# Statistical Reasoning

 Chong-ho Yu, Ph.Ds.

 Mr. X and Miss Y just got married. Their statistician friend Dr. Statistics says, "According to previous data, the divorce rate in the US is 60%. Thus, this couple has 60% chances that they will divorce." Their philosopher friend Dr. Human says, "You should not judge people by a probabilistic model. You should judge X and Y based upon what you know about them. They are our friends! You know that they are mature people and the chances that they will divorce is almost zero! Your approach is mechanical and formulaic." Who is right?
The ideas between Dr. Statistics and Dr. Human represent different probability models and different modes of reasoning. This article will briefly explain both of them. However, each school of thought here requires extensive study. For further inquiry, please review the references at the end.

### Probability models

In many textbooks, the concept of probability, which is the foundation of statistical reasoning and methods, is presented as one single unified theory. Actually, throughout history there are many different schools of probability (Galavotti, 2005). While it is important for learners to be aware of the diversity of probabilistic inferences, it is beyond the scope of this class to cover all of those theories. Rather, in this short lecture only three of them will be briefly discussed.

#### Direct probability

Dr. Statistics views Mr. X and Miss Y as members of a super-population, "the entire American population." The event "divorce" is a member of a super-set, "all marriages in America." In other words, Dr. Statistics treats Mr. X and Miss Y as everybody else. In the direct probability model, it is assumed that every event of the set is equi-probable and probability is derived from a statistical law governing the given population. Based on these premises, the probability of getting divorce is said to be 60%.

#### Bayesian probability

Someone may argue that it is unfair to judge this couple by the membership "American." Besides citizenship, there are many other dimensions in their lives. For instance, there are Asian Americans (race), Evangelical Christians (religion), middle class (social-economic status), master's degree holders (education), and Republicans (political orientation). Does this supplementary information change their probability of getting divorce? The Bayesian probability model uses new information as evidence. Even if there is no empirical divorce rate for those sub-populations, one can introduce subjective probabilities into the model (Berry, 1996).

#### Fiducial probability

In Fisher's fiducial probability model, the information of the sub-population does not negate the inference derived from the super-population if the statistical law governing the sub-population is unknown (Seidenfeld, 1992). Unless we know the divorce rate of each sub-category, the information about their race, religion, SES, and education is irrelevant. Since the super-set "all Americans" embeds all those subsets, a direct probability can be made to this couple.

Now examine Dr. Human's argument. Dr. Human hypothesized that mature people have less chance of divorce. However, there are some unanswered questions. First, what is the divorce rate of mature people? Second, how can Dr. Human measure the maturity of Mr. X and Miss Y? Third, is maturity a sufficient condition to sustain a marriage? Could a couple divorce for other reasons? In the Fisherian perspective, the absence of this information invalidates Dr. Human's argument.

### Modes of reasoning

Dr. Statistics and Dr. Human apply two different ways of reasoning. The former approach is called statistical reasoning or probabilistic reasoning while the latter one is rational reasoning or reasoning by direct evidence.

#### Statistical reasoning

In statistical reasoning, the judgment is made with reference to a class. Obviously, the inference made by Dr. Statistics about Mr. X and Miss Y results from statistical reasoning. Almost everyone applies statistical reasoning to some degree. For example, a potential car buyer says, "I prefer Japanese cars to American cars because most Japanese cars are better-built." By the same token, a taxi driver says, "I refuse to pick up people of certain ethnic groups, because many of them are criminals." Some people are resentful to statistical reasoning for its stereotyping effect.

Is stereotyping necessarily bad? In the first case, it may be a misapplication of statistical reasoning. Since almost every vehicle model's reliability information is available, a potential automobile buyer should find the information of particular models rather than making a blanket statement.

In the second case, it is impossible for the cabdriver to examine the background of each potential passenger. According to a report compiled by the US Labor department, driving a cab is the riskiest job in America, with occupational homicide rates high than policemen. When there is no available information about the potential passenger, statistical reasoning is the only tool for the cab driver to protect himself. That's why D'souza (1995) described the behavior of those cab drivers as having a rational base.

