chances of having the same birthday as any of the first three so this needs to be
added to the previous 3 to get 6 chances out of 365, and so on. By the time we exceed
183 chances in 365, which is just greater than our 5050 probability, we will have
checked just 20 people. Mathematically, this is more concisely expressed as follows:
We want the smallest integer N1 such that
(0)(1/365)+(1)(1/365)+(2)(1/365)+(3)(1/365)+...+(N1)(1/365) > 1/2 or
(1 + 2 + 3 +...+(N1))/365 > 1/2
The required N1 is 19 since 1 + 2 + 3 +...+ 19 = 190 but don't forget to add one for the first person checked even though a match can not occur with just one person. Thus N = 20 people are required according to this line of reasoning.
But hold the phone! We have a serious flaw in this argument. The number (1 + 2 + 3 + ...+(N1))/365 is NOT a probability but the EXPECTED VALUE or MEAN NUMBER of birthday coincidences for N = 20 people in a room. Thus, if many rooms of randomly assembled N's of 20 people were examined, the mean number of coincidences is just greater than 1/2. This is not particularly reassuring to a shrewd betting person!
Although N=20 is an incorrect answer to the original problem it does suggest an alternate approach for betting purposes. Suppose we wanted the expected value of coincidences to be just greater than one. We could continue the above computatation for several more terms until the ratio just exceeded one. With a calculator it is easy to see that we must only go out to N1=27 or N=28 for this to occur. Thus with many rooms of N=28 people we would have a mean number of coincidences just greater than one and many bettors would take greater comfort in this value.
Now let us explain the correct answer (c) N=23 for the original problem. The easiest approach is to find the probability of NO duplicate birthdays in a sample of size N and then subtract this result from ONE to get the probability of at LEAST ONE duplicate. Again we shall check the people one by one. After the first person establishes a birthday (P=365/365), the probability of the second person's birthday not duplicating the first is (365/365)(364/365). The probability of the third person not duplicating the first two is (365/365)(364/365)(363/365). This multiplicative process goes on and on and for a given N this product becomes (365/365)(364/365)(363/365)...((365N)/365). Since this term is the probability of no duplicates, our task is to determine the value of N that will cause this probability to be as large as possible without exceeding .50. Then when this probability is subtacted from one the probability of at least one duplicate will just exceed .50. With a hand calculator it is easy to show that when N=22 this product is .5243 and 1  .5243 = .4757 but when N=23 the product is .4927 and 1 .4927 =.5073. We can thus state that if a room contains 23 randomly assembled people, we stand a slightly better than 5050 chance of finding a duplicate birthday.
If you are a conservative bettor and flinch at the above chances, I have developed the following table that allows you to read in the desired probability level for a duplicate birthday and read out the required N in your room. Note that the probability levels should be interpreted as actually those just greater than the listed value.
Required Numbers of People For Selected Probabilities of a Birthday Coincidence 


Probability  N 
.50  23 
.60  27 
.70  30 
.75  32 
.80  35 
.90  41 
.95  47 
.99  57 
Thus if you are a real gambler, when the situation presents itself, you go with N=23 and impress the pants off everyone in the room by hopefully finding a duplicate. If you don't feel that you are an exceptionally lucky person, then you might select the comfortable 7525 chance of a duplicate and use N=32. On the other hand, if you fall at the other end of the continuum and only bet on close to sure things, then pick the .99 level and go with N=57. Here you are almost certain to find a duplicate but the people will not be that impressed and you won't elicit that wonderful "WOW!" effect.
I tried this experiment last semester in my Statistics I class with N=26 students in attendance that day. I knew my chances were below .60 but I put on an air of absolute certainty with my pronouncement. I confidently started around the room with students stating their birthdays. When I got to the 12th person I had a duplicate. The reaction was electrifying. You would have thought that I had just floated an elephant in midair! My student ratings skyrocketed for at least one day!
I hope you have enjoyed my presentation of the above topic and hopefully if you are an instructor you can have some fun and try this in your class. I must fess up to one other assumption that was not mentioned earlier. Not only must you assume a random sample of people are assembled in the room but theoretically you must assume that birthdays are randomly distributed throughout the 365 days of the year. This is probably not satisfied in any strict sense but that is a question involving a whole different ballgame. If you are turned on by the concept of chance and how pervasive it is in our society check out Chance News, a bimonthly newsletter letter published at Dartmouth University.
