David B. Howell
“Do you teach statistics?” asked a sampler.
“You mean beyond graphs and charts and mean, median, and mode?” responded several math teachers covering grades 7-12.
“Yes, I mean beyond those ‘basics.’ But I don’t mean theory.”
“Then, no, I don’t,” answered every one of a non-random sample of ten teachers.
“Should you?”
“Well . . . , yes . . . , probably. Advertising, polls and marketing surveys keep coming at us.”
“Could you?”
“I’m not so sure. Know any good resource books? I didn’t study much statistics. Didn’t like it, either.”
“The list of good resources is growing rapidly since the NCSM ‘Position paper on Basic Skills’ and the NCTM
Agenda for Action
appeared. Statistics is basic. And important at all levels. That’s what they said. Say, how about a unit on statistical sampling? Mo fancy theory. Just ‘seat-of-the-pants’ experience combined with a bit of mathematical stretching . . . ”
“
That
would help. Where can I find one?”
Here . . . .
Much of our daily experience with the news media and in the marketplace is impacted by statistical information based on sampling. Newspapers and magazines regularly headline lists of favorites—most popular singers, most admired leaders, best dressed men and women, finest chocolate ice cream. The invariable economic interest behind such lists often has a direct sales emphasis. It is important that we be able to analyze the claims: On what is the claim based? Who was sampled? How many were sampled? Does the sample represent the population
I
belong to? Do the sample results support the conclusion?
Quality control in manufacturing makes extensive use of statistical sampling. A company producing light bulbs, or audio tapes, or batteries must assure that no more than some small percentage of its products are defective. It
must
sample. It cannot test every item. How many items must be sampled? If some percentage of the sample is defective the company halts production to fix the problem. How does one determine the cut-off percentage?
Advertising for products, services, and even for political candidates increasingly is based on inferences made from opinion polls and sampling. How can pollsters make their claims for millions of voters on the basis of samples of one thousand? TV ratings are based on small samples. “Look out for
___
,” headlines a magazine ad. “It’s the ___ most often chosen #1 in the most recent . . . magazine tests . . . .” Could the claim be based on a single number one rating in one magazine in one month? Or two number one ratings in 20 magazines in two years? What was the sample? How was it chosen? What constitutes a claim we have confidence in?
It is possible, of course, to provide extensive theoretical mathematical answers to many of the questions above. possible, but not necessary for most of us. The role of probability in the theory of statistics is critical. Sampling, confidence limits, and so on depend on binomial and other probability distribution. It is not necessary, either, to have any background in theoretical probability to make reasonable analyses of many statistical claims. It is sufficient simply to have some experience with sampling and to have addressed a few of statistics’ major concepts. That is what this Unit is about.
It is the purpose of this Unit to offer students a series of lessons and activities which provide a concrete base for understanding statistical sampling. Students will conduct sampling activities. They will compare results across samples. They will predict based on samples. They will attempt to verify the predictions. They will explore the notion of “confidence.”
There are a few important skills assumed in the following lessons: counting, tallying, ratio, percent, graphing (especially histogram or bar graph), mean. Median and mode might be included in discussion, too. If students do not have all these skills in advance, the lessons may offer an interesting context in which to teach them informally. Use of calculators for converting ratios to percents and for finding averages is encouraged in order to keep the focus on the statistics rather than the arithmetic.
Let’s state the objectives for the Unit. The Unit is designed for any group of regular education students in Grades 7 through 12. Students will be able to
____
1. define sample of a population;
-
____
2. describe and give an example of a random sample;
-
3. Iist two or three factors which could have compromised the randomness of a given sample;
-
4. describe variations among several samples;
-
5. describe how changing sample size affects those variations;,
-
6. select a sample size to estimate a characteristic of a large population with confidence;
7. discuss ways to quantify confidence.
Let’s talk about a little bit of mathematics,
for the teacher
only
, to illustrate what we’re getting at.
The
Central Limit Theorem
is of fundamental importance in inferential statistics, and it is one theorem for which we are trying to build an intuitive sense based on concrete examples. The Central Limit Theorem may be summarized as follows:
Take samples of size, H, from
any
population:
1. The means of those samples have an (almost) normal distribution.
-
2. The mean of that distribution—i.e. the mean of the sample means—is (almost) the same as the mean of the entire population.
