Unit 5: The Normal Distribution

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Data Analysis & Activity

Activity

This activity is different from past ones. There are no questions to be answered; we just want you to observe.

What we want you to do is go to a web page and run an applet. The applet draws samples of various sizes from a normal distribution and makes a histogram. You can change the sample size to various values, and you can increase the sample size by repeated clicking.

What to look for: How large does the sample size have to be before a histogram "looks" normal? Also, notice just how much variety there are in the shapes of histograms that come from normal distributions.

Moral: It's hard to judge whether data are non-normal based on the histogram. There's quite a bit of variability in the shape.

What to do:

1) Go to this page:

http://www.rob-mcculloch.org/teachingApplets/iidNor/index.html

2) Click on "Show Histogram"

3) Click on "Draw 5". Notice how non-normal the histogram looks. Continue playing with different sample sizes.  You'll need to hit "reset" each time, otherwise you will be adding additional observations to the histogram.  But we want you to get experience with what histograms of small samples look like, so look at several histograms with a sample of 5.

Or do it in Fathom...

1) Start a new file (select "New" from File menu)

2) Drag the collection box into the file.

3) Double click on the collection box; an inspector opens up with 3 columns: Attribute, Value, Formula

4) Click on the space before Attribute and type in any name you want. For example, "fred"

5) Double click on the space below Formula. A formula editor window opens up.

6) In the window, type RandomNormal(0,1) or put in whatever values for the mean and SD you want. I chose 0 for the mean, and 1 for the SD.

7) click "OK" on the formula window.

8) Click once on the Inspector to select it.

9) Under the Collection menu at the top, select "New cases..."

10) Type in the number you want. For example, 5.

11) Viola. You now have 5 random numbers from a normal distribution.

12) Make a histogram of these by dragging the "fred" attribute onto a graph.

13) Now here comes the beautiful part. With the graph selected (by clicking anywhere on it once), hit control-Y (on the Mac) or select Rerandomize from the Analyze menu. You'll get another set of 5 numbers. You can do this again and again and watch the histogram change.

14) Instead of a histogram, you can also make a normal probability plot. Use the pull-down menu that is attached to the graph (upper right corner) and select "normal probability plot".

In addition to this activity, we also recommend returning to earlier data sets and comparing the normal probability plots with the histograms to help you develop a sense for what is "normal."