# The Measurement of Adolescents

## CONTENTS OF CURRICULUM UNIT 85.08.05

## Some Statistical Exercises

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After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

Our first activity is, “Do you know what all those jobs are?” Use a dictionary, give complete sentence definitions of each job. Can you name any other users of statistics? Can you tell how statistics are used by these people?

Here are some other fields that use statistics: communications and control theory, cybernetics, information theory, game theory, operations research, systems analysis.

So statistics is a useful subject. That is why we are asked to expose our students to it. By this activity our students come to see mathematics around themselves. This activity allows the use of nonmath skills in the math class. We are called upon to integrate writing into all our subjects, here is an opportunity.

While we are using our dictionaries, let us find the definition and etymology (another word for students to look up) of statistics. What word do you hear in statistics? State. Historically, statistics was the collection, organization and interpretation of census data, data about the nation. Today statistics uses any type of data.

Those interested in precise language may be interested to know that what many people call statistics are really data. Baseball and football “statistics” when they give number of times at bat, number of hits or football statistics when they give number of yards gained are giving data. A statistic is a number that is calculated to summarize a set of numbers. So a batting average is a statistic.

When I think of statistics, I think of averages and standard deviations. What do others think of or use? One list, the result of a survey of political science journals reported these seven topics, in order of use: relative frequency, frequency, mean, correlation coefficient r, index numbers, r
^{
2
}
and, lastly, standard deviation and variance.

Frequency is the count, the number of times something happens. We will see frequency in the dice activity and in the measurement project. Relative frequency is a fraction or a percent. A probability is a relative frequency. These two concepts are in the paper. I do not use the term relative frequency in the paper, however. I think probability is enough of an introduction. The vocabulary would be an intrusion. The student who knows a probability will have no trouble calculating a relative frequency at a later date.

As I said, the average is what I think of when I say statistics. What good is an average? I sometimes think some teachers just ask for it as a problem using two operations. Text books emphasize three properties of the mean. The mean is the most probable score, the mean is the number the sum of the deviations about which will be zero, and the mean is the number the sum of the squares of the deviations about which will be a minimum. This was the motivation for this project. I wanted to find how to demonstrate these properties to students. The third is a calculus problem so I did not write out the proof. The Fair Game activity is my attempt to answer these questions.

The correlation coefficient r and r
^{
2
}
are topics for a full course development. I want topics that can be covered in short periods of time as breaks from the normal flow of the course.

Index numbers are an arithmetic activity that some teachers might want to explore. When I read the calculations I thought my arithmetic students could do it, but I wondered how I could get them to do it. What would motivate index numbers?

The standard deviation and variance go with the mean. The mean gives us a measure of central tendency but what is the spread? The standard deviation gives us the dispersion. I have difficulty including standard deviation in an introduction. True, it is of fundamental importance, but the formula is intimidating and not self evident. Yes, we need a measure of dispersion, but why such a complicated one? Not a question to answer in an introduction. If we can get the students to see a need for a measure of dispersion we will have dome a good job. The measuring project is an effort to answer this desire.

If teachers want to give students practice in evaluating formulas and following directions, the standard deviation formula might be a candidate. Students can feel successful by working out a complicated formula, in class, in competition with their neighbors. Telling the class the formula is from a more advanced class would be all the motivation needed.

I wrote the units as if I were talking to my class. The Algebra class I had this past year is the one that came to mind most often. That does not mean this is for Algebra II students. If the dialogues were transcriptions there would have been interruptions. I would have had to check to see if the students understood what had been said. Students would have had questions that might have led to other issues. That is why I think of these stories as science fiction, not as a put down, but as a story that begins with certain assumptions. When these activities are done in a class room time will be called for, to wait for calculations to be performed, measurements to be made and recorded. All this time may cause the activity to take a number of days which in turn will call for review. Maybe it will become too long.

As you read this think of how you could modify it for your students. Try some different strategies. Finding probabilities when you have a sample space is no different than saying what fraction of the set is the circled subset? If it is asked in terms of probability it will be useful in some students’ minds rather than just fractions.

I did make the proposal of the dice game in class one day. The game is you get a dollar for each point appearing on the die. The students could see that some amount would have to be paid for the privilege of playing the game. However, the discussion turned to how bookies make money. The students were surprised that a bookie would take bets on any side.

How you use the material will depend upon your teaching style. I want discussions with my students. I need a blackboard and chalk. The measurement project will require metric rulers and the students’ texts. A form for recording the data would be helpful if it will take more than one day to collect and present the data.

The list of jobs and the page on shooting dice could be copied as hand outs to the students. Since, I think in terms of discussion, the questions in the articles are only suggestions. If you are going to use them as hand outs you might ask your students to add questions of their own.

While I am telling the reader to try new things, it is easier said than done. Historically probability came before statistics, statistics uses probability to interpret data, I was taught probability before statistics, my algebra text has probability in it but not statistics per se, so I wrote this paper starting with probability. So if you are interested in doing something different go directly to the article Measuring Paper Thickness. It is the article I see as the most statistical. It will call for the most student participation.

So let us get on with the activities. Our first wag Word Wealth using the dictionary on the job list. Our next activity is Role Playing.

Since statistics is concerned with real world problems we can share such problems with our students. The problems can be presented as role playing situations. To solve problems we must understand what is being asked for. To understand a problem we must ask questions. In role playing situations the students may be the managers who have to make the decisions. The students need not solve the problem, but asking the proper questions will be considered successful completion of the activity.

Consider this scenario. You are the manufacturer of pantyhose. At the start of each month you service your equipment. You run a batch of raw material. You know how much raw material is used each month so you know how many pairs you should have at the end of the month.

The problem is that you never have as many pairs at the end of the month as were predicted at the start of the month. It is always less, and by a good amount too.

Some of your fellow managers believe it is employee pilfering. They set up traps, but find nothing. They decide to call in a team of psychologists to find the coo-coo they believe is doing the stealing.

Can you save your firm from being told it has a suspicious mind, that in fact no stealing is going on?

The idea for this story came from
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How to Use (and misuse)
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statistics
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by Gregory A. Kimble. The psychologists were called in, but what they found was a “nonrepresentative” sample. When the machines were clean they produced a finer thread than later in the month after some loss of efficiency.. So as the month went on the thread got thicker taking more thread to make a pair of hose.

Problems like these are opportunities for students to think. One aspect of applying mathematics to real life situations is to know what is being attempted, what are the assumptions. These activities allow opportunity for discussion. Allow the students time to think, if they think you are asking riddles they may just wait for you to tell them the answer.

This idea, role playing, needs more work. There are books of applications of math to real life problems, however, the mathematics is at too high a level for students who have not had calculus. I think some of those problems can be presented to the class upto the solution step, all discussion and definition of the problem.

So we have two reasons for teaching statistics: it is used in jobs, it provides interesting problems. However, we can not teach a full years course in statistics, there is not enough time in the schedule. Furthermore if a full year course is to be offered a text would be purchased and followed. No need for these units. These units are to show places statistics may be integrated into the normal math curriculum. The teacher is already doing statistics, only it should be mentioned.