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# Variability Simulation

Module by: David Lane. E-mail the author

## General Instructions

The graph shows two normal distributions. Both distributions have means of 50. The red distribution has a standard deviation of 10; the blue distribution has a standard deviation of 5. You can see that the red distribution is more spread out than the blue distribution. Note that about two thirds of the area of the distributions is within one standard deviation of the mean. For the red distribution, this is between 40 and 60; for the blue distribution, this is between 45 and 55. About 95% of a normal distribution is within two standard deviations from the mean. For the red distribution, this is between 30 and 70; for the blue it is between 40 and 60.

You can change the means and standard deviations of the distributions and see the results visually. For some values, the distributions will be off the graph. For example, if you give a distribution a mean of 200, it will not be shown.

## Step by Step Instructions

Does changing the mean of a distribution change its shape? Change the mean of the blue distribution from 50 to 60 to see. You will find that the shape is the same. It is just moved over to the right.

How many standard deviations from the mean do you have to go to include almost all of the distribution? Take a look at the red distribution. Very few scores are below 20 or above 80. This means that almost all of the distribution is within 30 points of the mean of 50. Set the mean and standard deviation of the blue distribution to 60 and 5 respectively. What are the limits of the distribution. Are almost all the scores within three standard deviations of the mean as they were with the red distribution? In other words, is just about all of the distribution between 45 and 75?

Experiment with different means and standard deviations to see how the changes affect the shapes of the distributions. Try a distribution with a mean of 50 and a standard deviation of 20.

## Summary

The more spread out a distribution is, the larger its standard deviation. About two thirds of a distribution are within one standard deviation of the mean; about 95% are within two standard deviations, and just about all of the distribution is within three standard deviations.

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