The Above Was Archived on 5 April 1998
This is the season of good cheer and merriment. If you know of a lonely statistician
please tell him that you love him and truly appreciate all the wonderful methodologies
that he has perpetuated and enhanced throughout his career. I am sure any kind remarks
directed his way will warm his heart and point him toward the new year with a renewed
sense of vitality and dedication.
HAPPY HOLIDAYS to all my readers! May all your Summation
Sigmas be operational and all your means be true m's.
We at RAMO PRODUCTIONS are particularly thankful for this Holiday Season. In fact, we are ecstatic and even giddy over the honor that was recently bestowed on this Web Site. At the 1997 Annual Conference of the Society for the Preservation of Humor In Statistics (SPOHIS) held November 2022 in Las Vegas, this Home Page was awarded "The Golden Sigma Cup." This highly coveted award signifies the BEST contribution of any Site on the WWW toward the promotion of statistics as a humorous subject. The acceptance of this award was truly a defining emotional moment in my career. I would like to thank all the members of the SPOHIS Academy for the necessary and sufficient consideration given all the nominees for this award and the unbiased selection of this particular site. I will try to be a worthy recipient of this magnificent cup and redirect my energies toward uncovering new tidbits of humor that make statistics the enchanting field that it has now become.
The Above Was Archived on 7 February 1998
This month all my readers will be given a real treat. The World Famous Three Step Method (WFTSM) will be revealed. I have had many requests and pleadings through my guestbook and other personal email to present this marvelous technique to the World Wide Web. This procedure which I consider the Holy Grail of statistical methodology (just ask my students) is a threestep sequence for calculating the sample standard deviation. Some textbooks emphasize a multistep, direct or brute force procedure (see formula on right) which for most situations is very painful and tedious. That is, given a set of N raw scores (Xscores) you are advised to (a) compute the mean (b) subtract the mean from each score (c) square each of these deviations (d) sum the squared deviations (e) divide the sum of squared deviations by the number of scores N and finally (f) extract the square root of this result. While this technique works rather well when the number of scores is small and the mean is a nice whole number, it is a nightmare in other situations. When the number of scores is say 15 or more and the mean is a decimal (In practice this will be true about 95% of the time), this procedure involves repeated subtracting and squaring of decimals and gets extremely messy even when using a calculator. A better method is needed!
Now substituting the above results and applying WFTSM:
Thanks for reading the development of my favorite statistical procedure. Who says statistics has to be dull when it embraces world renowned technology cradled in elegant blue boxes!
The Above Was Archived on 20 December 1997
October is the month of goblins and ghoulies. Unfortunately, students of basic
statistics experience far too many of these creatures on days other than Halloween
night. As promised last month, I will offer some general suggestions for teaching
the course in basic statistics. Several caveats are in order. First, these ideas have
worked for me over several decades of teaching but I make no warranties they will work
for other instructors. Secondly, these techniques have been employed in classes with
enrollments of between 30 and 40 students and therefore are probably not appropriate
for large lecture sections. With this in mind, I present this short list of hints to
help rid the statistical learning environment of goblins and ghoulies:
The Above Was Archived on 11 November 1997
The fall semester has now begun at most universities across this great country. This means
that many students are experiencing for the first time an encounter with a basic applied
statistics course. Whether the course is taken in business, psychology, education, biology,
economics or some other discipline really does not matter. The frequency of application of
certain techniques will vary from field to field but the basic concepts remain amazingly the
same. If you are an upperclassman, this is the course you have postponed for several years
and with great trepidation must now meet head on. If you are an underclassman, the fear is
no less since the horror stories already hit the moment you arrived on campus. You must cope
with this perceived encirclement by dragons. Your mental outlook and approach to this course
will become the single most important determinant of a meaningful positive experience with
beginning statistics. I have attempted to put together, from several decades of teaching,
a short list of helpful suggestions for the student. I make no warranties that these will
work with everyone. I do know, however, that many students over the years have given me
feedback that these hints are quite useful. Here they are:
If you are a student, I hope the above suggestions prove useful. Next month I will present some tips for the instructor of basic statistics.
