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Histogram Sorting

Joan Garfield
Department of Educational Psychology
University of Minnesota
315 Burton Hall
178 Pillsbury Drive S.E.
Minneapolis, MN 55455

Statistics Teaching and Resource Library, June 24, 2002

© 2002 by Joan Garfield, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.

This activity provides students with 24 histograms representing distributions with differing shapes and characteristics. By sorting the histograms into piles that seem to go together, and by describing those piles, students develop awareness of the different versions of particular shapes (e.g., different types of skewed distributions, or different types of normal distributions), that not all histograms are easy to classify, that there is a difference between models (normal, uniform) and characteristics (skewness, symmetry, etc.).

Key words: Histogram, shape, normal, uniform, skewed, symmetric, bimodal

## Objectives

The objective of this activity is to give students experience with a variety of histograms of data and to help them better recognize different shapes and characteristics. Too often students only see one or two perfect examples (e.g., normal, right skewed) and have a difficult time describing and classifying histograms of real data. This activity also helps students determine which characteristics can appear together (e.g., skewed and bimodal) and which cannot be used together to describe a distribution (e.g., skewed and symmetric). This activity may be used to help students better understand the relationship between descriptions of data sets and the graphs that could be created from these data sets.

## Materials needed

This write-up includes a set of 24 histograms, generated by data on ActiveStats, and graphed using Data Desk software. One set of these graphs is needed for each group of students doing the activity. The pages need to be cut so that only one graph is on a piece of paper. These graphs can then be placed in an envelope or clipped together. A website  (http://app.gen.umn.edu/faculty_staff/delmas/gc_1454_course/distribution_file
s/distribution.html
) can be used for a follow-up debriefing activity.

Time involved

## 5 minutes to introduce the activity10-15 minutes for students to work in groups, sorting graphs10 minutes for instructor-led discussion of graphs5 minutes for follow up questions Teacher notes

Make sure you have enough piles of graphs for each group of students to use. Groups of three to five students work well for this activity. It is best to have students do this activity BEFORE they have formally study different shapes of graphs. However, they will still recognize familiar shapes and use terms like normal and skewed.

The groups the students will sort their graphs into will typically be: uniform, normal, skewed, and bimodal. There may be some smaller groupings such as right skewed and left skewed.

After the students have finished sorting and discussing, the instructor can lead a class discussion, asking the students questions such as:

1. What was the easiest group to sort? Which graphs are in that group?
2. How many different groups did you find? Which graphs are in each? What did you call them? What features did they have in common? Etc.
3. Which graphs were hardest to sort or classify? Why?

Students will often find the uniform graphs easiest to sort, and also the bell-shaped. They also find unimodal graphs easier to classify than bimodal graphs. They have more difficulty with the graphs that are skewed and bimodal.

The instructor can use the graphs on this website to refer to as the students suggest their categories:

http://app.gen.umn.edu/faculty_staff/delmas/gc_1454_course/distribution_file
s/distribution.html
.

This applet includes most of the graphs in the activity. There are buttons along the bottom that represent five different categories of distributions. When you click one, it brings up the set of graphs with that type of characteristic. You use the PREV and NEXT buttons on the right to view the graphs in each set.

The instructor can first ask the students to tell about one of their sortings and the words they used to describe them. The teacher can respond: "So, is this one of the graphs in that group?” and it usually will be one of them. A discussion can follow about the words they use for descriptions, then introduce the statistical term for the same characteristic (e.g. Statisticians use "uniform" to refer to what you mean by "even", "rectangular", or "steady state”).

The correct statistical terms for the graphs (uniform, normal, right and left skewed, bimodal) can be introduced if students have not yet learned these terms. Models (uniform, normal) can be described in terms of symmetry and shape (bell shape or rectangular). Other distributions that don’t fit these models can be described in terms of their characteristics (skewness, bimodality or unimodality, etc). A discussion of which descriptors can and cannot go together may follow.

These points may be included in the discussion of graphs following the activity:

 Ideal shapes: density curves vs. histograms Different versions of ideal shapes Idea of models, characteristics of distributions Statistical words vs. descriptors Normal, skewed, uniform, bimodal, symmetric: which can be used together? How well do they fit the graphs? Which fit best? Using judgment. Other ways to describe a distribution Why is it important to describe a distribution? Developing statistical thinking.

Assessment

To assess students’ ability to correctly describe graphs and understand the difference between graphs, these types of assessment can be used:

 Give students one or more histograms of data to describe in detail

For example:

For the graph below, of heights of singers in a large chorus, please write a complete description of the histogram. Be sure to comments on all the important features.

 Ask students to generate graphs for data sets such as:
1. The salaries of all persons employed by Northwest Airlines
2. The scores on a basic multiplication test for a group of college math majors
3. The scores on an art history test for a group of college math majors