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Mean, Median, Mode . . .

Mean, schmedian, schmode . . .

There is much that is wrong with modern math education. There is much that is missing, such as basic arithmetic skills (which is really what all math boils down to.) There is much that is unnecessary, such as matrices and logarithms and quadratic equations for students who will never see them in their lives and probably don't even have solid third-grade arithmetic skills. I could go on.

But there is also much that is detrimental, and actually destructive. There is a miserable emphasis on ugly inequalities when math is, first and foremost, the science, philosophy, art, beauty, and joy of equality. How about we get that down first?

There is far too much emphasis given to the special, oddball, or exceptional cases. For instance, when equations come along, the student is immediately hit with the exceptions - equations and systems of equations with no solutions, or an infinity of solutions. They even have official terminology for those things; who cares, who cares... Now what scientist or engineer or analyst of any sort is going to cobble up equations like,

x + 1 = x + 23 


             20 + 8r    
3r + 2 = r + ------- - 3

in real life??? The student should be having it impressed upon him that, unless something artificial and stupid thought up by math textbook writers is going on, one equation with one unknown yields a solution; two equations with two unknowns yields a solution; and so on.

In this category of destructiveness fall the mean, median and mode. Actually, I should use the singular verb there since mean, median and mode goes everywhere in math education as a three-headed monster.

Problem 1 is that the words have such a similar ring that students have a hard enough time just remembering which is which - even if they remember what the three notions are. All I can imagine is that a poet popped up at a math teachers convention one day and blurted, "Math needs alliteration!" If you think I'm being completely silly, see my web page on Maryland's standardized MSPAP test, in which all the "tasks" were multi-disciplinary, such as one combining poetry and biology.

Ok, mean, median and mode are not difficult concepts, and an average student should be able to hang on to which is which - at least until the test. But the bigger problem is that these three "measures of central tendency" are always packaged together in math education, even from grade 1 in our modern, oh-so-clever, "spiral" math curricula, where students get dribs and drabs of everything in every grade.

Now, if mean, median and mode were all useful, I'd say I guess that's the way it has to be. But the reality is that two of them are worthless, or less, while one of them is perhaps the most important and useful concept to come out of math, after the numbers themselves, and the basic operations, plus, minus, multiply, divide.

Which is it? You got it, the AVERAGE!

Well, that's what everybody calls it outside of schools. It's the old-fashioned word for our poetic "mean".

Everyone knows what "average height" means. Or, average age. Or, average income. Or, average house price. Or, average test score. Or, average number of hairs on a 3-year-old's head. Or, average anything.

We hear it almost everyday, and it has a pretty clear meaning even to those who couldn't hope to add 48 and 45 and divide by 2 in a power outage.

The trusty average is the one number that could stand in for all the others in the batch. If Farmer Brown takes 63 hogs to the meatpacking plant, and we're told their average weight is 217 pounds, you know it's the same as if each and every one of them weighed 217 pounds. It's such a handy description that most of us probably don't even waste time pondering the reality that some were over 217 lb., some were under, and maybe none of them weighed 217 lb. on the dot.

So what's wrong with the other two? Consider an example.

One day, 19 customers patronize a certain shop. Ten of them spend $1 each. Eight of them spend $2 each. One of them spends $183. What is the mean, median and mode for these purchases?

First of all, the average (mean) is just the total spent divided by the number of purchases, 19.

(10 · $1) + (8 · $2) + $183 = $209

---- = $11

The average purchase is $11, and we could comfortably view the sales activity for the day as 19 customers coming in and spending $11 each.

For the median, the student lays the data out from smallest to largest:

1   1   1   1   1   1   1   1   1   1   2   2   2   2   2   2   2   2   183

The median is the number smack in the middle of the lineup. For 19 data points, it's number 10. (Note that you will never directly get the midpoint directly by dividing by 2. Bummer. Math should be prettier than this.) So if we count up to the 10th data point, we find the median purchase is $1.

Now, I ask you, does that do a good job of describing the "typical" purchase here? Would we sensibly view the sales activity as 19 purchases, at $1 per purchase, indicating a total of $19 in sales for the day?

The mode for these purchases is the value that appears the most, and in this case is also $1. Yippee.

Now I see you jumping up and down in your seat, flapping your hand about, bursting to tell me, "Outliers! Outliers! Throw out the outliers, and then watch the median and mode shine!"

Ok, let's throw out the outliers. What I didn't tell you is that this is an upscale art gallery in a well-to-do area. Typically, four or five artworks are sold in a day. But today, a bus going to a picnic got lost. The driver stopped in town to let the passengers mosey about while he fiddled with his gps. A bunch of them stopped in the gallery and bought up post cards and note cards which had been gathering dust for years.

Oh, yeah, I almost forgot to mention the bus driver didn't even know he bumped a few rich old men on their morning walk to the art gallery. Not fatally, thank goodness.

So, after tossing out outliers, the average purchase works out to... $183! Much more impressive, but still wofully below the proprietor's expectations.

