Too Far from the Average

By Bobby Neal Winters

Buster Williams was sitting in his Elementary Statistics class struggling not to go to sleep.  Sometimes he bit his lip, sometimes he slapped himself on the neck--a trick he’d learned while driving all night to get to Panama Beach--but nothing he did seemed to work.  The problem was at the front of the classroom: the Standard Deviation, or Dr. Bottlebutt as he was known in polite society.

Professor Bertram Bottlebutt, covered from head to foot with chalk dust, wrote one last equation on the board.  He turned for the first or second time during class and asked, “Are there any questions?

Twenty pairs of glazed eyes, Buster’s included, didn’t even blink in response. This was a trick considering he’d been drifting off to sleep before.

“In that case,” he said, “class dismissed.”

Buster felt a rush of adrenaline. His heart soared.  Existence was worth continuing again.

For the first time in an hour, the students in the room showed life. Notebooks were shut, slid into backpacks, and motions were made toward the door.

“Wait, wait, wait,” he called out.  It was something students had heard before.  Indeed several of them did passible imitations of Dr. Bottlebutt, and competed against each other with their “Wait, wait, waits”.

“I need to talk to ... uh... Williams and ... uh ... Young,” he said.

Buster, hearing this, uttered an expletive to himself.

Since being in Dr. Bottlebutt’s class, he’d gotten used to being called “Uh...Williams.” Everyone who took a class from Bottlebutt had the same first name. Students also worked uhs into their imitations of Dr. Bottlebutt.  This was a man who could give hour upon hour of lecture of mathematics composed entirely of abstruse equations on the blackboard without any notes at all but could remember Bob, Emily, or Buster.

The room emptied of students with the expressions of those who, having eaten their morning live frog, knew everything would be better for the rest of the day.  All but Buster and Young who remained.

“Uh...Young” was actually named Betty.  Buster had been sitting in a desk one behind and to the left all semester just because he wanted to be able to see her.  She was...gorgeous. Everything that nature equipped young women with to get the attention of young men was present in Betty not in abundance but in perfect proportion.  And everything was presented in such a way that Buster could find no way to ignore.  As he stood, working on an inventory of Betty starting at her ankles, working up her calves to her thighs...

“Mr. Williams,” came Dr. Bottlebutt’s voice. “Could I please have your attention?’

“Uh...” oh God it must be catching he thought, “Sorry.”

“I was saying my assistant lost your and Miss Young’s grades,” said the professor, seeming quite chagrinned. “I always put the papers in alphabetical order, and apparently he lost the two on the bottom.”

Professor Bottlebutt had his own way of dealing with exams.  He would grade them himself and work up the statistics for the class.  He would then hand back the papers to the class for a few minutes to allow the students to look them over.  Then he would put them into alphabetical order and give them to his assistant to record.

“If you tell me your correct scores, I will have my assistant record them,” he said.

Buster was astonished.  Dr. Bottlebutt was willing to take his word.  This was an opportunity for him to improve his grade.  He was straining through his brain for a number when he heard Betty’s voice from next to him.

“Ninety-eight,” she said.

Mesmerized by the sweet sound of her voice and not being the sharpest tool in the said, Buster followed suit.

“Yeah, me too, ninety-eight,” he said.,

The professor, who was sitting at his desk, glanced upward at both of them with a look on his face that was not indicative of credulity.

“Ninety-eight,” he echoed.

“Uh...,” Buster began but he didn’t get far.

“You know,” Bottlebutt said, “I don’t think that is exactly believable.  Let’s see.  The class average, which I calculated before your papers were lost, was about 70 and the standard deviation was about 6.  This means 98 is just over 4 standard deviations from the mean. That is unlikely for any population, but we have only 20 students in class.  By Chebyshev’s Rule, only 1 over 4 squared of the data can be that far from the average, that only 1 in sixteen. Now 20/16 is between 1 and 2.  So there is some chance that one of you can have that grade, mathematically speaking, but by Chebyshev’s it is mathematically impossible for both of you to have a grade that far from the average.”

Buster stood with his mouth open.  He noticed that Betty didn’t move a muscle. Not even her eyes moved.  Her chest was still moving in and out with a beautiful rhythm...

Butster looked down at the professor and noted that he was still staring up at them.

“Does either of you have anything more to say?”

Buster glanced Betty’s way and her expression was as innocent as a dove.  It made him ashamed of lying, but before he could own up, the professor continued.

“It might be well for you to know that since I’d figured up the statistics for the class  I was able to figure out some information from the rest of the papers,” he said. “The sum of your scores is 140.”

Buster thought of the score he’d really made, 70, and then, in spite of not being the sharpest tool in the shed, figured what Betty’s score had to be. She’d made 70 too.  Two thoughts raced to share his mind. One: They’d made the same score and had something in common.  Two: She was lying her gorgeously perfect backside off.

Buster looked at Betty.  The expression was so slight that he didn’t know whether it was there or whether he was imagining it.  It was in the eyes; it was around the curve of her beautiful lips. Please don’t give me away.

