Fall Semester 2003
The "mean" is the arithmetical average - the one with which we are (likely) most familiar. If, for example, we're discussing the income of a population, we total the income of all people, then divide that figure by the number of people, and that's the "mean" income. Maybe. Including children can be misleading, so we might want to total the income of all people, then divide by the number of adults, and that's the mean income. Well, maybe. Including unemployed people can be misleading, so we might want to total the income, then divide by the number of adults who are employed, and that's the mean income. But wait. Shouldn't the unemployed people be counted as having zero income? If so, what about the voluntarily unemployed?
But let's suppose there are no decisions like those to make. Suppose we just
want to know the average income of the employees of the
Here's the accounting of salaries:
They must whistle while they work!
$470,000 earned by the president of the company 100,000 earned by his wife 80,000 earned by each of his wife's three brothers 50,000 earned by his wife's best friend from high school 30,000 earned by his plant manager 25,000 earned by each of the six production workers
Here are the facts:
Let's suppose that the thirteen employees of
Corporation consumed 182 aspirin tablets last week. Then let's suppose
that we are going to buy the supply for next week. Of course, a calculation
is unnecessary - we'll just purchase the same amount. But the underlying
reasoning goes like this:
Those 182 tablets divided by thirteen employees equals a mean of fourteen aspirins per employee. Assuming that we anticipate the same number of employees to be with the company the following week, we get a projected aspirin consumption of 182 tablets. Obviously, this makes sense. That's what the employees consumed in the previous week. We'll get back to this example, but in the meantime, read on.
Part Two. The Median.
The "median" is another kind of average that is commonly used by those
who want to influence our opinion. Strictly speaking it is the numerical
middle. In the example of the
Brand X corporation, the median
income is $30,000. That is, six employees earn more than that, and six
employees earn less. Most people believe that the arithmetical average
should be used to calculate all averages and are likely to reject the
possibility that the numerical middle is ever representational, but it's
clear that this notion is flawed. After all, the mean income of the
Brand X employees is $80,000, and the median income is
$30,000. In this case, then, the median probably comes closer to telling
us what we want to know.
Not that the median is all that good. On the contrary, let's get back
to that example of how the thirteen employees of
Corporation consumed 182 aspirin tablets last week.
Here's the breakdown, ranked in order of the aspirin tablet consumption:
The median consumption of aspirin tablets is two. That is, six employees take more than two, and six employees take fewer than two. But if we base our purchase of next week's supply on the median consumption of two aspirin times thirteen employees, we'll buy only twenty-six tablets, clearly too few. Here comes the second batch of questions (answer all three):
40 taken by the wife's brother Tom 40 taken by the wife's brother Dick 40 taken by the wife's brother Harry 30 taken by the wife 20 taken by the president of the company 5 taken by his plant manager, Doc 2 taken by the production worker Grumpy 1 taken by the production worker Sneezy 1 taken by the production worker Bashful 1 taken by the production worker Happy 1 taken by the production worker Sleepy 1 taken by the production worker Dopey 0 taken by the wife's best friend from high school
The "mode" is yet another kind of average that is a convenient
tool of manipulation. Simply put, it is the most common of the
numbers cited. In the
Brand X Corporation, the modal
number of aspirins taken is one.
That is, more employees (five) take one aspirin than take any
other number of aspirin. The next most common number of aspirin
taken is forty (three employees). So if the five employees who
take one aspirin take a brand called Exasperin, the manufacturer
can claim that "more people take
(Exasperin), than any other brand,"
at least at the
Brand X Corporation.
Suppose, then, that similar numbers are obtained in a much broader study. Five thousand employees take one aspirin, and three thousand employees take forty, and so on. Again, the manufacturer can claim that "most people take Exasperin than any other brand," and in a comprehensive survey, yet. But what if only those employees took Exasperin, and we're considering buying stock in the Exasperin Corporation? Wouldn't it be interesting to discover that the sales of Exasperin accounted for less than three percent of the market?
Not that the mode is all that bad. On the contrary, let's go back to our
aspirin example. In the
Brand X Corporation, the modal number
of aspirins taken is one. But the modal aspirin taken is something
else entirely. Let's suppose that we're going to purchase the next week's
supply of aspirin for the
Brand X Corporation, and we've wisely
decided to buy at least 182 tablets (and perhaps
a few more, in the event that
the annual report is nearing publication). If we limit ourselves to only one
product, which should we select?
Let's say we've already decided not to consider the cost of the product, concerning ourselves with employee satisfaction, instead. Here's how their favorites stack up:
There are two modal choices, and either one may suit our purpose. Roboprin is the aspirin consumed in the greatest number (forty each by the wife's three brothers, a total of 120 tablets), but Exasperin is the aspirin consumed by the greatest number (the president of the company, his plant manager, and the six production workers - a total of eight employees).
40 Roboprin taken by the wife's brother Tom 40 Roboprin taken by the wife's brother Dick 40 Roboprin taken by the wife's brother Harry 30 Nouveauprin taken by the wife 20 Exasperin taken by the president of the company 5 Exasperin taken by his plant manager, Doc 2 Exasperin taken by the production worker Grumpy 1 Exasperin taken by the production worker Sneezy 1 Exasperin taken by the production worker Bashful 1 Exasperin taken by the production worker Happy 1 Exasperin taken by the production worker Sleepy 1 Exasperin taken by the production worker Dopey 0 taken by the wife's best friend from high school
That is, we can select either Roboprin (because it will be taken in the greatest number) or Exasperin (because it will be taken by the greatest number). Or, if we come to our senses in time, we can always select Nouveauprin (perhaps a wise choice, regardless).