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Institute for Advanced Study/Park City Mathematics Institute Summer 2009 Secondary School Teachers Program/Reasoning with Data and Chance

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Was Leonardo Correct? (continued)

Drawing on the work of Marcus Vitruvius Pollio (born c. 80–70 BCE, died after c. 15 BCE), Leonardo da Vinci (1452-‐1519) drafted guidelines for artists on how to proportion the human body in painting and sculpture. Included in his directions were guidelines for the ratios of:

the length of a man's outspread arms to his height (equal)

the maximum width of the shoulders to a man's height (one quarter)

the length of the hand to a man's height (one tenth)

the length of a man's foot to his height (one sixth)

In this activity we will:

• Use data to examine the value of these ratios in small and large samples • Study different ways of determining the “typical” value and examine the implications of each

DAY ONE: On the first day, you will gather data from the people in your class, enter it into Fathom, and use some of the tools that Fathom offers for exploring data.

DAY TWO: On the second day, you will use a much larger set of data containing body measurements from almost 4,000 individuals. You will use this data to make charts and consider how measurements are distributed in the population. Through this, you will develop ways to decide what is “typical” and to make comparisons between different measurements.


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DAY ONE INSTRUCTIONS

For the first part of this activity, taking measurements, you will need to work in groups of 3 or 4. Instructions on how to take each measurement are given below1.

Group Member Responsibilities: Each person needs to be measured. Each measurement needs to be taken 3 times, and the values have to be averaged. While one person is being measured, one person should be doing the measuring and the other should be recording the data. For groups of 4, the fourth person should do the average calculations. For groups of 3 the recorder should do the calculations. It is up to the entire group to rotate responsibilities as each person gets measured.

ARM SPAN -‐ The distance between the tips of the right and left middle fingers when the subject’s arms are extended out to the side as far as possible

SHOULDER WIDTH -‐ The maximum horizontal distance between the widest part of the shoulder muscles. The subject sits erect looking straight ahead. The shoulders and upper arms are relaxed and the forearms and hands are extended forward with the palms facing each other.

HAND LENGTH -‐ The length of the right hand between the bony point on the thumb side of the wrist and the tip of the middle finger. The subject’s fingers should be together and the thumb should be sticking out.

FOOT LENGTH -‐ The length of the foot from the tip of the big toe to the back of the heel while the subject is standing2.

HEIGHT -‐ The vertical distance from a standing surface to the top of the head. The heels are together with the weight distributed equally on both feet. The shoulders are relaxed.

Your teacher will assign a number to each person in the class, because we are interested in the measurements themselves – not who they came from. You should only write your number on your individual Data Table3. After everyone in your group has been measured and the measurements have been averaged, one member of your group should write the average measurements for each member of your group on the large table that your teacher will post at the front of the room. You will then record the data for your entire class in the Class Data Table.

1 These directions are a simplified form of the official measurement definitions used in the US Army’s ANSUR study. The full definitions are in the document CORRESPONDENCE OF ACTIVITY DATA SET TO ANSUR DATA. 2 This may not be exactly correct – I was unable to find a clear definition of the “long axis” of the foot, per the ADAS definitions. 3 Note that da Vinci’s prescriptions are for “men”. You may want to record the gender of each subject to do further analysis on your own. In this activity, however, we will pool all of the subjects regardless of gender.


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Before we turn to using Fathom to study the data from your class, answer the following questions:

Q1 a) What did you notice about the numbers on different trials of the same measurement? Were the numbers close to each other? Were they identical? Explain.

b) What are some possible sources of error in your measurements? c) How did your measurements compare to the measurements within your group? Is this

what you expected? d) Without looking at more than the numbers in your own group, make a hypothesis about the typical relationship between people’s arm spans and their heights.

After you answer these questions, open the Fathom software by clicking the icon on your desktop.

Once the software opens, follow the instructions below, stopping to answer the questions when they come up.

