Wednesday, November 13, 2013

How People Misuse the Transitive Property

In math the transitive property states:

Whenever A > B and B > C, then also A > C

Whenever A ≥ B and B ≥ C, then also A ≥ C

Whenever A = B and B = C, then also A = C.

Hence, on a test if student A scores 7 points better than student B and student B scores 8 points better than student C, then we can conclude student A scored 15 points better than student C. However, most people make the mistake of thinking they can use the transitive property to evaluate trends and prognosticate future outcomes. This is what I like to call “eyeball statistics” and wrote about it here:

For example, if team A beats team B and team B beats team C, then we like to assume that team A will beat team C. This is not necessarily going to be the case. And if team A beats team B by 10 points and team B beats team C by 10 points, it is highly unlikely team A will beat team C by exactly 20 points. The reason for this is simple, in the real world there are a lot of variables that go into deciding a particular outcome – injuries, illness, matchup styles, mistakes, and of course people have good and bad days.

We see people try to use the transitive property incorrectly in politics all the time. For education the government is using standardized testing to decide if students pass a minimum set of requirements for reading, writing, and math. The standardized test encompasses a few hours of one day. Bureaucrats make the wrong assumption that children doing well in school will translate to higher test scores and conversely children doing poorly in school will translate to lower test scores. But this is not necessarily true. A good student can have a bad day due to extenuating circumstances such as trouble at home, lack of sleep, or other variables that may turn a good student into a poor one on any given day. The same can be said of a bad student; they can have a good or lucky day and pass the test. To assume a few hours of one day correctly assesses the aptitude of students is an insane proposition. The same test can be administered on a different day with drastically different outcomes for each student. It makes more sense to evaluate the student’s overall body of work over the course of the year instead of over a few hours. This is the only way to average out good and bad days.

On climate change we assume if there is a trend of higher global temperatures for a few straight years then we automatically assume the upcoming year will be warmer. If the earth is hit with a few violent storms we all the sudden assume that storms are increasing with intensity. This is human nature because it is what makes sense from a simple mathematical concept – the transitive property. However, climate change is not a simple problem to statistically evaluate using one or two variables – there are thousands of variables that must be considered. The same can be said about education, there are hundreds of unique variables and extenuating circumstances that affect each student.

A day does not pass when I see some genius try to extrapolate world issues by using the transitive property on data – this is wrong and it has to stop.

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