Columbia University College of Physicians and Surgeons

Journal of Statistics Education Volume 14, Number 2 (2006), jse.amstat.org/v14n2/ancker.html

Copyright © 2006 by Jessica S. Ancker all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

**Key Words:** Statistics education; Statistical language.

Like Humpty Dumpty, statisticians often use words to mean just what we choose them to mean. So “significant,” to a statistician, no longer means “important” but rather “unlikely under the null hypothesis.” Like Alice, students can understand the concepts only by learning to use the new language.“I don’t know what you mean by ‘glory,’” Alice said.

Humpty Dumpty smiled contemptuously. “Of course you don’t – till I tell you. I meant ‘there’s a nice knock-down argument for you!’”

“But ‘glory’ doesn’t mean ‘a nice knock-down argument,’” Alice objected.

“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean – neither more nor less.”Lewis Carroll, Through the Looking Glass, The Dial Press, Toronto, 1931. p. 238.

Such specialized statistical vocabulary can be a powerful tool for experts, but familiar terms can sometimes be better for helping beginners understand new ideas in applied statistics courses. In particular, the tricky concept of a conditional event can be clarified if teachers avoid the unfamiliar technical phrase “A given B” and instead describe it as “A within B,” a wording that encourages an interpretation in the context of set theory. To further encourage a set-theory interpretation, teachers can describe conditional probabilities as “subset probabilities.” Introducing the ideas in easy-to-understand set-theory language can help new students focus on the important concepts and avoid several of the most common mistakes with conditional probability.

These mistakes stem from the same underlying problem, which is failing to recognize when the sample space – or denominator of the frequency calculation – has changed. The events A|B, B|A, and A are all thought to be the same event, and students try to describe them all with the same techniques. Effective teaching methods should thus draw attention to whatever features distinguish the events as different. The phrase “A given B” may not be sufficient to draw attention to the existence of a new category of events, perhaps because the phrase is rarely used outside of statistics. Describing conditional events as “A within B” or “A in B” can encourage students to visualize one event as a subset of another, thus emphasizing the existence of a new category. This language thus emphasizes the concept of the “reduced sample space” frequently used to teach conditional probability in probability texts (Ross 2002).

Describing conditional events as “A in B” is particularly helpful when referring to Venn diagrams or contingency tables. Both are powerful tools for visualizing event spaces and probabilities, possibly because they allow students to conceptualize them as frequencies, which are cognitively simpler to handle than probabilities and tend to lead to more accurate conclusions (Cosmides and Tooby 1996; Hoffrage, Lindsey, Hertwig and Gigerenzer 2000).

Unfortunately, conditional events in statistics sometimes become confusing if conceptualized as sequential. Consider the following probability problem from a biostatistics text (Pagano and Gauvreau 2000). United States life-table data in a certain year shows that the probability of a male birth is 0.512. If two pregnant women are chosen at random from that population, what is the conditional probability that both will give birth to boys given that we know that at least one did so?

If A is the event that at least one child is a boy, then the problem can be solved by applying the definition of conditional probability as follows:

Unfortunately, if we encourage students to focus on the sequence of events, then it appears that the sex of one child magically influences the sex of other. Even students who correctly apply the formula are likely to reject their solution as wrong because they know that for independent events such as births among randomly selected women, one event cannot affect the probability of another event. (In fact, b1 and b2 remain independent, although the events and A are not.)

Set-theory language de-emphasizes the time sequence of the events by drawing attention to the fact that revealing the sex
of one birth reduces the sample space. For problems like this one, the conditional probability of A given B can be
expressed as the probability of A *within* B. The problem can be stated: “What is the probability that both children
will be boys, within the subset of events in which at least one child is a boy?”

This rephrasing of the P-value becomes problematic if students conclude that the complement of the P-value is either the
probability that the null hypothesis is false or the probability that the alternate hypothesis is true
(Rossman and Short 1995). For example, some students reason, “If P = 0.01 for
the difference between two means, then there’s a 99% probability that the difference is not due to chance.” When students
learn about the P value as “the probability of the event *in* the world described by the null hypothesis,” they are
less likely to be misled into thinking that its complement is the probability of the alternative hypothesis.

A teacher should consider students’ vocabulary as part of this body of preexisting understandings. Introducing conditional probability in familiar language rather than the traditional statistical terms can be a way of linking new ideas to existing knowledge, especially when the familiar language emphasizes a visual representation of the conditional events in the context of set theory. The set-theory phrasing can be used with traditional statistics texts or with a variety of elegant and useful teaching methods that have been proposed for demystifying conditional probabilities (Warner, Pendergraft and Webb 1998; Hoffrage, et al. 2000).

Students in introductory statistics classes often feel like so many Alices in a disconcerting statistics Wonderland. By using familiar language, statistics instructors can prevent this feeling by helping students seeing new concepts as relevant to the real world.

Hoffrage, U., Lindsey, S., Hertwig, R., and Gigerenzer, G. (2000), “Communicating Statistical Information,” Science, 290, 2261-2262.

Knowles, M. S. (1988), *The Modern Practice of Adult Education: From Pedagogy to Andragogy*, Englewood Cliffs, N.J.:
Cambridge Adult Education.

Lovett, M. C. and Greenhouse, J. B. (2000), “Applying Cognitive Theory to Statistics Instruction,” *The American
Statistician*, 54, 196-217.

Pagano, M. and Gauvreau, K. (2000), *Principles of Biostatistics* (2^{nd} ed.), Pacific Grove (CA): Duxbury
Thompson Learning.

Ross, S. (2002), *A First Course in Probability* (6^{th} ed.), Englewood Cliffs, NJ: Prentice Hall.

Rossman, A. J. and Short, T. H. (1995), “Conditional Probability and Education Reform: Are They Compatible?” *Journal of
Statistics Education* [Online], 3(2).

jse.amstat.org/v3n2/rossman.html

Warner, B. A., Pendergraft, D., and Webb, T. (1998), “That Was Venn, This Is Now,” *Journal of Statistics Education*
[Online], 6(1).

jse.amstat.org/v6n1/warner.html

Jessica S. Ancker, MPH

Department of Biomedical Informatics

Columbia University College of Physicians and Surgeons

New York, NY 10032

U.S.A.
*jsa2002@columbia.edu*

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