Standard Deviation with Rat Terriers

Created: Summer 2012
Uploaded: January 3, 2013

Introduction
So here is a question that I get from folks who are interested in teaching statistics with dynamic data software packages. Should I get TinkerPlots or Fathom? My answer is BOTH. Since TinkerPlots Version 2 is the new kid on the block, I want to talk about how TinkerPlots can and should be used with learners of all ages.
TinkerPlots provides an excellent (and fun) environment to foster an understanding of the different measures of variation (range, interquartile range, mean absolute deviation and standard deviation). In this post, I will illustrate how the distribution curve device in the sampler engine can be used to foster conceptual understanding of standard deviation and the normal distribution.

The normal distribution can be modeled in the sampler by using the distribution curve device. The context used here is the weight, in pounds, of rat terriers (medium sized dogs). I set the endpoints of the curve to a range of 4 to 35 pounds and traced a symmetric mound shaped distribution with the mouse. The distribution curve represents the parent population of all rat terriers. Disclaimer: For educational purposes, I am assuming that weight for this breed is normally distributed with a shape similar to what I sketched. Also, I ran the Sampler until I got some nice numbers for the graphics.
In the sampler, I set the repeat value to 500 and the draw to 1. The resulting case table contains the weights of one sample of 500 randomly selected rat terriers. It is interesting to see that the dot plot created with the sample data resembles the general shape of the curve, but it has some spikes and dips. Running the sampler again will show a different pattern. Comparing the smooth curve on the left with the bumpy dot plot on the right helps my students make a connection to the distribution curves printed in most textbooks!
Rats1.jpg
The mean tool, from the plot menu, shows the mean weight for this sample of dogs. Next, overlay the dot plot with the -1.00 to +1.00 Standard Deviation Hat Plot from the hats plot menu. The ruler tool can be snapped to the crown of the hat (shown above) or between the mean and the right crown of the hat (as shown below). The number sentence in the plot indicates that the sample mean is approximately 19 pounds. The sample standard deviation is about 4.5 pounds, so the weight at one standard deviation above the mean is 23.5 pounds.
Rats1a.jpg

Since the sample standard deviation is 4.5 pounds, I placed vertical reference lines in the plot at 10, 14.5, 19, 23.5 and 28 pounds. These lines represent weights at -2, -1, 0, +1 and +2 standard deviations from the mean. It is important to note that these weights are based on this sample of 500 dogs and the curve that I drew in the sampler.
Rats2a.jpg
According to the Empirical Rule, 95% of the population will fall between -2 and +2 standard deviations from the mean. In this example, the probability of a randomly selected dog weighing between 10 and 28 pounds is 95%.
As a result of using this activity, my students have developed a good understanding of the normal distribution curve. For example, they can approximate the location of the mean, as well as -1 and +1 standard deviations from the mean in a plot of data that appears to be normally distributed (see the example at the bottom of this page). Reference lines can be added to the dot plot, or students can sketch lines with the draw tool. This is crucial when we get to the unit on hypothesis testing.
As you might guess, this activity was developed in a graduate level statistics course and my goal for the day was to introduce z-scores. The next graphic shows how I used the rat terrier weight context to bring in the formula. All of the in-service teachers "got it."
rats2b.jpg

Hopefully this TinkerPlots activity on standard deviation and the normal distribution sparks a number of follow up questions like, How well does this sample of 500 dogs represent the population of all rat terriers? Will the statistics change with a new sample? What if we draw the distribution curve with a flatter or steeper mound? What would happen if we increase or decrease the sample size?




Sample Quiz question with solution.Sketch reference lines for -1, 0 and +1 standard deviations in the plot below. Rats3.jpg