# The Measurement of Adolescents

## CONTENTS OF CURRICULUM UNIT 85.08.05

## Some Statistical Exercises

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## Measuring Paper Thickness

This story will take a number of days. I have not tried it with students yet so I do not know how many days will be needed. I can anticipate some delays. It takes longer to do the measuring than one would think. How are the measurements of all the students going to be shared? Provide the students with data recording and calculating forms. Have a box at the bottom edge to be filled in with the five student thicknesses. You may then stack the forms on a copier so only the results show getting all the class data on one page. Having all the results the class can then arrange them in order, set up intervals, do frequency counts, think about what they have and so on.

That implies we do the measuring one day the analyzing next. However, we must be sure the students know how to use a ruler. What are the marks mm or cm, what are the numbers? So first have the students do some gross measurements. Measure the length, width, and thickness of their text including the covers. I can hear students saying, “We did this in general science.” Maybe they will be able to use a ruler.

Since we want to show that measurements are estimates, we need more than one measurement, to show the variation. When I tried this with the seminar the adults felt one measurement was enough. When the measurements of others were pointed out the response was “I’m right , they’re wrong.” That is why I ask the students to do me a “favor” in the script. If you get two different values you are not going to say that you are wrong. Do not pass out the recording forms until you have agreement that a number of measurements need to be done.

So having our text, a ruler, and knowing how to use it, we ask the following questions. “How thick is a single page in your text? How can we find out?” We can not put our ruler upto a single page, a page is too thin. Measure the thickness of the book and divide by the number of pages in the book. Not so fast, are you including the covers, the forward, the table of contents, unnumbered pages? How are you holding the pages? We have to standardize our technique. If we are going to compare answers we better all be doing the same thing. You want your pack to be square, as your wood working teacher would say. If there is a slant tot he pack you will not get the shortest distance across. Let us do the following. Hold your book up with its back resting on your desk. Release one cover and some pages then do the same with the other cover, you now have a pack of pages between your hands with the two covers and so me more pages resting on your desk. Measure the thickness across the head (top edge) or the tail (bottom edge). The curvature of the back will curve the front edge so that is why we measure the top or bottom.

So you know how thick the pack is. How many pages are in the pack? Are you going to count them? My, you are ambitious. Here is a chance for mathematics to do the work for us. How many page numbers are there on each sheet of paper? So we will be dividing by two. What will be divided by two? Pretend we had one page with the numbers 6 and 7. If we subtracted we would get one, but dividing by two would give us a half, so we will have to do something to the one. Add one and then divide by two. Try it with two sheets, pretend the numbers are 7,8 and 9,10. Ten take away seven makes 3, add one makes 4, divide by two makes 2 like it should. So if BPN stands for “back page number and FPN stands for “front page number”’, then the number of sheets in the pack will be (BPNFPN + 1)/2.

Go to it. Record your data, do your calculations. Now before you tell me your results, please do me a favor. Do it again, use different page numbers at the beginning and end. You came up with different answers each time. Why? Some variation may be due to the difference in pressure when the pack was held for measuring. The difference in an individual’s measurements will be due to the precision of the ruler. The thickness of a mm mark is thicker than a sheet of paper. How well are you at estimating distances between the marks?

Accuracy is how close you are to the true value. How can you tell how accurate you are if you do not know the true value? If we could get our measurements closer together then we would say they were more precise. If our measurements agreed for a certain number of decimal places we could say the measurement was accurate to that number of places. How can we get our measurements closer together? Use more precision instruments. A ruler with finer and more frequent markings, a vernier caliper or a micrometer would give more accurate results. There would still be variation but farther out in the decimal places.

That might well be one day’s work. We established the need to make a number of measurements. The next day pass out the data form and collect the measurements. Some teachers might have the class fill out the forms as home work. If a computer is available a program could be written so all the students had to do was enter the data and the machine would calculate the thicknesses.

