StudyTime+MAD

Finding Variability in Study Time Data
Return to Mean Absolute Deviation Source: Developing Essential Understanding of Statistics: Grades 6-8, NCTM, 2013; Pages 38-41

This activity examines three different data sets for 28 students in three different classes. The data in each class is symmetric with one difference. Class one has a uniform distribution, class two is mound shaped and class three is U shaped. The goal is to explore the variability of the three distributions noting the effect of the shape.

The Study Time data for three separate classes given on page 39 are shown in three separate frequency tables and three separate plots. Plots created in GeoGebra can be found at the bottom of this page; GGB will not automatically compute the MAD. I provided the ggb and tp files below.

When creating a plot in TP, the data is entered differently. The numeric attribute is study time and the categorical attribute is class. Because the variable class is categorical, in TP use text so you do not get a scatterplot. To create the histograms in TP set the x-axis to begin at 43 minutes and the bin width to 5 minutes.



Compare the shape and center shown for the three classes. Make a conjecture about the ratio of the variation between the study times for class one to class two and class one to class three.
 * GeoGebra || TinkerPlots ||
 * [[file:3Classes_StudyTime.ggb]] || [[file:3Classes_StudyTime.tp]] ||
 * Task 1 **

Create a dot plot and use TinkerPlots to find the MAD for each class.
 * Task 2 **

//MAD for class one is 8.57143 minutes// //MAD for class two is 6.07143 minutes// //MAD for class three is 10.3571 minutes//

Compute the ratios between classes with the MADs. Compare the calculated ratios to your conjectures.
 * Parallel Histograms ||
 * [[image:3Classes_StudyTime_Hist.png width="382" height="290"]] ||
 * Parallel Dot Plots with MAD calculated ||
 * [[image:3classesMAD.png]] ||

First, we notice that the uniform shaped data set (class one) has a MAD of about 8.6 minutes. This is greater than the bell shaped data with MAD of 6.1 minutes, but less than 10.4 minutes for the U shaped data. This makes sense if we look at how many data points are 15 minutes away from the means of 60 minutes.
 * Discussion **

The variability for class one to class three is about 4:5. The variability for class one to class two is about 3:2.

An extension to this problem would be to investigate the variability shown in parallel box plots. The ranges are identical at 30 minutes so that does not tell us too much. The IQRs for class two is 10 minutes while the IQRs for classes one and three are 20 minutes. An analysis of the IQR’s shown in the box plots would indicate the following ratios for variability: The ratio from class one to class three is 1:1. The ratio from class two to class one is 1:2, as well as from class two to class three.
 * Extension (not in the book) **

Now the question becomes this: Which statistic provides a better measure of spread for these three classes; Range, MAD or IQR? Without looking at the histograms (or dot plots) we do not know the shape of the data.

Graphs in GeoGebra Parallel Box Plots Histogram for Class 1 Histogram for Class 2

Histogram for Class 3