Fall Semester 2003

Homework Assignment One: Mean, Median, Mode
Part One. The Mean.

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 `Brand X` Corporation.

Here's the accounting of salaries:

 \$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
They must whistle while they work!

Here are the facts:

1. The average employee of Brand X Corporation earns \$80,000. (The total annual payroll of \$1,040,000 divided by thirteen employees equals \$80,000).

2. The president's wife can truthfully say that her brothers earn no more than the average employee. And,

3. Think how happy everyone will be if the president earns \$990,000 next year! Why, the earnings of the average employee will jump to \$120,000. (The total annual payroll of \$1,560,000 divided by thirteen employees will equal \$120,000.) And the president's wife could now truthfully say that she earns less than the average employee.

Question 1
What measure of central tendency is better suited than the mean to describe the "average" employee salary in this situation, if any? Please justify your answer.

Not that the use of the mean is all bad; quite the contrary. We use it daily.

Let's suppose that the thirteen employees of `Brand X` 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 `Brand X` Corporation consumed 182 aspirin tablets last week.

Here's the breakdown, ranked in order of the aspirin tablet consumption:

 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 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):

Question 2
When is the median not that useful as a measure of central tendency? When is it useful? What makes the mean more suited than the median in this example, in describing the "average" consumption (here, of aspirin).

Part Three. The Mode.

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 their aspirin (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:

 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
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).

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).

Question 3
When is the mode useful as a measure of central tendency?

With all these caveats, it shouldn't surprise us that none of the three types of average is necessarily germane to our interests. We can easily drown crossing a body of water that averages two feet deep, whether that average refers to the mean, the median, or the mode.

Last updated: Oct 28, 2003 by Adrian German for A201