Shoe Size vs. Height
Statistical Analysis and Linear Regression
with FATHOM

Jay Yohe
Susquehanna Township High School
Grades 10-12
Algebra II

Prerequisite skills:

    Algebra I (basic skills in graphing linear functions)
     Basic knowledge of statistics (mean, median, mode, range, standard deviation, histograms and box plots)

Objectives:
 

PA Math Standards:
 

Materials and Resources:
 

Instruction Mode:

    *Discovery learning in cooperative groups
    *Individual graded explorations for closure activity

Anticipatory Set:

• Students will complete surveys (click here to download a sample survey in WORD) on the opening day of school recording height (in inches), shoe size and sex.
• Students will receive an unsorted list of all collected data in table form.
• Before analyzing the data, students will be asked the following questions:
• What is the population you wish to explore?
• Does the sample surveyed match the desired population?
• How could you improve the sample?
• Give examples when you can record information from an entire population and give examples when you cannot?

Instructions/Procedures:

(A) Entering Data into FATHOM
The sample data (click to download this data set in EXCEL) for the instructions presented in this lesson was collected at the Eisenhower Institute at Shippensburg University in July 2001.  Of approximately 85 total mathematics teachers attending the institute, 67 of them returned completed surveys.

In FATHOM, drag down a new collection by pulling the "open box" icon off the menu bar to a blank area of the document as illustrated:

Highlight and copy the data from the attached EXCEL spreadsheet (enter male data and female data into separate collection boxes).  Right click on the collection and select "paste".  With the collection still highlighted, choose "Case Table" from the "Insert" menu.  The data should have copied nicely into FATHOM (double click on Collection 1 and rename it):

You may also highlight the collection and choose "Case Table" from the "Insert" menu to bring down a table.  Create the headings by clicking on <new> and entering in the category names.  You can manually type in the data into these new columns or copy and paste information into columns from another source.  Observe the collected data below after it was placed into FATHOM.
 
 

(B) Analyze 1-Variable Statistics  (use Male data for class discussion and assign Female data for lab work)
 

• With the collection highlighted, pull down the graph icon from the menu bar and drag it into the empty space:

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• Drag the Male Height category from the Case Table onto the x-axis of the empty graph (the graph automatically creates a dot plot complete with axis and labels.
• Pull down the graph menu and select Histogram:

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• Answer the following questions (Reminder -- students will be analyzing their own data sets):
• If the histogram was a seesaw, where would you place the fulcrum to balance it?
• From the graph, estimate the mean, median and mode.
• Describe the distribution.  What kind of shape does it have?
• Are there any outliers?  What might explain these outliers (if any exist)?
• What do you expect to happen with this data over different populations?
• Now calculate actual statistical data using built-in FATHOM formulas.  Right click on the graph and choose "Plot Value".  Observe a sample for entering the formula for "mean" below:

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• Observe the Histogram with formulas added:

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  • How close were your predictions to the actual values?
  • How close are the measures of center (mean and median)?
  • Later you will be analyzing 1-variable data regarding shoe size of both males and females. Whom might be interested in the measures of center with regard to shoe size and why?
• Now Analyze the Male Height data using a Box Plot.  Observe below:

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Enrichment (Normal Curve)

1. You can form the "Normal Curve" distribution over your original Histogram.  Since the total area under a normal curve must add up to "1", first rescale the Histogram.
2. Rescale the Histogram by choosing "Scale" from the "Graph" window and choosing "Density":

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4. Now form the Normal Distribution curve by highlighting the Histogram and right clicking.  Choose "Plot Function".
5. In the dialog box click "Functions", "Distributions", "Normal" and double click on "normalDensity".
6. Since you must know the mean and standard deviation to form the Normal Curve, enter the following into the formula:

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8. You should see the following:

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10. Apply the empirical rule to your data set to see if it appears to have a normal distribution (the sample for this exercise has too few men to be a good sample of a normal distribution).  What is the empirical rule?
11. Do you think that the population graph of height for all men in the world forms a normal distribution?  Justify your response and/or provide other distributions that might be more accurate!