The point I'm trying to make here is, you cannot do better than the average. If you do not have insider information on outliers in the data set, you have no reasonable option but to throw all the data in the pot and crank the average. And it may very well turn out to be a perfectly acceptable "measure of central tendency". If for some reason it's not, you have no way of knowing that, so don't sweat it.

If you do have insider information on outliers, go ahead and toss them out before calculating the average. Massage the data to your heart's content. Cook up whatever final answer you want. Like they say, "Numbers don't lie; but statistics do." Of course, I'm being facetious here. Please think long and hard before you mess with the raw data.

I'd like to think that what I've said so far is enough to deal the median and mode a death wound, but let me twist the dagger a little harder.

From now to doomsday, you will never see or hear reference to the mode (outside of schools, of course.) If it ever is of interest which value occurs the most, you could always refer to the "value which occurs the most" (sort of like people who call the grassy area between the curb and the sidewalk the "grassy area between the curb and the sidewalk.") It doesn't require, or deserve, any place of honor in the pantheon of math concepts.

When all the data points in a set are different we formally, and politely, say, "There is no mode." As opposed to, "All the data points are different." Or, "Golly, the mode is worthless."

And then there's the problem of different values occurring an equally great number of time. I'm not sure there's a fully agreed upon convention here. If two data points occur the same large number of times, we call the data set "bimodal". You'll agree, that's a pretty shaky "measure of central tendency" when the two modes are the smallest and largest values in the set, say. And I'm not sure how far up the line from bimodal to trimodal to quadrimodal... to dodecamodal and beyond we go before we throw in the towel and say, "Forget it!" How about we just kick the mode out of math?

If the mode is stupid and worthless, the median is downright insidious. It's something we do hear frequently, such as "median income" and "median house prices". Take the latter, for example. When we hear, "The median house price is $180,000," it sounds so mathematical and intelligent that it must be good, right?

But it really tells you nothing for sure about the price of houses in that area. As we saw from the example above, the best you can say about the median is that it falls in the range of values somewhere, perhaps even at one extreme end or the other. This might be a subdivision with 15 houses, eight of which are assessed at $180,000, the other seven being trailer homes ranging from $35,000 to $55,000. Who knows?

All you can do when you hear the word "median" is to hope that it's somewhere near the average. That's the best that anyone can do, including the most brilliant mathematical and statistical minds. Which begs the question, so if everyone has to fall back on the trusty average when they hear "median", why not just drop the median for good?

And I'm not through bashing it yet. The calculation of the median frequently (half the time) depends on an average! It can't even stand on its own two feet as one of the "measures of central tendency"!

Whenever there is an even number of data points in the set, no one of them is positioned exactly in the middle. For instance, if there are 8 data points, the midpoint falls halfway between data points 4 and 5. The student is told to take the average of those two data points in the middle.

If we really had faith in this notion of stepping up to the middle of the pack as a "measure of central tendency", we shouldn't flinch at establishing a convention about which of those two data points to choose, the first or second, and leave the average out of it. (Actually, it's only the schools that would need such a convention. This is analogous to rounding; in real life it doesn't matter a whit whether you round a trailing "5" digit up or down, but students have to do it a certain way or else they'll get the "wrong" answer.)

And I'm still not done with the median. After they've got you finding the median, which splits the data set in two, they'll have you finding the median for each half, thereby splitting the data into four parts called "quartiles" so you can construct a "box and whiskers" plot! A what? Don't worry, no one has ever seen a box and whiskers plot in real life.

Had enough?

Now that we're all in agreement about abolishing the median and mode, I propose that schools funnel some of the recouped time and effort back into our faithful average, revisiting it as often as necessary until it's in every student's bloodstream.

In particular, I'm hoping everyone can be brought to see the average as a sort of "balance point"; not in the usual sense of balancing as much on one side as on the other, but the point at which there is as much overage on one side as there is underage on the other. So, getting back to Farmer Brown's hogs, we know that if we added up how much over 217 lb. the fat ones are, and added up how much under 217 lb. the skinny ones are, those sums will be exactly the same. No surprise at all; it's what the average is.

Having attained that familiarity, we can often knock off average problems with a bit of inspection - no algebra required. For example, if you have test scores of 86, 89, and 92, what do you have to score on the next test to bring your average up to 90? The typical A+ student will set up and solve:

86 + 89 + 92 + x     
---------------- = 90

But if he really knew the average, he could have mentally weighed up a few overages and underages relative to 90 and produced the answer in seconds.

I don't pretend for a moment that this is any sort of revelation to anyone who has stuck with me this far. (Thank you, by the way.) But I know from working with students at various levels that not many are fully "one" with this simple and most useful and wonderful of mathematical concepts - the Average!


*** Big Picture? ***

It's hard enough making the tiniest, most specific, contribution to anything in this world, but, in case there's a bigger issue here, I suggest it may be to the math education establishment's benefit to lend an ear to the outside world about what's good, what's bad, what's missing, what needs more, what needs less, and so on, in math education.

And a final freebie to mull over: the first school system in the nation to require mastery of basic math skills for all students instantly jumps to the No. 1 school system in the nation.

How could it not?


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