Reasoning With Mathematics

Lesson Five - Patterns: Leading and Misleading

Back in the 1950s and '60s stories circulated in the U.S. about a supposed connection between prices on the New York Stock Exchange and the length of fashionable women's skirts. According to the stories, stock prices went up when skirts were short; stock prices went down when skirts were long.

No one knows where the story originated, but it made the rounds in newspapers, magazines, radio, and person-to-person conversation.

Some versions hinted that if investors watched women's fashions, they could make more money ­ or at least avoid losing money ­ on stock investments. Others hinted that if people watched the stock market they could predict what length skirts would be fashionable in the next season. In other versions the point seemed to be that both the stock market and fashion dictates were seriously crazy.

Sample Explanations

When telling someone else about a pattern most of us say at least a little about how or why the pattern works. Four different explanations of the stocks and skirts pattern come to mind right away:

1. Short skirt length made stock prices go up

1. Rising stock prices made skirts get shorter

1. A trend toward taking more risks caused all kinds of things ­ including skirts getting shorter and stock prices going up; a trend toward avoiding risks did the opposite

1. Skirt length and stock prices just happened to move up and down at the same times

The first two explanations assume that one thing directly causes the other. Most of us are drawn to these kinds of explanations because they answer ­ or seem to answer ­ questions that are very important to us: How does this work? What's going on here? Why?

But those first two explanations also seem far-fetched. There just doesn't seem to be any way that changes in skirt length would influence stock prices or vice versa.

The third explanation assumes that changes in skirt length and stock prices are both examples of a more general trend in risk-taking. According to this, short skirts and rising prices happen because people are willing to take more risks; long skirts and falling prices happen because people want to avoid risks. In other words, all this business about skirts and stocks are merely part of a larger pattern of risk-taking.

The fourth explanation says skirt length and stock price are unrelated, that the pattern is a coincidence.

As you go along in this course you'll use some mathematical approaches to thinking over explanations like these, including ways to test them out.

Leading and Misleading

Patterns can lead, no doubt about it. Many of the examples mentioned so far in this course include some description of how a pattern helps explain something . Laws of physics explain how planets move ­ giving the patterns used to discover another planet like Neptune; how human beings develop antibodies helps explain a pattern leading to smallpox vaccine. These and countless other patterns lead somewhere, often to the solution of a problem.

Patterns can mislead, and there's no doubt about that either. The skirts and stocks story, simple and a little silly sounding now, illustrates some of the ways we can be misled.

We can get the pattern wrong, for one thing, and we can do that in many different ways. For instance, the people who saw a pattern in skirts and stocks may have used bad or irrelevant information. (In more mathematical language, their data might have been off.) Could be they used some irrelevant measure of skirt length or a questionable measure of stock prices. In the early days of computer use there was a phrase for this kind of mistake: Garbage in, garbage out.

There are also many ways to misinterpret patterns. We're tempted to assume that a pattern we see one time or place applies to all times and places. The supposed connection between skirts and stocks was seen during the late 1940s and the 1950s. Some people who spread the story said the same pattern held during the 1920s and '30s, too. But how about the 1970s, '80s, and '90s? There's a phrase to remind us of this potential problem: That was then, this is now.

We can get into seemingly endless tangles when trying to interpret what a pattern might tell us, for instance about cause and effect. Is there any cause and effect relationship between skirt length and stock prices? And if there is, which is the cause and which the effect? One of our old but sometimes useful clichés warns us about this problem: Which came first, the chicken or the egg?

We can get tangled up thinking about whether a pattern means anything at all. Skirt length and stock prices seem so different from each other that it's hard to imagine any reason they are alike or would affect each other. Another of our clichés cautions us Don't compare apples and oranges.

And we can run into cases where someone misrepresents a pattern and its meanings to us, either intentionally or unintentionally. Two closely related clichés caution us about this: There are lies, damned lies, and statistics and Figures don't lie but liars can figure.

A Misrepresentation

Cases of someone misrepresenting a pattern to us deserve some special attention. Most of us are consumers of mathematical information far more often than we are producers. We frequently must look at what someone else claims is a pattern, then try to understand and interpret the claim, a process which includes checking how accurate and valid it is. Although it's impossible to say how often patterns are misrepresented to us, it happens often enough to matter; and misrepresentation sometime are about problems or topics important to many people.

Many people care about divorce rates. According to most accounts, divorce rates have been increasing in the United States and many other societies as well. For instance,

The New York Times for August 29, 1995 reported the following facts under the headline

In China, Rapid Social Changes Bring a Surge in the Divorce Rate:

"The divorce rate in Beijing leapt to 24.4 percent in 1994, more than double the 12

percent rate just four years ago, Beijing Youth Daily reported this month. Although

statistics can be misleading -- the divorce rate is measured by comparing the number

of marriages and divorces in a given year -- officials say it is rising all over China, and

faster in cities than in the countryside. The national divorce rate is now 10.4 percent,

still far behind the United States, where the divorce rate rose sharply in the 1970's to

around 50 percent, where it has remained."

