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Wednesday, August 31, 2011

More About the Normal Distribution: the 68-95-99.7 Rule

In the last post, we covered the Normal distribution as it relates to the standard deviation. We said that the Normal distribution is really a family of unimodal, symmetric distributions that differ only by their means and standard deviations. Now is a good time to introduce a new term: Parameter. When dealing with perfect-world models like the Normal model, their major measures -- in this case, their mean and standard deviation -- are called parameters. The mean (denoted by the Greek letter "mu" (pronounced "mew"), µ, and the standard deviation is denoted by the Greek letter sigma, σ. We can refer to a particular normal model by identifying µ and σ and using the letter N, for "Normal:" N(µ, σ). For example, if I want to describe a Normal model whose mean is 14 and whose standard deviation is 2, I use the notation N(14, 2).

Every Normal model has some pretty interesting properties, which we will now cover. Take the above model N(14,2).  We'll draw it on a number line centered at 14, with units of 2 marked off in either direction. Each unit of 2 is one standard deviation in length. It would look like this...
Now let's mark off the area that's between 12 and 16; that is, the area that's within one standard deviation of the mean. In a Normal model, this region will contain 68% of the data values:
If you've ever heard of "grading on a curve," it's based on Normal models. Scores within one standard deviation of the mean would generally be considered in the "C" grade range.

Now, if you consider the region that lies within two standard deviations from the mean; that is, between 10 and 18 in this model, this area would encompass 95% of all the data values in the data set:
From a grading curve standpoint, 95% of the values would be Bs or Cs. Finally, if you mark off the area within 3 standard deviations from the mean, this region will contain about 99.7% of the values in the data set. These extremities would be the As and Fs in our grading curve interpretation.
In statistics, this percentage breakdown is called the "Empirical Rule," or the "68-95-99.7 Rule."

What about the regions up to 8 and beyond 20? These areas account for the remaining 0.3% of the data values? That would make 0.15% on each side.
(Self-Test): What percent of data values lie between 12 and 14 in N(14,2) above?
(Answer): If 68% represents the full area between 12 and 16 and given that the Normal model is perfectly symmetric, there must be half of 68%, or 34% between 12 and 14. Likewise, there would be 34% between 14 and 16.

(Self-Test): What percent of data values lie between 16 and 18 in N(14,2) above?
(Answer): Subtract 95% minus 68% = 27%. This represents the number of data values between 10 and 12, and between 16 and 18 both. Divide by 2 and you get 13.5% in each of these regions.

If you do similar operations, the various areas break down like the following:
(Self-Test): In any Normal model what percent of data values are greater than 2 standard deviations from the mean?
(Answer): Using the above model, the question is asking for the percent of data values that are more than 18. You would add the 2.35% and the 0.15% to get the answer: 2.50%.

This is all well and good if you're looking at areas that involve a whole number of standard deviations, but what about all the in-between numbers? For example, what if you wanted to know what percent of data values are within 1.5 standard deviations of the mean? This will be the subject of a future post.

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