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Sampling Distribution Laboratory Module

The hypothesis testing model of inferential statistics is built around the concept of Sampling Distributions. This is a difficult concept for beginning stat students to understand. With my colleagues, we have developed a series of exercises which allows students to work with the concept of sampling distribution.

Critical Definitions: Statistic, Frequency Distribution, Sampling Distribution

Step 1: Realize there is a problem

[Picture of puzzled instructor]

"Gee, my students don't understand why I think sampling distributions are important."

"Of course!"

Let students draw repeated samples from a known population....duh.

The first "known population" was 5,000 slips of paper.


Step 2: Get yourself a population.

"My two kids finished 5,000 popsicles over the weekend."

Psychologist with "Known Population"

The 5,000 popsicle sticks constitute Normal population with a mean of 28 and a standard deviation of 5 aka N(28, 5).

Pokerchips would have been nice and have you ever priced 5,000 pokerchips?

Basically the laboratory project involves drawing samples from this population, calculating a statistic and plotting the results in a histogram.

After drawing a sample or two from the population, calculating a particular statistic, and plotting the result, students are quite happy to let software repeat the process for them.


Step 3: If 15 are good then 10,000 is better.

"Boy, is that ever a good idea."

Automate & Animate

If drawing 15 samples of 10 observations helps students understand sampling distributions, then drawing 10,000 will really help them understand. Right?


Step 4: Write yourself a Laboratory Handout

Or...use ours

Click the document ("pdf" format) to download a nine-page guide that gently walks students through the conceptual forest of sampling error and standard error using the capabilities of RippleSoftware's Statistics Tool.

Incidently, the information you need to create your own "Known Population" is included in the handout.


Step 5: Go to the laboratory task in "Statistcs Tool"

Begin the Lab

This is the beginning of the lab.

When a student clicks on the field of blue numbers, a value is randomly selected and appears on a "poker chip". The poker chip can be dragged into the "Sample" window.

The student calculates the mean and standard deviation. After drawing one or two samples, the process can be automated.


Step 6: Here's the first task completed.

The completed card

Here's the tasks for the first card completed. The card shows the mean and standard deviation of 15 samples of 10 observations.

In addition, the value of the standard deviation of the 15 means is also shown. This value is the standard error of the mean and is shown to be smaller than the standard deviation of any one of the samples.


Step 7. Repeat this whole process again and again and again.

Next Card: Further Automation

Now the process of drawing 15 samples of 10 observations can be repeated again and again by simply clicking a single button.

Students are asked to identify 3 ways to assign a value to how much the mean of a sample will change from one sample to the next. For the results displayed on the left, the three "estimates are,

  1. The population standard deviation divided by the square root of sample size: 1.5811
  2. The standard deviation of the 15 means: 1.513
  3. The standard error of the mean for any of the 15 samples. The values for the first 6 samples are: 1.593, 1.692, 1.009, 1.067, 2.206, and 1.526.

Step 8: Let's get really serious...

LIke, draw 30,000 samples

Using another card in the Statistics Tool application, students can quickly draw thousands of samples from a variety of populations.

It took about a minutes (on a Mac powerbook) to draw 30,000 samples of size 10, calculating the sample mean of each value.

The mean of the 30,000 samples is 28.016 and the standard deviation is 1.4994.

When a sample of size 10 is drawn from this population, it comes from this sampling distribution.

Now, copy the means to the clipboard (click on the paperclip; then click the "Histogramer" button.


Step 9: Plot the 30,000 means

Well, Here's the "Central Limit Theorem"

How about that, sample means are normally distributed.

This plot illustrates the "sampling distribution" of sample means. The mean of the sampling distribution is the mean of the parent population. The standard deviation of the sampling distribution is the standard deviation of the parent population divided by the square root of sample size.


Step 10: What does the "Sampling Distribution" of the sample variance look like?

The sample variance isn't normal.

Here's the sampling distribution of 30,000 sample variances; each variance calculated from a sample of n= 10.



© 2005 by Burrton Woodruff. All Rights Reserved. Modified Fri, Dec 28, 2007