Examining Relationships Between Variables: Charts and the Coefficient of Determinations

01-03-2023 03:02 PM
Esri Frequent Contributor
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In this activity, which requires one or two class periods, and can be used in upper secondary and in higher education, you have the opportunity to advance your mathematics knowledge by creating and interpreting charts and plotting coefficient of determinations.

Open the following web map in ArcGIS Online:


The map opens to an extent showing the state of Colorado in the USA with 2 layers visible:  Colorado 14ers (peaks over 14,000 feet in elevation), and ACS (American Community Survey) county level housing and income data from the US Census Bureau. 

To the left of the map, use Layers > turn off the ACS layer > so that only the mountain peaks are visible.  If you open the table of data behind this layer, you will see that 58 records exist, corresponding to the number of peaks that meet this elevation.  You will also note that a field exists called Elevation, and another field exists called Difficulty.  The difficulty are ranked from 1 (most difficult) to 58 (least difficult).  The criteria used to generate the difficulty includes (1) climbing duration and challenges inherent in the trek, (2) terrain stability, (3) cliff exposure, (4) presence of a trail, (5) elevation gain, and (6) total roundtrip hike distance.  Thus, the difficulty is really an index that incorporates 6 different criteria.



Is there a relationship between elevation and difficulty?  To find out, you will create a chart and create a coefficient of determinations fit on that chart.  To the right of the map > Configure Chart > Add Chart > Scatter Plot > Data > X-axis:  Difficulty.  Y-axis:  Elevation > Show Linear Trend (the coefficient of determinations) > observe your chart (shown here).  Hover your touchpad or mouse over some of the points on the scatter plot to observe their values (elevation and difficulty). 




Based on what  about you know or may need look up about the R squared value and what it means, is there a relationship between difficulty and elevation?  How strong or weak is it?  Why do you suppose this is the case?

Next, turn on the layer ACS Household Income Distribution Variables > Expand > County > Configure  Charts > Add Chart > Scatter Plot > X-axis:  Median Home Value for owner-occupied units.  Y-axis:  Median Household income in past 12 months > Show linear trend (shown here).


The map of this data is here:



Based on what you learned in your courses about the meaning of the R squared value, is there a relationship between median home value and median household income?  How strong or weak is it?   Why do you suppose this is the case?  Do you think the same relationship would exist for other states?

Note the presence of mountains in Colorado from the presence of 14-ers and the shaded relief base map that is in use.  Traditionally, it has been more expensive to build homes in the mountains and to be able to afford to live there.  Use your mouse or touchpad to click on a point in the scatterplot where the home value is high and the household income is high.  Use the shift key to select additional points (or use shift-and-band an area on the scatterplot to do so).   Use the “lower” symbol to minimize the area occupied by the table so you can see the map.  Where in Colorado are these higher income and higher home value areas?  Do they correspond with the most mountainous areas of the state?   Conversely, repeat this procedure for the lower income and lower home value areas in the state, noting any geographic region(s) in the state in which these are concentrated.  Do these areas correspond to the Great Plains region of Colorado to the east and the canyons and high plateaus of the northwest part of the state?

Note that the above activity does not require you to sign in to ArcGIS Online to work through it.  This is one of the instructionally appealing aspects to modern GIS as evidenced in ArcGIS Online:  You and your students can engage with it in a variety of ways; here, without signing in.  To save your work, however, and to explore additional functionality, you would need to sign in as a named user using your school, college, university, or other account.

For more activities that bridge the boundaries between geography, GIS, and mathematics, see the book that Dr Sandra Lach Arlinghaus and I authored, Spatial Mathematics.

I look forward to hearing your reactions to the above short lesson.

--Joseph Kerski



About the Author
I believe that spatial thinking can transform education and society through the application of Geographic Information Systems for instruction, research, administration, and policy. I hold 3 degrees in Geography, have served at NOAA, the US Census Bureau, and USGS as a cartographer and geographer, and teach a variety of F2F (Face to Face) (including T3G) and online courses. I have authored a variety of books and textbooks about the environment, STEM, GIS, and education. These include "Interpreting Our World", "Essentials of the Environment", "Tribal GIS", "The GIS Guide to Public Domain Data", "International Perspectives on Teaching and Learning with GIS In Secondary Education", "Spatial Mathematics" and others. I write for 2 blogs, 2 monthly podcasts, and a variety of journals, and have created over 5,000 videos on the Our Earth YouTube channel. Yet, as time passes, the more I realize my own limitations and that this is a lifelong learning endeavor and thus I actively seek mentors and collaborators.