(C) Analyze 1-Variable Statistics  (use Male data for class discussion and assign Female data for lab work)

Have students complete all activities for part (B) above with male shoe size.  Answer the same questions that were posed in part (B) for shoe size.

(D) Analyze 2-Variable Statistics  (use Male data for class discussion and assign Female data for lab work)

1. Now drag a new graph onto a blank portion of your FATHOM screen.
2. Drag the Height from the Case Table to the x-axis of the graph and drag the Shoe Size from the Case Table to the y-axis of the graph.  This will form a scatter plot:

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   Does it appear that the data points form a line?
   What type of correlation best describes this data:  positive, negative or zero?
   Is the degree of relationship of the points with regard to forming a line weak, moderate, strong or non-existent?
   
   

4. With the graph highlighted, from the "Graph" menu, select "Movable Line".  Slide the line up and down by dragging the middle.  Change the slope of the line by grabbing the ends.  Drag this movable line to estimate the line of best fit!

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6. Choose "Show Squares" from the Graph menu.  Your goal is to form a line with the smallest sum of squares formed by the square of the distances from the actual y values to those generated by your movable line.  Observe:

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8. Now choose the "Least Squares Line" from the "Graph" menu to see how well you did with the movable line:

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10. This is a perfect opportunity to demonstrate the meaning of the least sum of squares.  Compare the sums from the movable line you created and the least squares line generated by FATHOM. The line that fits the data best has minimized the sum of squares from the actual values to the values on the regression line.
11. You can also observe the residuals line which shows the difference between the actual y value and y value generated by the regression line.  First select "Movable Line" from the "Graph" menu to turn it off.  From the "Graph" menu, choose "Make Residual Plot":

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13. Drag one of the points on the graph.  What does this do to residuals?  What does this do to the "Least Squares Line"?
14. Based on the regression line, what is the predicted value for a man 6'3" tall?
15. Based on the regression line, what is the predicted height for a man with a shoe size of 12?
16. What does the slope of the regression line really mean?
17. What does the y-intercept of the regression line really mean?
18. What human traits might have a better correlation than shoe size vs height?  How would you test your idea?  What population do you wish to include?  If you can't measure the entire population, how would you determine an appropriate sample to measure these traits?
19. Write a plan (include the sample/population) to measure the correlation between two human traits other than shoe size vs height?

(E) Final Exploration (Optional)

Combine all the data for males and females and make a histogram of height (sample of combination table below):
    
What kind of distribution occurs?  Create a summary table of results also (select "Summary Table" from the "Insert" menu and drag appropriate data to the down arrow and horizontal arrow):
    
It is obvious that in the sample data above, there are a majority of females in the institute.  Do you think the graphs would have the same traits if the sample sizes of females and males were of sufficient size (>30) and were identical?  Explain?  What do the combined heights of your data show?  Explain!  Do you find any interesting traits regarding mean vs median for all heights of males and females?

Assessment:

(1)  Have students work in groups to duplicate all information for the females in the sample population.  Answer all the same questions for females that were posed in lessons A through E for males.  Students will work in small cooperative groups on the female data in the computer lab utilizing FATHOM.  If students don't have access to FATHOM, they will conduct most of the data analysis on TI-83 calculators or EXCEL.

(2)  Students will submit a plan for analyzing another set of human traits for homework (refer to the questions at the end of part (D) above).  Students will receive homework points for submitting their proposal.  Students will share their ideas in class during a follow-up session.

References:

    *Eisenhower Institute presentations July 2001
    *Advanced Algebra by Holt, Rinehart and Winston ©1977 (page 28)
    *FATHOM (Key Curriculum Press)

Click to download FATHOM samples for all exercises and homework based on the Eisenhower sample data. Of course, you need the FATHOM software package from Key Curriculum Press to view these documents.

1. Male Lab 1-Variable Statistics
2. Female Lab 1-Variable Statistics
3. 2-Variable Scatter Plots and Linear Regressions Equations for males and females.
4. Least Squares Determination and Residuals

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