We have all these measurements. What are we going to do with them? What is the smallest measurement? what is the largest? How can we tell? If the numbers were arranged in order we might see some pattern. If a computer or calculator were used we will have so many decimal places that rounding will be called for. We have all these measurements which one is right? Which one is in the middle? That is called the median. Which one appears most frequently? That is called the mode. To see what other questions can be asked we need a set of data. Here are my results in summary form. Since my measurements seam to say the thickness is 0.08xx mm, I tried to determine the next decimal place so I grouped the data in intervals of length .0010.

I expect that someone will suggest calculating the mean.

____
____
____
*
Frequency
*

Measurement | Interval | times | ||

Interval(mm) | Frequency | mid-value | mid-value | |

.0800 .0809 | 0 | .0805 | ||

.0810 .0819 | ******** | 6 | .0815 | .4890 |

.0820 .0829 | * | 1 | .0825 | .0825 |

.0830 .0839 | *********** | 9 | .0835 | .7515 |

.0840 .0849 | ********* | 7 | .0845 | .5915 |

.0850 .0859 | **************** | 12 | .0855 | 1.0260 |

.0860 .0869 | ************* | 10 | .0865 | .8650 |

.0870 .0879 | *** | 3 | .0875 | .2625 |

.0880 .0889 | ** | 2 | .0885 | .1770 |

.0890 .0899 | **** * | 5 | .0895 | .4775 |

.0900 .0909 | 0 | .0905 | ||

.0910 .0919 | 0 | .0915 | ||

.0920 .0929 | * | 1 | .0925 | .0925 |

.0930 .0939 | * | 1 | .0935 | .0935 |

total | 57 | total | 4.8785 |

We can treat the chart as a sample space. What is the probability that a measurement will be between .0850 and .0859? 12/57. If we did all the measurements again it is doubtful we would get the same ratio so instead of probability we ask for the relative frequency of measurements between .0850 and .0859. 12/57 or 21%.

Ask each student to compare his individual scores against the class average. Were they higher, lower or equal to the class average. count the number of times for each case. Most students will be either high all the time or low all the time. This is called bias. It shows that some used more pressure than others when they did the measuring, but they were consistent. Bias is to be expected in measurement our job is to be aware of it.

Ask each student to compute the average of his scores. You will find the range of the averages will be less than the range of the scores.

The asterisks give us a histogram. The normal way to show a histogram is to have the equal interval along the x-axis and vertical bars rising up for the frequency.

We may be able to go farther, but I think we would overwhelm our students if we did so. We have seen an application of statistics, a need to use statistics, and the statistical techniques. A large amount of data had to be organized. The central tendency gave us an idea as to what the correct measurement was. The dispersion gave us an idea as about how close we were to being correct.

I would like to hear how it goes when you try it with your class. Ask the questions. Teachers ask questions, students answer them. If students do not answer one question ask them another question.

Summary and Conclusion

In this paper we have seen six activities covering many statistical concepts.

The Word Wealth, Shooting Dice, and Measuring Paper Thickness can be done with any class. The different level classes will take different amounts of time to do the job, because of different levels of efficiency. deciding what to do and then doing it. Different classes have different levels of interest and attention spans. How the activities are sold to introduce them will determine the interest of the class. If Word Wealth is presented as “Look these words up” it will go over as “busy work”, “boring!” Emphasize doing something different.

- 1. Word Wealth: Jobs and subjects using statistics.
- 2. Role Playing: Do the assumptions fit the facts?
- 3. Shooting Dice: Sample space and probability.
- 4. Multiplying Polynomials: An example of a generating function.
- 5. A Fair Game: Mathematical expectation and the properties of the mean.
- 6. Measuring Paper Thickness: Collecting and organizing data, median, mode, mean, frequency, accuracy, precision.

Do try putting statistics in your curriculum. Have the class generate its own data. Work for class participation in active activities. We all learn more by doing than by listening.

Do look at the literature. There are authors presenting statistics as an interesting subject. Texts are being written that engage the student. One example is sampling. One college professor has his class count its money, then he takes a random sample and predicts the class total. How big a class is needed?

There is a lot that can be done. Let’s get started. Let me know about your results.