“Uh...Dr. Bottlebutt, did I say 98? I must have misspoke,” Buster said as he frantically tried to do mental arithmetic. “My real score is...”

“Forty-two?” the professor interrupted him.

Buster did a quick 98+42=140 and answered, “Uh...yeah.”

“There is a problem with that,” Dr. Bottlebutt said. “Forty-two is also more that four standard deviations from the average.  There can only be one that far away, high or low.  Do you want to stick with that?”

Buster looked at Betty, who was the most beautiful example of the human female that he’d stood in the presence of.  She was still holding that expression. It was so subtle, but so compelling.

“Yes,” he said, “sure.”

Bottlebutt looked up at both of them.

“You know,” he said, “I could take ten or twenty minutes and work this out. It’s not hard. I’ve got all the information I need.  But I think there is a learning opportunity to be had here, so I am going to take both of your words as true.  Good day.”

With that, they were dismissed.

Buster walked into the hall with Betty.

“Betty,” he began.  His next words were going to be about maybe going to get a latte or something, but no one was listen.

Betty walked over to the tallest, most broad shoulder man Buster had ever seen.  She put her arm around him and he put his hand in her back pocket. And the strolled away.

Buster, more than a bit crestfallen, got kind of a bitter taste in his throat. He turned and saw Bottlebutt who had clearly taken in the whole scene.

Buster stood there wordless and didn’t expect the professor to say anything, but he did.

“There are some lessons harder in the learning that math,” he said.

Elementary Statistics

By Bobby Neal Winters

The Learning Phase

Mathematics and Statistics are separate disciplines. This is something that both mathematicians and statisticians are insistent upon. Statisticians use mathematical tools much in the same way physicists and chemists do, and many mathematicians take statistic for the same reasons they take physics: as an application of their mathematics and to make themselves more marketable when it comes time to look for a job.

My undergraduate degree included a course in Probability and Statistics, but it was very theoretical in nature without much indication of how one might actually apply this in the real world. This is common among undergraduate math degrees. Also commonly offered is a class in elementary statistics that is rather short on theory but long on practical applications.

Fifteen or twenty years ago, as a part of my responsibilities for teaching service courses in the math department, I was assigned elementary statistics. It was a course I’d never had. The first time I taught the course, being a team player at heart, I used the text that had been adopted by the department and, as I taught, staying a few sections ahead of the students, following the syllabus that had been determined by the department.

It was in this way I learned statistics. As I learned it from the book the department was using, that book must have done something correctly, but--and you saw this coming--I hated the book.

Why?

You may be familiar with the French post-Impressionist painter Georges Pierre Seurat. If you are, then you probably know more about him than I do. What I know is that he painted pictures using dots. If you stand with your eye just an inch or two away all you see are dots; if you stand back there is a park or dogs playing poker or something else.

This text was written like that. When I was done teaching the course, I took a step back and saw the whole picture at once, but I was not convinced that any of the students would do that. For them, it was all dots. Or, if I my phrase it another way, it was just one damn thing after another without any connection between them. Nevertheless, being a team player, I continued in this way.

The Synthesizing Phase

In thinking about the course, it occurred to me there were four parts: descriptive statistics, basic probability theory, applications of the Central Limit Theorem, and advanced applications.

The descriptive statistics consisted of the topics that one, for the most part, could cover in a high school or even middle school class: Make a frequency distribution; list numbers from largest to smallest; find the average; find the standard deviation; draw a bar graph; draw a pie chart. Students, even the ones who were ultimately going to fail the class, rocked on this. Among those who showed up for the exam, it was rare to see anything less that a B. This made the second part of the class even more tragic.

Students find what I refer to as basic probability theory to be difficult. Topics in this section can be anything from if you have a can that has 300 red beads and 700 blue beads, then what is the probability of drawing a single blue bead (Yes, it’s 0.7.) to the hypergeometric distribution. This section cleaned the students clocks, which is to say they uniformly found it to be quite difficult.

The portion of the course I describe as “applications of the Central Limit Theorem” is actually the meat of the course. This is the part of the course that our clients in the university, i.e. those who use statistics in their classes, want their students to know. This includes estimation and hypothesis testing.

The final portion of the course consisted of the sample proportion, which is used to estimate the percentage of the vote a politician will get, and the chi-square test, where the question of whether particular models fit is discussed.

Teaching as Literature

After I’d taught the course a number of times, it was clear that it all did fit together like one of Seurat’s paintings, but that the students might be better served if it were taught like a classic work of literature. By this I mean, I thought I should take advantage of certain opportunities to foreshadow concepts.

For example, there is something called Chebyshev’s Rule. It is the most difficult topic covered in the first section. To use layman’s terms, it says that only a small proportion of the data can be very far away from the average. This opens up the opportunity to mention proportion, which has already come up with relative frequency, and foreshadows the central means of hypothesis testing. In terms of literature, it helps to tie the course together.