The Above Was Archived on 7 October 1997
Merry Christmas everyone! Contrary to popular belief statisticians also believe in Santa Claus and have their wish lists. I thought you might want to see what desires I have had for many years. These are far out so be prepared!
Hope you enjoyed the above. Have a happy holiday season! If you are a statistician don't take yourself seriously and laugh at yourself. If you are a student make a New Year's resolution to attempt to understand the poor statisticians of this world who are only trying to eke out a living.
The Above Was Archived on 28 February 1997
For the month of November we have a very special report for you! From our home office high atop the grain elevator in Fooseland, Illinois we are proud to bring you: THE TOP TEN REASONS WHY STATISTICIANS ARE MISUNDERSTOOD. These are not listed in any particular order of importance but represent all those nagging suspicions you have always harbored against statisticians but were always afraid to ask about. Fasten your seat belts and here we go!
The Above Was Archived on 20 December 1996
We are quickly approaching election day and throughout the entire month of October you can expect to be bombarded with the results of many presidential polls. The pollsters of today (Gallup, Roper, etc.) use highly sophisticated techniques that employ samples of about 1600 or less registered voters who are likely to vote. If these samples are drawn at random, the public can expect the percentages that favor the candidates to fall within a 3% or 4% margin of error. This all sounds great to the typical citizen (except if your candidate is trailing). What happens, however, if the sample is biased or in some pernicious way, nonrandom? In short, incorrect inferences may be drawn and widely disseminated, the public may lose faith, and entire polling organizations or their sponsers may go out of business! Following is what I consider to be the worst case in history of a biased presidential poll which resulted in such a devastating effect (No folks, I am not going to rehash the TrumanDewey election of 1948).
Here are the basic concepts of a random sample and bias. A sample is considered random if each member of the population from which it is drawn has an equal chance of of being selected. I like to think of a random sample as an equal opportunity employer. A table of random numbers or a computer is usually employed to draw a random sample. A sample becomes biased when certain members of the population have a greater chance of being selected than others. The sample then tends to systematically overestimate or underestimate a certain characteristic of the population such as the percentage of a particular class of individuals. Thus, serious inferential errors may occur.
Now go back to the year 1936. This was the year that pitted the Republican, Alf Landon, against the Democratic incumbent, Franklin Roosevelt. This year was during the great depression. It also should be remembered for the record extreme temperatures and dust storms that hit the midwest. In fact, it was so hot that a statistician could not even calculate a standard deviation without working up a sweat. Moreover, the high humidity that year forced the Goudey Baseball Card Company to print only black and white cards and many of the cards came off the printing press hopelessly bowed.
A prestigious weekly news periodical called The Literary Digest continued that year a tradition of conducting a national presidential poll through the mail. Supreme faith was placed in a humongous sample of 10,000,000 prospective voters drawn primarily from telephone directories. When the returns were tallied, an easy win for Landon was indicated. This highly respected periodical staked its reputation on this outcome. With such a huge sample how could anything go wrong?
Well, as strange as it may seem, the poll was a miserable failure. Two sources of bias that were unfortunately in the same direction raised havoc with the results. First, the poll excluded nontelephone owners and hence also included a disproportionate number in the older age groups. Since many more Republicans (the wealthy) in these depression years owned phones than Democrats (the poor), it is not surprising that the returned ballots would favor the GOP candidate. Secondly, it is a fairly well known fact that members of the party out of power are far more likely to return ballots through the mail than members of the "in" party. This again supported more Republican ballots. These two sources of bias formed a potent combination that pointed toward a Landon victory. It is interesting to note that in the actual election, Roosevelt swept all states except two and won in a landslide. It is also of historical note that several years after this debacleThe Literary Digest went out of business.
As an ironic footnote to the above story, The Literary Digest, using the same sampling technique, correctly called the outcome of the 1932 election. Recall that Roosevelt ran against the incumbent Republican, Hoover. Again, economic problems were the prime issues in this campaign. However, the above two sources of bias were in opposite directions and tended to cancel one another out. That is, the use of telephone directories favored Republican returns but the "out of office" phenomenon favored Democratic returns. Thus, through sheer luck,The Literary Digest correctly predicted a win for FDR.
Here are some important lessons from these historic presidential polls:
The Above Was Archived on 10 November 1996
If you have taken a basic statistics course, when the topic of probability was introduced you no doubt heard the instructor mention the timehonored coin flipping example. Flip a penny and the chances of getting a tail (or head) is 1/2. No problemthis is a concept that a primaryaged child can understand.
