# The Measurement of Adolescents, II

## Some More Statistical Exercises

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I said to put statistics in the curriculum and look at the literature. Have the students do statistics. “Let’s get started.” Well I am still getting started, but here is what I have accomplished and what I have found.

Students will do statistics, they enjoy it. In fact I now believe they would find the standard deviation, especially if calculators are available. In fact, the properties of the mean and standard deviation of modified data sets can be used to study transformations on functions. I cover this in the section called “Working with the Mean and Standard Deviation”.

The section “What Happened” gives a report of my experiences in the past school year. It includes a computer program to do the steps of a classroom demonstration for drawing a histogram, finding the mean, median and mode.

I am still looking for the ideal books for students to read. I have found some statistics stories to share with students they are found in the obvious section “Some Stories”. I also found a set of materials that have the students do statistics.

The set is four 8 1/2 x 11 paper backs called
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Statistics by Example
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by the Joint Committee on the Curriculum in Statistics and probability of the American Statistical Association and the National Council of Teachers of Mathematics. The four volumes are
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Exploring Data, Weighing Chances, Detecting
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Patterns
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and
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Finding Models
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. The Chairman of the committee is Frederick Mosteller the professor I mentioned last year who samples his class to predict how much money it has. He also says students should be able to find out if runs of heads on coin flipping experiments or runs of dice are independent. In two of the articles he wrote for these volumes he looks at the fractional parts of stock prices to see if they are uniformly distributed, equally likely. All together there are 52 articles.

So what is statistics? What are the key ideas of statistics? Statistics is the scientific method. We do experiments, record our data and then ask, “How significant is it?” It is inductive thinking, not deductive. Since inductive thinking does not give certainty we would like to have some idea of what our chances are of being right. We calculate the probability of our results as if chance alone were responsible. If the probability is small then something else must be the cause. Here is an example.

“Turning the Tables” by Joel E. Cohen is a three page article in
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Statistics by Example:
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Exploring Data
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that makes this point very well. A psychologist observes people sitting at square tables in cafeterias. He counts the numbers of times people sit “opposite” each other and the number of times they sit adjacent to each other, “corner” seating. He also observes whether they are interacting or working independently and just happen to be at the same table. His counts were

The psychologist concluded people preferred corner seating for interacting and they preferred it because it allowed then to avoid eye contact.

- ____ Number of pairs observed

Corner Opposite Interacting 134 63 Noninteracting 2 16

What if the people had sat down at random? What is the probability of sitting at a corner? What is the probability of sitting across from one another? How many corner seatings are there? There are four corners. How many ways to sit opposite one another? There are only two opposite seating plans. Here is another way to look at it. The first person sits down, the next person has three choices two of them will result in corner seating only one will result in opposite seating. So we should expect to see twice as many corner seatings as opposite seatings. That is, out of 197 interacting pairs we expect 131 to sit at corners and 66 to sit opposite each other if they sit at random. 131 versus 134, 66 versus 63 that seems to be what one would expect. Look at the noninteracting pairs, however, out of 18 pairs we expect 12 to be corner seatings and 6 to be opposite if the seating is random. 12 versus 2 and 6 versus 16 seem to be going just the reverse of what is expected.

Something must cause people to sit opposite one another when they are working independently. Eye contact must not be the issue with the interacting pairs either since, I assume, noninteractors would be more likely to want to avoid eye contact than interactors.

Can we be sure? How do we know that the difference in the first case does not matter and the difference in the second case does matter? To answer that question we need to know about a statistic called the chi-square statistic. At this point in my experience of showing statistics to high school students I do not believe I should teach chi-square.

Our job is to give students insight into what is going on. Statistics is uncertainty. If students are to use chi-square they should know that it is a distribution, not a number one looks up in a table. These points are well covered in a number of articles in
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Statistics by Example
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. If your students show interest and ability let them use these books to find out.

For the reader who may not know how to calculate a chi-square, I did not when I read we should test to see if runs of heads were random, here is the formula.

The sum for i=1 to n of terms of the form

(Oi Ð Ei)
^{
2
}
/ E2 Where Oi is the i-th observed count Ei is the i-th count expected by probability
Perfect agreement with chance would give a chi-square of zero. How probable one’s chi-square is when it is not zero is determined by looking up a value in a chi-square table.