Spending more time on Chebyshev’s Rule also serves to make that first section a bit more challenging. If it is de-emphasized, as I tried a few times, there is little to give the students that they are now in a college course because so much of the material in that first part of the course is at a middle school level. Chebyshelv’s Rule, by way of contrast, can be challenging in a number of ways.

There are other opportunities to create ties they run the length of the course. I use the jar of colored beads as a recurring example. It’s used when I introduce probability, when I talk about Bernoulli Trials, and when I talk about the sample mean. By this means, and others, the student has a better opportunity to see that the course is unified and that it’s not just one damn thing after another.

Embedding in Time

Shortly after I’d separated the Elementary Statistics course into four parts, I took a year of sabbatical a Brigham Young University. They had a testing center that was very popular with the students. Teachers left copies of the exam with the testing center for an period of time--I think as much as a week--and during that interval of time, students could come and take the exam when they felt as if they were ready. Staff at the center proctored the exam, so pressure to be a policeman was removed from the professor. As I said, the students loved it and I grew to love it, but there was a catch. In order to use it, you had to schedule your exams ahead of time.

Up until that point in my career, I had not. I had exams when I felt we’d covered enough material, and the idea that this could be a predictable thing was foreign to me. In order to use the testing center, I scheduled my exams and discovered that this was not such a difficult thing to do.

When I can back from sabbatical, I scheduled my exams for Elementary Statistics and discovered a number of things. One of them was that the students didn’t complain. Indeed, while my students have never said so directly to me, I’ve read studies that support the idea that students like structure.

More important to me, however, was the discovery of how easy it made everything. Instead of having to weigh the decision of whether to have a test on a particular day, there it was on my calendar! I could make out the damn test and not procrastinate. Once those dates are set, they serve a similar function for the course as the arbor does for a grape vine: the course just grows around them.

Quite frankly, in some very real sense, when the dates are set for the exams and other assignments there is nothing much else for me to do.

Assessing and Modifying

As you’ve seen, course design is a dynamic process. I’ve modified my course as I’ve learned new things. I have changed as I’ve observed that worked and what didn’t. I was doing course design before I new the name for it. I was doing assessment before I heard the word.

That having been said, those words and phrases have power. When we teach a course, there are certain things that we want students to learn or we wouldn’t be teaching the course. Those items are called student learning outcomes. When we’ve taught those things we want to determine whether the student has learned them. This is called assessment.

When we assess, we are assessing a communication channel. The channel has two ends, the transmitter and the receiver. We give a lot of attention to the receiving end of the channel in that we assign a grade of A, B, C, D, or F to the student, and this is right in that it is the student who is paying money to learn and those who hire that student will want some measure of how well the learning has taken place.

But.

But there are two ends of that channel. For my own sake as a teacher, I need to know how well I am doing and I need to change what I do in order to make it better. When I learn what I need to change, I should carry it out.

Technology

Over the last twenty-odd years there have been numerous changes in technology. When I began teaching, I had to walk two blocks across campus to check my e-mail. I did it once a week, and frequently I didn’t have any. Now, God help me if I skip a day of clearing out the junk mail.

Regardless of the downside, technology has provided more tools to reach out to the student. I would like to mention three of these: PowerPoint, lecture-capture, and the learning management system.

Much has been made of PowerPoint and the effect it has had on bullet-pointing the learning process. There is a danger than the medium will affect the message. That is a legitimate concern. PowerPoint is not a panacea. I still use the chalkboard in many classes as I believe that the students need to see the practitioner at his (or her!) craft so that they may be empowered themselves.

If I’ve gotten my hands dirty, then they know they might have to get their hands dirty.

Yet there is so much material that simply needs to be put in front of the student and talked about. This can be done badly with PowerPoint, yes, but it can also be done badly with acetate slides and with chalk on the board. What I put on PowerPoint, I used to put on sheets from a yellow-pad and just transfer it to the board. There is no loss of empowerment and there is a great gain in efficiency.

As far as the bullet-point-ization goes, this is taken care by talking and adding meaning while you lecture over the slides. The audio can be recorded and synchronized with the slides on your computer by what is called--straightforwardly enough--lecture capture. It can be done in an almost effortless fashion and uploaded to the Internet.

On the Internet, we has access to our university’s learning management system (LMS). We can upload the PowerPoints ahead of time for the students to print-off and take notes on. We can put the captured lectures there by them for the students to listen to.

I have my whole course online, organized in four parts. In each part there is a schedule of what I plan to do everyday. All of the deadlines are there. I’ve interspersed quizzes for the students to take online, where they are automatically graded and recorded.

For the students who want to learn, there is unparalleled opportunity. The students who don’t want to learn are forced to be more creative in their excuses. It’s win-win in other words.

The Human Touch

Yet with all of this, we cannot forget the human touch. If we don’t care, the students won’t care. If we aren’t excited, the students won’t be. If we don’t think the material is important, then the students won’t.

In other words, you still have to teach.