3. The distribution of the sample means clusters more tightly around its mean as the size, N, of the samples is increased.
Example: What is the percentage of foreign-made passenger cars on the road in the U.S? By counting at pseudo-random (beginning at the next overpass) stretches of turnpike I drive, I take samples to approximate answers to that question regularly. Here are the results of four such recent samples:
TABLE I.
|
|
1
|
2
|
3
|
4
|
|
|
|
|
Number of
|
percent of
|
|
|
Sample group
|
N
|
Foreign Cars
|
Foreign Cars
|
|
|
A
|
50
|
7
|
14
|
|
|
B
|
50
|
18
|
36
|
|
|
C
|
50
|
11
|
22
|
|
|
D
|
50
|
13
|
26
|
Let’s form groups of 100 by making all possible combinations of two of the four groups above.
TABLE II.
|
|
1
|
2
|
3
|
4
|
|
|
|
|
Number of
|
percent of
|
|
|
Sample group
|
N
|
Foreign Cars
|
Foreign Cars
|
|
|
AB
|
100
|
25
|
25
|
|
|
AC
|
100
|
18
|
18
|
|
|
AD
|
100
|
20
|
20
|
|
|
BC
|
100
|
29
|
29
|
|
|
BD
|
100
|
31
|
31
|
|
|
CD
|
100
|
24
|
24
|
The Central Limit Theorem assures us of three things.
1. The means—percent of foreign cars—of all the possible samples that could be in Table II are distributed (almost) normally.
-
2. The mean of Column (4) of all the possible samples in Table II is (almost) the same as the mean of the entire population. Directly, the mean of the samples listed is 24.5%. That is probably close to the mean of the entire population. Assuming my sample was random (O.K, it was taken on the Maine Turnpike in April:), approximately 24.5% of the passenger-car population in the U.S. is foreign-made.
3. The larger the sample size N, the more the means cluster centrally. The range for samples A,B,C, and D (N=50) is 36 14 = 22. The range for the six samples in Table II (N=100) is 31 18 . 13. The means are, of course, the same: 24.5%.
Now, if we apply the Central Limit Theorem and if we standardize scores to the normal distribution and if we formulate null and alternate hypotheses and if we calculate variance and standard deviations and t-ratios and degrees of freedom, then we can quantify the notions of confidence intervals and confidence limits which are also among the major foundation stones of inferential statistics. But hold on: You, the teacher, may still be here, but most students in Grades 7 12 aren’t.
Consider an analogy. When Geometry students reach the pons Asinorum—the theorem that says the base angles of an isosceles triangle are congruent—the reaction to the claim at least is one born of familiarity. “Sure,” thinks the student, “I’ve measured it every year since the fifth Grade, and this teacher draws it that way every time. Of course it’s true.” And now we’re free to focus on the logic of possible proofs.
But not so with the statistics of the foreign car population. What we need to do is take samples. Compare samples. Combine samples. Examine possible inferences from samples. Declare our faith that while we’re not too sure about there being exactly a 24.5% foreign car population out there, we’d bet a whole lot that there’s more than 14% and less than 36%.
That
’s confidence—intervals, limits, and all. And that’s what this Unit’s lessons are about.
Each of the following Lessons should take from one to three class periods depending on task efficiency in sampling, sophistication of discussion, the skill levels for charting, graphing, finding percent, combining smaller samples into larger ones, etc. The Lesson outlines deal with content, not management or individual differences or testing. All of the Lessons are described in the same format:
-
A. Objectives
-
B. The experimental question—a question which presents a statistical sampling opportunity.
-
C. Issues, and some possible resolutions—a series of questions which should arise, which the teacher will probably pose (with some possible student responses in parentheses); a hint of dialogue between teacher and class. There is occasionally a direct comment to the teacher in brackets. This section really defines the activity.
-
D. Observations and discussion to Objectives—more questions (possible student responses in parentheses) and summary relating specifically to the stated objectives.
