Teaching Bits: "Random Thoughts on Teaching"
Deborah J. Rumsey
The Ohio State University
Journal of Statistics Education Volume 17, Number 1 (2009),
Copyright © 2009 by Deborah J. Rumsey,
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 author and advance notification of the editor.
"Watching our Language When We Teach Statistics"
One of the things Iíve struggled with over my years of teaching is the language that we use to teach certain statistical concepts and how it affects my ability to get ideas across. In any situation, using language and terms that are more complicated than needed lowers our †ability to process information and to see it in a broader context. I believe the same is true for statistics.
For example, I always struggled with the word "spread," as in "shape, center, and spread." Since this term appears in many textbooks, I suspect many other teachers feel the same. Students would ask me what I meant by spread. Iíd say
itís the amount of variation in the data - then they would ask what that means. We would talk about differences and distances around a central point, and I would end up practically spelling out the idea of standard deviation
before I wanted to. There had to be a better way.
Now instead of using the term "spread," I describe variation as "diversity" in the data using various contexts. For example, suppose you have two ponds of fish; in one pond the fish are all the same age and of the same species. In the
second pond the fish are of different ages and species. You collect a random sample of fish from each pond and measure their lengths. Which data set has more diversity? The second one does. So the amount of variability is higher in the second pond than the first. Using the term "diversity" helps my students make an intuitive connection regarding the critical concept of variability.
The word "spread" is just one example where having to explain a particular term gets in the way of my teaching the underlying concept. Over the years Iíve collected many such examples. Here is my top ten list of phrases and terms
whose names and usage I believe impede the teaching and learning of statistical concepts. I certainly donít expect you to agree with all of the points on this list (or with any of them for that matter); in fact, some of you might become downright outraged by my opinions herein. My goal, however, is to get us thinking about the impact of the terms and language we use in statistics, and
to be bold enough to consider changes where needed.
- "Sampling distribution of the sample mean." We all agree that this concept is
hard no matter how you slice it, but the language we have chosen to use
here doesnít exactly help matters. There has to be a simpler way to describe
the result of taking all possible samples of size n and plotting their
means. Now add the fact that weíre supposed to talk about the mean of the
sampling distribution of the sample mean! I wonít pretend to have an easy
answer for this one, but for starters we can eliminate the word "sampling"
because it seems redundant to say the SAMPLING distribution of the SAMPLE
probability. The common definition of †P(B|A) is the probability of B occurring
given that A has occurred; the term "conditional" doesnít come up. In real-world terms the word "conditional" is used in an "if then" context. For
example "You can have a cookie on one condition - that you finish your
homework." Itís not a direct analogy because in statistics we are talking
about the probability of B depending on A, not the occurrence or
non-occurrence of B depending on A (which is a special case.)
- "Given." You see this word and you already know what Iím talking about Ė our use of the
word "given" in the context of conditional probability problems. How did
this term come about? My opinion is that this word was coined by teachers
who were frustrated that students couldnít answer questions about
conditional probability. Using the word "given" easily cues the students.
But then when the teacher changes the wording even slightly, students canít
do the problem and accuse the teacher of being tricky because the word
"given" is not there. By reducing the process to looking for a certain
word, weíve lost the opportunity to explore a very commonly used practice of
breaking down data. The media doesnít say "Given a voter was a democrat,
the probability that he/she voted for Obama is Ö" They say "For democrats,
the breakdown of their voting pattern looks like thisÖ" Big difference.
- "A or B." When talking about the probability of A union B, we use the phrase "A or
B." In common language, "A or B" means you either have A or B but not
both. (For example, you ask someone if they want a cookie or a piece
of cake - this does not imply that they can have both.) However in
statistics, when we write P(A union B) or P(A or B) we really mean the
probability of A or B or both. Moreover, the formula for the
probability of A union B, stated as P(A or B) = P(A) + P(B) Ė P(A and B), can
be deceiving; students often look at it and think we are doing the†
subtraction to eliminate the chance for both A and B to occur, which is
incorrect. We can fix this problem by simply saying that the probability
of A union B is the probability of A or B or both.
A student once asked me why we use the term "histogram" for this
particular type of graph, and I had no idea. I did a little research on
the history of statistics and this is what I found:
- Karl Pearson was the first known user of
the term "histogram" in a statistical context. The root could be from the word
"history" since a histogram provides a record. The Greek root of
history is from histor, which means "a learned man." The implication is
that a learned man is aware of history.
With all due respect to Karl Pearson,
that was then (the 1880s) and this is now; perhaps itís time to revisit the
word "histogram" and come up with a more meaningful term. One possibility is to
call it a "quantitative bar graph" as compared to a "qualitative bar graph"
(our current bar graph.) This broadens the well-known term "bar graph" and
helps students to see that histograms and bar graphs are not complete apples
- "Variance" of a data set. Teachers and textbooks alike typically include the
variance of a data set as one measure of spread (there goes that word
again!) Variance is defined and calculated; we get a number but we donít
talk about what it means. "Find the variance of this data set of studentsí
agesÖ okay, the variance is 5; on to the next question." What the students
donít realize is that the variance is not in terms of the original units,
but rather in original units squared. (For example, for the data
set of studentsí ages, the variance is 5. This means 5 years squared, which
makes no sense.) Why discuss a statistic that we canít even interpret? Itís
a part of the well-worn path that for years and years has led us to
calculating the standard deviation. Why not just find the standard
deviation in the first place and not even talk about the variance Ė are we
that afraid of a little old square root?
of "Error." The word "error" means mistake Ė itís as simple
as that. That said, why do we take a common term and change its meaning in
a statistical context? What teacher wants to explain that in statistics
"margin of error" doesnít actually measure the chance of making an error? Maybe
we could use the phrase "margin of variability" instead.
To those outside the statistical community, correlation simply means two
things are related, as in "There appears to be a correlation between
gender and political affiliation." Correlation is likened to the word "pattern."
But in statistics, the word correlation has a very specific meaning and we
fight to make sure everyone knows it. We define correlation as a number
that measures the strength and direction of a linear relationship between
X and Y Ė so in our book itís wrong for anyone to say "political
affiliation and gender are correlated." Instead of fighting the street
use of correlation, we could present our version as a special case under
the correlation umbrella. How about using the term "linear correlation," or
better yet, "statistical correlation" to get at what we are talking about?
Websterís dictionary defines the word "regression" as "to move backward," †as
in "He is regressing back to the way he was in high school." How does this
term help teachers explain that we are creating a model for a certain type
of relationship? Itís important to remember that if our goal is to
communicate with our students, we should use the language that is most
effective. The word "regression" to me is not effective, at least not for
introductory statistics students.
In general terms, the word "infer" means to generalize conclusions to a
larger entity, and it means the same thing in statistics. However, I donít
think the word "infer" is used enough in the real world to help students
make the intended connection. To clearly make the point that we are moving
from a sample to the entire population it might be better to use the
phrase "generalizing to the population;" or we can talk about the process
of "drawing conclusions" rather than the process of "making inferences." I
donít think the word "inference" is helpful or needed.
Those are my random thoughts on teaching for this time around. Now what do YOU think?
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