1.) Drag a new Collection off the shelf. (See Figure 1)

Figure 1

2.) Select “New Cases” from the Collection menu (See Figure 2) and enter the number of

observations you have (i.e. the number of people in your class) in the box that appears (See Figure 3)


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Figure 2

3.) Double-‐click Collection 1 to open the Inspector (See Figure 4), name two Attributes by typing in “ARMSPAN” and “HEIGHT” and enter the data that you collected from your class, using the arrows in the lower left corner of the inspector to scroll between cases (Fathom calls each of the individuals in your data set a “case”.)

Figure 4


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4.) After you have entered all of your data, create a new Attribute called “SPAN_HT”, which we will define as the ratio of armspan to height for each individual. Double-‐click the formula box to open the formula editor (See Figure 5). Type “ARMSPAN/HEIGHT” in the formula editor – Fathom formats it automatically into the form shown below – and click OK.

Figure 5

5.) Drag a Summary Table from the shelf (See Figure 6a, Step One) and drag the attributes SPAN to the table (Figure 6a, Step Two). Drag the other two attributes to the table (See Figure 6b) so that you get the summary table in figure 6c (your numbers will be different.)

Figure 6a Figure 6b Figure 6c

QUESTIONS Q2: a) What is the average arm span in your class?

b) What is the average height in your class? c) What is the average ratio of arm span to height? d) Is the average ratio larger or smaller than da Vinci’s ideal? Express

the difference between the average and the “ideal” as a percentage.

Calculate the ratio of the average armspan to the

average height. Is the ratio of the averages the same as the average of the ratios?

Why?


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Drag a graph from the shelf (See Figure 7) and drag the variable “ARMSPAN” to the ___-‐axis and “HEIGHT” to the ___-‐axis.

Figure 7

Go to the “Graph” menu and add a Movable Line and Show Squares (See Figures 8a and 8b). When you think that you have minimized the area of the squares, add a Least-‐Squares Line and see how close you got (See Figure 8c).

Figure 8a Figure 8b

If you think of the ratio of arm span to height as the slope of a best fit line, which attribute would be the “rise” and which would be the

“run”?


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Figure 8c QUESTIONS Q3: a) What does the slope of the Least-‐Squares line tell you about the relationship between

the arm spans and the heights of the individuals in your class? b) What does the y-‐intercept of the Least-‐Squares line tell you? c) Go back to the Graph menu and select “Lock Intercept At Zero”. What is the new value

for the slope of the Least-‐Squares line? (Compare this to your answer in Q2c) Q4: Leonardo da Vinci was interested in describing “ideal” proportions for a human figure, rather

than giving an empirical description of what people actually look like. Nonetheless it may be interesting to consider the relationship between an “ideal” and what we observe in reality.

In our next two activities, we will use a much larger collection of body measurements, taken

from 3982 male and female members of the United States Army in the late 1980’s. Before we do this, consider the following questions:

a) Leonardo da Vinci provided the ratios described above as a guide to painters and sculptors who

wanted to represent the human figure. Are we, in modern times, offered any ideals of how people should look? Where do we find some of these? What is their purpose?

b) Would you expect an “ideal” of human appearance to be the same as what is “typical”? Why or why not?

c) Do you think that the average value(s) for the students in your class will be the same as the average value for the people in the US Army study? Why or why not?

d) Do you think that the average value(s) for the 3982 people in the study will be close to Leonardo da Vinci’s ideal? Why or why not?


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DAY TWO INSTRUCTIONS

1.) Open the Fathom file titled ANSUR Average Value (See Figures 1a and 1b).

QUESTIONS

Q1: a) What is the slope of the Least Squares line relating arm-‐span to height in the sample from the ANSUR study? What is the percentage difference from da Vinci’s ideal value of one? b) In the next step you will create a summary table and find the average ratio of arm span to height for the individuals in this collection. Do you predict that this value will be greater or less than the slope of the Least-‐Squares line? Why?

2.) Drag a Summary Table from the shelf (See Figure 2, Step One) and drag the attribute SPAN_HT to the table (Figure 2, Step Two) – open the Inspector by double clicking the Collection first.

Hint: Is the intercept of the Least-‐Squares line

negative or positive?


