Sunday, February 20, 2011

Statistics For the Mathematically Challenged

When I applied to U Mass Boston's MBA program I didn't expect to get any waivers on any courses. At the informational seminar that I attended the Dean said that they only gave waivers for courses taken in the past ten years. I graduated from college in 1997 with a BA in French Literature, so I didn't expect to have anything waived. To my surprise, they decided to waive two classes. I didn't have to take the class on How to Use a Computer (well, that wasn't really a surprise--I did send them my CV) but I also got out of the basic Math class. Huh? They didn't waive the Macro Econ class, but they waived the Math class? I took Econ in my fall term Freshman year and got a B- (and I got a 5 on the AP test in Micro.) I took Calc 2-my last Math class ever-the next term and got a C+. Also, I scored a 47th percentile in Math on the GMAT (The test is adaptive, and you can tell how you're doing by whether the questions get easier or harder. By the end of the Math section I could hear the computer thinking "I can't *make* them any easier--did someone send their dog in to take the GMAT?")

Maybe the powers that be thought "Well let's not make her take a straight-up Math class because then she'll drop out. Let's collect tuition fees from her for a few terms and see how it goes."

And now I'm taking Statistics. I haven't spent any time on Statistics since my Freshman year of high school and I did not love it then. Taking this class was one of the thought-barriers I had to get over before applying to MBA school. But I have given myself room--I'm only taking this one class this term and I one of my friends knows a girl who tutors statistics.

My Dad has taught Statistics. He is not particularly mathematically inclined, but then again he's a psychologist. Stats are necessary to his profession. Just before this term he told me about how he felt in his first Statistics class as an undergrad "There were all these Math people and me and a Biology major. I was scared." Evidently he got over it.

I understand the usefulness of Statistics. If you have a hypothesis, you must test it. And after you have tested it Statistics are a tool to help you understand your results. I understand the concept of "statistically significant." If you send out a questionnaire to 500 people, but only 10 return it than no matter what your results are, they are not statistically significant because they come from a very small sample of your population.The only thing that you have learned is that of your population of 500 only 10-2% are the sort of people who answer questionnaires.

This weeks lesson was about mean, median, mode, variance and standard deviation. Mean, median and mode are 7th grade math. Variance I still don't get, but I work in financial services so I am familiar with the concept of Standard Deviation (it means Risk and Return.)

When I opened the textbook last Sunday to get a head start on this weeks homework, I saw some awful formula in sum notation and I quailed and put the book away for a few days. When I started reading the chapter, it became apparent that the formula that had scared me was just the mathematical notation for mean-average. Add everything up and divide by the number of things you had, can in fact be expressed as a sum series--but why would you want to do that?

That, however was not the interesting part of this week's homework. The textbook gave the formula for Variance and Standard Deviation (the square root of Variance-I never knew that.) But because this is a course in Statistics in Excel, I still haven't absorbed what the formulas are or how they relate to the data. Why? Well, I wouldn't really want to do the math--even with a calculator but as painful and error-prone as that is, it actually teaches you how to relate the variables to each other in a way that selecting a column of numbers and choosing "STDEV" from the Excel function menu fails to do.

Is knowing how to do the Math important? YES-even though I'm bad at it. I'm the sort of person who does not put shortcuts on her desktop to network resources because it's important that I remember where they are. When I crunch numbers in Excel I make all the numbers, the sub totals and any side calculations I made visible because I want to make it clear how I got from A to B.

I find myself quoting/paraphrasing the Russian movie Nightwatch. "What is more important--what goes into the potion or the effect?" The answer is "The effect." But I actually care about "what goes into the potion" and why. Perhaps it's just me being a bit OCD. Perhaps it's because Math is so foreign to me that I want to see all steps to make sure I understand them.

"Do the Math" is kind of like "RTFM*" Anybody possessed with a modicum of intelligence should be able to do either. I am uncomfortable about the way that this course does not help me Do The Math.

*Read The Fucking Manual


dtm said...

What text are you using?

I know that with calculus classes, there's often something that happens where students will end up learning mechanics but not understanding why those mechanical actions do what they do because the teacher hasn't emphasized the why of it. (Because "why" isn't tested, and many students are actively hostile to being told things that aren't going to be on the test)

However, the text almost always contains a sometimes terse but usually quite thorough explanation of the why. In other words, it's there in the text if you want to devote the time to unpacking the terseness.

Now I know that this isn't always the case with stats texts - some follow this model, of having a full albeit terse explanation in the book, but some simply have references off to some other more complete text. (In these days of institution-specific customized abridged textbook editions, the more complete text may actually be the same text, unabridged)

dtm said...

As for the question of "but why would you want to do that?", I'm not entirely sure I can explain why in this case, but let me tell you a little bit about something called "procepts".

"procept" is a portmanteau of "process" and "concept" and is a word invented by academics in the field of math education to describe this thing that math does frequently where it takes something that had been a verb and packages it up as a noun.

For example, my daughter has recently learned to count by fives, and can recite "five, ten, fifteen, twenty, twenty-five, ...". Counting by fives is right now something she experiences as a process, as something she does. Eventually, this will be packaged into a concept of multiplication by five. However, this will be a difficult mental transition, as difficult as the mental transition from the process of counting ("1, 2, 3, 4" said almost as a chant) to the idea of "4" as a thing by itself, and not just the counting process cut off at a certain point.

Anyway, many subjects do something like procepts, but math is just chock full of them. It's the natural thing a mathematician does when he sees a repeated action - abstract the verb somehow into a noun so that the noun can be manipulated by other verbs.

The people looking into how math is learned (or not learned) came up with the word "procept" because they found that many stumbling blocks in mathematics are tied to one procept or another, and a failure by the student to convert the verb to a noun in their head. I haven't read too much of the literature so I don't know what techniques they recommend as the general way to trigger this internal mental shift. It does get a little bit easier with practice, and I think it becomes easier once you understand that you're being asked to do a internal mental verb-to-noun shift.

Anyway, that's why I'd write the formula for average with summation notation - because that's the notation for the (noun) concept of "sum". For the same reason, if I wanted to represent

n + n + n + n + ... + n (m times)

Sure, I could write a bunch of ns and plus signs, and put a sideways curly brace over the whole thing and write "m times" at the top, but I'd much more likely write n×m, because × is the name for the process of repeated addition.