# Difficult Concept: Teaching Sampling Error and Sampling Distribution of the Means

I am currently teaching sampling distribution of the means and sampling error to my students. They are difficult concepts to convey to students, and unlike much of my teaching, where lecture comprises a fair portion of my teaching time, I find myself “slowing down” the progress at this point by putting more of the activities in the hands of the students, forcing   them to participate in activities during class time, and requiring them to generate ideas in and out of class.

There are three activities that I use to help students learn the concept of the sampling distribution of the means and sampling error.

(1)    Generating hypothesis, then identifying “individual differences in extraneous variables”

• First, I model for them, using the Socratic method (asking them questions as a means of leading them to the answer), how to identify individual differences. I first do this when introducing extraneous variables, during the first week or two of class, and periodically do so throughout the first half of the semester, anytime I speak of Independent, Dependent, or Subject Variables, I have students generate the extraneous variables as well. This task, repeated early on, and especially as we approach sampling error, not only helps students to understand sampling error, but it makes the teaching of confounds easier as well. (Sampling error are random variations in extraneous variables, while confounds are systematic variations in extraneous variables.)
• I assign for homework, that students have to generate a hypothesis (by this point, they have been doing this throughout the semester), then generate a list of 10 individual differences in extraneous variables.
• During class time, they form groups, to discuss and critique each others’ list, then generate another list, as a group, that gets graded as a quiz. Truthfully, I have too many students (and no TA)  to grade all 80 of these assignments, by working in groups of 5, I have little trouble grading the list.

Notice how much time I spend on the concept of individual differences and extraneous variables. But, as a critical concept, it is time well spent. Truthfully, it comprises about 50 minutes, but it typically takes place over the course of weeks, helping build students’ thinking.

(2)    M&E creation of a pseudo empirical distribution of the means.

• I formally model sampling distribution in class with the M&M demonstration.  Though I’ve described this activity before, I’ll describe it again here.
• I get plain M&M’s whose proportion by color is: 24% blue, 14% brown, 16% green, 20% orange, 13% red, and 14% yellow.
• Each color receive a value (e.g., 1 – 6).
• I calculate what the mu would be given the stated proportions.
• I have students randomly sample N=X (that value depends on how many M&M’s I have to share with the students, 10 should be the smallest value).
• Students then calculate the mean for their sample.
• Then I have them report their sample means, I enter them into Excel and do a very quick (and sloppy) empirical sampling distribution, and tell them what mu is.
• We compare our mean of the mean to the mu, and talk about the variability in the rest of the sample means.
• We talk about how their individual sample means differ from mu and why.
• It seems so obvious to the students, that I can then switch over to other examples, like dog weight or performance on at recall for a list of words.
• Students generate the extraneous variables that serve as sampling error, just as the colors of the M&M’s can serve as sampling error.

(3)    I end with having students participate in a Mathematica Demonstration, both in and outside of class.  If you haven’t used Mathematica Demonstrations, start with  reviewing this prior blog https://statisticalsage.wordpress.com/2011/01/08/before-the-semester-starts-im-playing-with-pictures/ or this one https://statisticalsage.wordpress.com/2011/05/24/using-mathematica-deomnstrations-to-visualize-statistical-concepts/.

If you have used Mathematica, this demonstration works well in helping students to understanding the sampling distribution of the means

This year, I am requiring that student answer a series of questions about each mathematic demonstration to see if focusing them on the activity will increase what they are gaining from it.

For this demonstration the questions are as follows:

1. Try three different sample sizes. Which ones did you select? Draw the sampling distribution of the means by each N. What happens to the shape of the sampling distribution of the mean as N gets larger? Explain why this happens.

2. Using N = 15, change mu. What happens to the shape of the sampling distribution of the means as mu changes? Explain why this happens.

3. Write the symbol for standard error. Change the standard deviation. What happens to the standard error as sigma gets larger? Explain why this happens.

4. Define Sampling Distribution of the Means. Define sampling error. What value do we calculate to find sampling error. Write down that formula. Why is this such an important part of statistics?

As with all of our difficult concepts. If you have any recommendations, I encourage you to  first work on getting it published in http://www.teachpsychscience.org/ and then let us know about it!