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3.) Go to the Summary menu and choose “Add Formula” (See Figure 3a). Enter the formula “percentile(95,SPAN_HT)” in the formula editor. This will tell you the value of SPAN_HT for which 95% of the cases in the Collection have a smaller value of the attribute (i.e. Your value for S2 should be 1.082, so if the ratio of your armspan to your height is 1.082 then 95% of the people in the collection have a smaller armspan than you relative to their height.)

CHECK FOR UNDERSTANDING: What percentage of the individuals in this collection have an arm-‐span to height ratio less than 1.082?

4.) Drag a graph from the shelf and place the attribute “SPAN_HT” on the horizontal axis. This will give you a Dot Plot of the collection in which each dot represents one individual.

QUESTIONS Q2: a) Look at the Dot Plot that you just made for SPAN_HT. Approximately where on the horizontal axis does the center of the Dot Plot appear to be located? Is this about the same as the average that you found for your class, or is it quite different?

b) Notice that, when you pass the cursor over a point on the Dot Plot, the SUBJECT_NUMBER and value of SPAN_HT appear in the bottom left corner of the screen (See Figure 8). Try to find a point representing an individual with an arm span to height ratio of exactly one (there are eighteen such individuals), and describe its horizontal location in the Dot Plot (e.g. is it in the middle? in the right or left tail? somewhere in between?)

c) Try to find an individual with a ratio close to your own, and describe that individual’s horizontal location in the Dot Plot.

d) Try to find an individual with a ratio close to 1.096 (the slope of the Least-‐Squares line in that you looked at in Q1), and describe that individual’s horizontal location in the Dot Plot.


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5.) Now we will use a slider to explore how the ratio of arm-‐span to height is distributed in the collection.

Drag a slider from the shelf (See Figure 5, Step One) – we are going to use this to control the percentile reported in the summary table, so double-‐click the scale on the slider (See Figure 5, Step Two) and change the Lower_bound to 0 and the Upper_bound to 100.

Rename the Slider variable “SPAN_HT_PCT” (See Figure 6 – you can type the new name in the Slider box, or in the Inspector.)


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Then go back to the second formula in the Summary Table and replace the number “95” with the name of the Slider variable “SPAN_HT_PCT” (See Figure 7). Add a third formula in the Summary Table to report the value of the Slider variable – click on “Add Formula” in the Summary Table and enter the name of the Slider variable for S3. When you finish your Summary Table and Slider should look like Figure 8.

The 50th percentile has a special name –

what is it?


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QUESTIONS Q3 a) Adjust the slider to find the 50th percentile of the collection. What is it?

b) Adjust the slider until you find the percentile at which the ratio of arm span to height is exactly one. What percentage of the individuals in the collection have a ratio less than one? What percentage have a ratio greater than one? c) Adjust the slider until you find the percentile at which the ratio of arm span to height is approximately 1.096 (the slope of the Least-‐Square line). What percentage of the individuals have a ratio less than 1.096? What percentage have a ratio greater than 1.096? d) What is the percentile of the ratio of your own arm span to height? e) Would it make sense to find the percentile of your class average in this group? Why or why not?

Q4 If you wanted to answer the question “What is the typical relationship between a person’s armspan and a person’s height?” how could you use the numbers here to answer the question? Are there any other statistics that you might want to calculate? Is there any other information from the collection that you might want to look at? [HINT: Try dragging the attribute “GENDER” to the y-‐axis of your dotplot] You can also use formulas such as “mean(SPAN_HT,GENDER=”male”)” to calculate statistics for a subset of the collection.

Q5 If we define the “typical range” of measurements for a physical attribute as the range such that no more than 2.5% of individuals will be below that value, and no more than 2.5% of individuals will be above that value, what would the typical range of values for the ratio of arm span to height?

EXTENSION QUESTIONS

Q6 In the collection there are four more ratios for which da Vinci prescribed ideal values. Copy the collection into a new Fathom file and repeat the steps above for the other three ratios.

Q7 It can be difficult to determine exact body measurements from photographs, but it is relatively easy to determine the ratios of body measurements. Use magazine and newspaper pictures to take one or more samples of individuals from groups that one might expect to have ratios different from the average, for example professional athletes or fashion models. Find the percentiles for these individuals.

The 50th percentile has a special name –

what is it?

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