An Activity-Based Statistics Course

Mrudulla Gnanadesikan
Fairleigh Dickinson University

Richard L. Scheaffer
University of Florida

Ann E. Watkins
California State University, Northridge

Jeffrey A. Witmer
Oberlin College

Journal of Statistics Education v.5, n.2 (1997)

Copyright (c) 1997 by Mrudulla Gnanadesikan, Richard L. Scheaffer, Ann E. Watkins, and Jeffrey A. Witmer, 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.

Key Words: Active learning; Data analysis; Experiment; Sample; Simulation.


So that students can acquire a conceptual understanding of basic statistical concepts, the orientation of the introductory statistics course must change from a lecture-and-listen format to one that engages students in active learning. This is the premise underlying an effort of the authors to produce and use a collection of hands-on activities that illustrate the basic concepts of statistics covered in most introductory college courses. Such activities promote the teaching of statistics more as an experimental science and less as a traditional course in mathematics. An activity-based approach enhances learning by improving the students' attention, motivation, and understanding. This paper presents examples of the types of activities that work well in various classroom settings along with comments from colleagues and students on their effectiveness.

1. What Does an Activity-Based Lesson Look Like?: Cents and the Central Limit Theorem

1 Ms. Jones walks into the room and asks her introductory statistics students, "Did you bring your 25 pennies today? Let's start making a histogram of their dates here on the floor." Each student has collected 25 pennies over the past few days. The students come to place the pennies in groups above a number line on the floor, according to the years in which the pennies were minted. This creates a histogram on the floor that looks something like Figure 1.

Figure 1. Dates of Pennies (19 suppressed)

         Interval     Count
           58-61        1  *
           62-65        6  ***
           66-69        6  ***
           70-73       14  *******
           74-77       23  ************
           78-81       22  ***********
           82-85       28  **************
           86-89       16  ********
           90-93       53  ***************************
           94-97       69  ***********************************

2 As the histogram is created, Ms. Jones enters the data into a computer (or calculator). She asks the class to estimate the mean and standard deviation of the distribution, which is a bit difficult because the distribution is strongly skewed to the left. She then uses the computer to find the values of the population mean, mu, and standard deviation, sigma.

3 Now Ms. Jones says, "Suppose we took random samples of size n = 5 and found the average age within each sample. What would happen?" She walks onto the pennies and starts to shuffle and slide her feet, destroying the histogram and creating a big pile of pennies. After mixing the pile thoroughly, she invites the students to come up and take samples of five pennies each. Each student randomly selects five pennies from the pile and calculates the average date of those five. Ms. Jones tells them to take a nickel from a stack of nickels she has put on the table and place it above the number line at the date that corresponds to the average they got for their sample. As the students do this, Ms. Jones enters these data into the computer.

4 The nickel histogram looks something like Figure 2. The class estimates the mean and standard deviation of the dates. Ms. Jones finds the actual mean and standard deviation on the computer and points out that the mean is close to the population mean, but the standard deviation is smaller than the population standard deviation of the original values.

Figure 2. Average Dates of Five Pennies

         Interval     Count
           76-77        1  *
           78-79        3  ***
           80-81        2  **
           82-83        9  *********
           84-85        4  ****
           86-87        6  ******
           88-89       12  ************
           90-91        3  ***
           92-93        2  **
           94-95        3  ***

5 The students return their pennies to the pile, which is remixed. Then they take samples of n = 10 each and find the average date for those 10. This time they use dimes to represent their sample averages as they create a new histogram above the nickel histogram on the floor. Ms. Jones finds the mean and standard deviation of the dime histogram using the computer. In the final part of the activity the students take samples of n = 25 and make a histogram using quarters. The class can now compare the three histograms on the floor and can use the computer to compare these to the original penny histogram. Ms. Jones points out that the nickel, dime, and quarter histograms are all centered at about the population mean, that the standard deviation not only gets smaller as n increases, but that it is roughly equal to the theoretical value of sigma/sqrt(n), and that the quarter histogram looks pretty much like a bell-shaped curve, even though the original penny histogram was far from normal. As the semester unfolds, Ms. Jones will often refer back to these histograms when she makes use of the central limit theorem.

6 As homework, Ms. Jones asks the students to answer the following questions.

  1. What can you say about the shape of the histogram as n increases? What can you say about the center of the histogram as n increases? What can you say about the spread of the histogram as n increases?
  2. The distributions you constructed for samples of size 1, 5, 10, and 25 are called sampling distributions of the sample mean. Sketch the sampling distribution of the sample mean for samples of size 36.

7 This scenario parallels the way several of us present an introduction to the idea of sampling distributions for means. Other teachers have tried similar forms of the activity with equal success, as attested to by the following comments.

The students indicated they really felt they understood what the CLT (central limit theorem) was all about after this demonstration. The non-normality of the original population helped greatly to convey the meaning of the CLT.

The students enjoyed working with the pennies so much that they wouldn't leave the room when the class ended.

Excellent idea to show the CLT in action using a non-normal population generated by the students.

2. Activity-Based Statistics is Not a New Idea

8 "How can I get students to see this important concept?" Statistics teachers, including the authors of this paper, have repeated this phrase, or a variation on it, often over the years, but few answers have been forthcoming. We decided that experienced teachers of the subject must have at least a few answers. Over the last few years, then, we have been collecting their activities for illustrating the concepts covered in most introductory college courses. This work formed the basis of an NSF-funded project entitled Activity-Based Statistics that led to a book by the same name (Scheaffer, Gnanadesikan, Watkins, and Witmer 1996).

9 We first decided to see if leaders in the development of applied statistics had anything to offer. After all, many of them have been successful at explaining statistics to scientists and engineers who work in business and industry. These leaders did have some very good ideas. William Cochran had his famous box of rocks sitting by the coffee pot in the lounge. He would ask any taker to estimate the average weight of the rocks in the box, by any method they wanted to use, and then bet the price of the next cup of coffee that he could come closer to the truth by using the mean of a random sample of rocks. It is reported that Professor Cochran paid for very few cups of coffee. Geoffrey Jowett, in a speech given at the Third International Conference on Teaching Statistics, told of getting students actively involved with experiments as early as the 1940's. [See Jowett and Davies (1960).] One favorite activity involved making a small slingshot to propel a coin across a table, with the response measure being distance from a target. Multiple factors on the slingshot could be adjusted by the user. In his speech Professor Jowett remarked, "A statistics course at a university should have as many laboratory hours as physics or chemistry." Frederick Mosteller had students toss thumb tacks to estimate the probability of a tack landing point up, among many other activities that he used successfully.

10 These master teachers and great statisticians had success with hands-on activities that got people involved with statistical principles. What they observed has been borne out by recent research into the learning of statistics: activities work. Garfield (1995) reviewed numerous research studies. She found that this research suggests the following ways to help students learn statistics.