How to Convince a Student that
an Estimator is a Random
Variable
KERSTIN VANNMAN
an Estimator is a Random
Variable
KERSTIN VANNMAN
At the University of Lulea, all students of engineering subjects take
a compulsory course in statistics during their second semester (out of
eight). The course contains the standard concepts, and we try to emphasize
the inference part. The students are taught, in classes of about 30, in
a student-activity kind of way. This means that the teacher tries to avoid
talking constantly and instead presents the theory and examples in a form
of a dialogue. I have found it most important to start every lesson by
not
talking. Instead I present a small problem to the students on a prepared
over-head transparency, and they immediately begin to solve it on their
own. The problem usually has a connection with the theory to be discussed
during the lesson but can also be a review of the previous lesson. The
most important thing is that the lesson begins with student activity rather
than student passivity. We all know how easily they fall asleep during
lectures.
Since most students have poor, or almost no, pre-knowledge in statistics from secondary school most of the concepts in the elementary course are new and difficult. One such concept to get across is that of an estimator being a random variable. To help the students get the appropriate feel for this situation I have found the following way fruitful. It is inspired by Professor Gottfried Noether and his book "Introduction to Statistics. A Nonparametric Approach".
First the students are asked to prepare this lesson by reading a few
pages in their text-book (Vännman (1979)) in advance. There they are
introduced, by an example, to the idea of a random sample and an estimator.
Then I begin my lesson by showing the following transparency (see Figure
1). I, of course, use a Swedish version.
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Figure 1.
The two questions on the transparency always start a lot of activity. Some students press their calculators finding a figure, others start to doubt everything. I try to lead the discussion into the direction of making assumptions and finding a model. After a while, and some convincing discussion, we all agree on the most simple model, i.e. a uniform distribution on 1, 2 to some unknown number N. This is an important discussion since the students realize that you usually start by simplifying reality with the aid of a lot of assumptions, some of which might be unrealistic.
But now we want to estimate N, the unknown number of cars. There are always a lot of suggestions from the students. Very often the first one is N 168. When I ask where this number comes from the answer is, "I took two times the average of the observed numbers, minus 1", i.e.
= 2
- 1.
(1)
Sometimes they skip the "minus 1", and I do not bother. The reason the students give for (1) is that the average of the observed numbers ought to be approximately equal to the average of 1, 2 N, i.e.
is approx = (N
+1)/2
But when they see that 168 is smaller than the largest observed number they reject the suggestion and go on with others. The next suggestion often is two times the median minus 1, but that is also smaller than the largest observation. Then usually the suggestion N = 237 occurs. This comes from
N = Xmax + Xmin ? 1, (2)
which in turn comes from
(Xmax +Xmin)/2 is approx = (N+l)/2
This will at least always be greater than or equal to the largest observation.
At this moment I draw a number line on the blackboard indicating the taxi numbers. We find that suggestion (2) is Xmax + the first gap. That is a good idea, but we also have other gaps. Let us use all of them. After some discussion the students usually get N = 266, which comes from
= Xmax + the average
of all gaps. (3)
Suddenly we have many different suggestions on the blackboard. What is the true number of cars? Here the students get an aha-experience. Their teacher does not know the answer. Some students react with anger. "Why didn’t you ask a taxi-driver?" Others may propose a trip to Sheffield. It is good that the students get emotionally involved.
Then some of them realize why I have brought a micro computer into the classroom. We can now simulate the situation and maybe see which method is to be preferred. I have a prepared program that first simulates the number of cars, N, using a uniform distribution on 200, 201 600. Then it simulates a chosen number of observations from a uniform distribution on 1, 2 N, and calculates the three estimates given in formulas (l)?(3). One may repeat the simulation of the observations for the same "unknown" N as long as one wants.
Since the display of the micro computer is small I ask one student to act as a secretary at the blackboard and to write down the three estimates after each simulation. At this moment the true number N is unknown and all students can follow how the estimates vary and make guesses. The part above the thick line in Figure 2 is what they might see emerge, row by row, on the blackboard. They decide themselves how many observations to be simulated, 20 is a common suggestion.
TAXI SIMULATION
20 observations per
simulation
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| True number = 489 | |||
Figure 2.
Here they can see that the value of the estimator varies, and hence the estimator may be considered as a random variable. They also realize that if we go on for a while with the simulation we will find the distribution of the estimator. We usually stop after 10 simulations. The program also calculates the average and the standard deviation of the estimates introducing the idea of unbiasedness and small variance (see Figure 2). When I ask which method they prefer the answer is immediately (3) because of the small standard deviation. Then we get the true value from the program, in this case 489.
I usually do one series of simulations on the blackboard and then sow some others prepared in advance, including an example with a larger number of simulations. (See Figure 3 and Figure 4, showing 100 simulations.) We have introduced stem-and-leaf displays and boxplots (see e.g. Velleman and Hoaglin (1981)) earlier in the course.
There are many interesting loose ends to be pulled here, depending on
how deep you want to go. I talk a little bit more formally about the matter
and then the students begin to solve exercises. But we are not spending
time checking whether estimators are unbiased or not or calculating variances
for different estimators. Instead the students use simulated values from
different distributions to compare estimators, as we did in the taxi example.
I have found many advantages with a lesson like this. The students understand, for example, that an estimator is a random variable, since they have seen how the value of the estimator varies when we repeat an experiment. They also realize that in reality we have only one observation of the estimator. Furthermore, since we want to come close to the true number they understand that unbiasedness and small variance are desirable properties. They find that model assumptions are important. Since students are active during such a lesson, they remember it. Therefore I can remind them of the taxi simulation, which is convenient, for instance, when treating. confidence intervals. Still another advantage is that the lesson provides an efficient way of using the short time available.
Sometimes I hear teachers say, "But I have no time for such spectacular shows". I do not believe that. On the contrary, you need them. I and some colleagues have used the taxi show successfully for several years, and we will go on using it for many years to come.
Acknowledgement
The cartoons, The Footies, appearing in this article are by Andrejs Dunkels, to whom the author wishes to express her sincere thanks.
University of Lulea, Sweden
References
Noether, G. (1976). Introduction to Statistics. A Nonparametric Approach. Second Edition. Houghton Muffin Company, Boston.
Vännman, K. (1979). Matematisk statistik. Hbgskolan i Lulea. (In Swedish.)
Velleman, P. and Hoaglin D. (1981). Applications, Basics, and Computing
of ExploratoryData Analysis. Duxbury Press, Boston.
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