But change the scenerio slightly. Suppose a penny held vertically on a table by the index finger of one hand is spun vigorously with a flick of the other index finger and allowed to come to rest flat on the table. Is the probability still 1/2 of either a head or a tail facing up? Well, the headnodders in the first few rows generally smile warmly and shake their heads up and down in agreement ( By the way, all you aspiring statistics instructors should enlist at least five headnodders prior to the second week of class to offer you constant supportive feedback during the entire semester :)). However, the more the students reflect on this situation, the more uncertain they become. Is spinning really the same as flipping? Finally, a carefully planted confederate toward the back of the room timidly suggests that maybe we should replicate the experiment a number of times and see what happens. Yes! Yes! Yes! Just what you want as an instructor. You quickly seize this opportunity to introduce the class to Monte Carlo type probability. You announce an extra credit assignment for everyone in the class. Each student is instructed to select a relatively shiny penny without noticeable wear, spin the penny on a table 100 times, and record the number of tails that face up. A deadly silence settles over the classroom! The students now realize they have been hoodwinked into performing a rather embarrassing act, particularly if their dorm roommates are watching that evening. Spin a penny 100 times on a table and watch it land what type of looneyness is this? Before any student utters another word, you quickly remind them that all the results will be posted and discussed next meeting, and then you grudgingly dismiss them two minutes early.
Next meeting the fruits of your wellplanned operation are realized. Thirty of your 35 students complete the experiment. Wow! you gleefully think to yourself. That is 3000 replications of the pennyspinning experiment. Methodically, you start around the room asking each student to report the number of tails he or she obtained on the 100 trials. The numbers roll in and you write them on the chalkboard: 65, 59, 57, 64, 52, 70,... The students sit in utter amazement as a definite pattern unfolds. Overall, the numbers appear to be much larger than 50! When all the numbers are collected, a mystified student in the rear of the room suggests that we average the 30 results. Obligingly, I ask a student with a fancy calculator and a pocket protector in the front row to add up the results and find the mean. In a wink of the eye, the student blurts out 62.12. This is totally unreal! Can we place any faith at all in this finding? Does this mean that if we spin a penny on a table many times the coin will fall with tails facing up about 62% of the time?
*DISCUSSION*
This is no abberation. Experts refer to this phenomenon as the "pop bottle cap
effect". Find a cap from an old 16 oz. bottle of Pepsi or Coke and spin it in the
same fashion we did the penny. About 90% of the time or more, the cap will fall
with its top facing down and the sides facing up. Now how does this relate to the
penny? If you examine closely a relatively new shiny penny, you will observe that
the edge around the penny protrudes further on the tail's side than on the head's
side. Thus, the extra edge on the tail's side simulates the side of the pop bottle
cap although certainly not as pronounced visibly. The experts proclaim that the
extra edge produces results that in the long run converge on 60% tails facing up.
Of course, if you use a worn penny, this advantage in favor of tails disappears.
I can indeed attest to these results. In the four or five years that I have used
this experiment in class, the results have hovered right around 60%. Amazing but
true. I then tell my students they have a sure way of winning some money. Engage
a friend ( maybe your nosey roommate from last night) in a four or five hour
pennyspinning game and bet on tails each time!
Gosh Henry! It really does work!
The Above Was Archived on 4 October 1996
The field of statistics is replete with technical terms or jargon that I
prefer to call "club words" in my classes. We have a lot of fun with these since
I tell my students that they can derive much satisfaction from mastering these
and joining a very unique club. They are then able to throw these terms around
in casual conversation and blow the sox off of their friends who are not in
the "club". Let me give you a few examples of some real humdingers.
The Above Was Archived on 4 September 1996.
Visit the Best Collection of Annotated Stat Jokes in the World With Over 200 entries. First Internet Gallery of Statistics Jokes
For a Discussion of Questions You Always Wanted to Ask in a Statistics Class But Were Afraid of Looking Foolish See Sticky Stat Wickets.
Also, If You Want Information About the Author That Created This Set of Pages Check Home Page of Gary C. Ramseyer.
Please email comments about this page to vmorrow@wowway.com
Page last revised on 17 July 2010
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