#### Reasoning by direct evidence

Reasoning by direct evidence is based on the relevant information extracted from involved individuals. Dr. Human adopts this approach by examining the relevant qualities of Mr. X and Miss Y. He may go even further by talking to the couple's friends, coworkers, and parents to gain a deeper understanding of their backgrounds.

#### Case study: White vs. Brown

Schoeman (1987) observed that people have more confidence in direct evidence than in statistical probability. He provided a counter-argument to this bias in the following cases: In the first case, there are 500 bunnies. 499 are white and 1 is brown. The color difference is unrelated to behavioral differences in rabbits. One of the bunnies has overturned the pellet dish, but no one observed which bunny upset the dish. Another case is similar except that a person was watching the rabbit overturned the dish. The witness claims that the brown bunny upset the dish. The first case is judged by statistical reasoning while the second case is investigated with direct evidence.

Statistical reasoning in Case 1

Since there are 499 white rabbits and 1 brown rabbit, it is reasonable to believe that a white bunny was responsible. So the error rate is 1/500.

Reasoning of direct evidence in Case 2

The witness is given a color identification test and he identifies the right color in 95 percent of the cases. It means that he will misidentify the color of the offending rabbit 5 out of 100 times.

Given a series of 500 rabbits for color identification, he will mistakenly identify 25 white rabbits as brown (5 * 5 = 25) and the brown one as brown. The error rate can be as high as 25/26!

Intuitively, the judgment in the second case is more convincing than the first one. Indeed, the opposite is true. When Dr. Human predicted that Mr. X and Miss Y will not divorce based on their maturity, the error rate of Dr. Human's assessment of maturity should be taken into consideration.

In a similar vein, Dawkins (2009) prefers scientific, indirect evidence to direct evidence from eye-witnesses by citing the following experiment: In a study the subjects were asked to observe players' performance in a video clip of a basketball game. During the game a man wearing a gorilla suit was wandering around the field. Amazingly, the majority of the subjects could not see the man in the gorilla suit. As a matter of fact, every second our sensory channels are bombarded by many pieces of incoming information, and therefore we have to employ selective perception. It is not surprising that our selective perception and recalling are highly fallible.

### Conclusion

No matter what the statistics indicates, many people refuse to be identified as a member of a certain reference class. For example, when I talked to parents about the problem of lowering academic standard in American schools, many of them admitted the problem but also claimed that their children are exceptionally bright. Someone says, "Well, 97 percent of American high school students have above average academic performance!" According to the survey results of a Christian university, a large percentage of incoming freshmen rate themselves as above the average in spirituality comparing with their peers.

This "above-average fallacy" is a common blind spot and thus sometimes we cannot trust individual information. In a study when the researcher asked the female participants to estimate the probability of being attacked if a woman walks alone in the Central Park, New York, most subjects reported a relatively high probability. But when the question was reframed to "how likely that YOU will be attacked," the estimated probability became much lower (Shermer, 2011).

Very few couples look forward to getting divorced. Instead, many expect to live happily together forever, and believe they belong to the 40 percent group while their next door neighbors belong to the 60 percent group. By talking with Mr. X and Miss Y, I may not find any evidence that they will divorce a few years later. Actually, every couple is subject to the law of probability, except my wife and me!

Last updated: 2013

### References

• Berry, D. A. (1996). Statistics: A Bayesian perspective. Belmont: Duxbury Press.

• Dawkins, R. (2009). The greatest show on earth: The evidence for evolution. New York: Free Press.

• D'Souza, D. (1995). The end of racism. New York: The Free Press.

• Galavotti, M. C. (2005). A philosophical introduction to probability. New York: Center for the Study of Language and Information.

• Schoeman, F. (1987). Statistical vs. direct evidence. Nous, 21, 179-198.

• Seidenfeld, T. (1992). R. A. Fisher's fiducial argument and Bayes' theorem. Statistical Sciences, 7, 358-368.

• Shermer, M. (2011). The Believing brain: From ghosts and gods to politics and conspiracies---How we construct beliefs and reinforce them as truths. New York, NY: Times Books.

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