This excerpt is carefully written, probably more careful than most news stories and much more careful than most conversations. For one thing, the article makes clear that it is passing along information from another source, Beijing Youth Daily. And it says "Although statistics can be misleading ." before describing how the divorce rate is measured: the divorce rate is measured by comparing the number of marriages and divorces in a given year.

Yet it would be a misrepresentation if someone looked at this news story, then said "My gosh, according to this article over 24% of marriages in Beijing end in divorce and around 50% of marriages in the United States do!"

It's misrepresented because the statement doesn't follow from how the divorce rate is measured. How was that rate arrived at? The divorces in Beijing during 1994 were added up; the marriages for 1994 were added up; the number of divorces was divided by the number of marriages; that decimal was converted to a percentage.

Let's make up a number of divorces and a number of marriages to see the calculation work out (with numbers picked to get exactly 24%, just to make the arithmetic shorter):

Divorces in 1994: 11,712

Marriages in 1994: 48,800

11,712 / 48,800 = .24 = 24%

So why is it a misrepresentation to say "According to this, 24% of Beijing's marriages end in divorce"? Well, to say that you would need to know how many of the 1994 marriages (or any year you wanted to check) eventually end in divorce.

You would have to do something like this: 1) Record the number of marriages in 1994. 2) Wait some period of time ­ say a year later ­ count how many of those marriages had ended in divorce, then calculate the percentage. 3) Wait some more ­ say one more year ­ count the total number of divorces in the group you're tracking, then calculate a new percentage. 4) Keep going this way as long as you think it's useful or until every marriage is ended.

You can spot some other problems with the measure of "divorce rate" reported in the news story, especially if you use a table or graph to look at the results.

Divorces	Marriages	"Rate"
11,712	48,800	24%

But imagine a situation where the number of divorces stays exactly the same but the number of marriages changes from one year to another:

Year

Divorces

Marriages

"Rate"

1989	11,712	117,402	9.98%
1990	11,712	96,790	12.1%
1991	11,712	73,200	16.0%
1992	11,712	50,921	23.0%
1993	11,712	44,704	26.1%

The table makes a couple of things clear right away. If the "divorce rate" is measured as reported in the newspaper article, the rate could change because of changes in the number of marriages, even if the number of divorces stays steady or goes down, too. To see an extreme version of this situation, imagine some year in the far future where the table looks like this:

Year	Divorces	Marriages	"Rate"
2045	11,712	13,608	86.1%
1990	11,712	11,801	99.2%
1991	11,712	11,605	100.1%

Which raises the question of how there can be more than 100% of marriages ending in divorce.

Don't conclude that China is the only country where "divorce rates" are measured this way and the only place where you will encounter this misrepresentation. Official divorce rates in the United States and Canada are measured the same way; most other nations might use that measurement, too. One obvious reason for using it is that it takes too long and costs too much to do it as described earlier.

Visual Misrepresentation

Charts, illustrations, and other graphics can communicate mathematical information quickly and powerfully. Visual tools are widely used all our mass media ­ television, print, online, posters, billboards. Most of us are so accustomed to them we are barely aware of their presence.

Like other tools, they can be misused, again either intentionally or unintentionally. And because visual presentations can be so powerful, misrepresentations can be powerful, too.

Edward Tufte has studied and written an great deal about visual presentation of information. In fact, one of his books about these matters is titled The Visual Display of Quantitative Information and another is called Envisioning Information.

Tufte sets out some standards for graphic quality and one of them is present the information honestly. He provides many examples of graphics that fail to meet this standard in The Visual Display of Quantitative Information, including this one from a New York City newspaper:

[Insert gas mileage graphic from N.Y. Times]

This graph claims to show the mandated fuel economy standards set by the US Department of Transportation. The standard required an increase in mileage from 18 to 27.5, an increase of 53%. The magnitude of increase shown in the graph is 783%.

Tufte worked out a short formula for scoring misrepresentations like these, a score he calls "the lie factor." The formula starts by calculating the percentage change or difference in the basic data, then calculating the change or difference shown in the graphic. (You can see those in the paragraph above. In the original data the difference is 53% and in the graphic it's 783%.) The last step is to divide the second percentage by the first one. Perfect agreement between the two would be a score of 1. When the graphic overstates the difference, the score would be greater than 1; when the graphic understates the difference, the score would be less than one.

The graph shown here has a whopping lie factor score of 14.8, which means it's badly exaggerating the difference.