STANDARD SCORES

If I tell you that I own 250 books, you probably will not be able to make very much of this information. You will not know how my library compares to that of the average consultant. Would it not be much more informative if I told you that I own the average number of books, or that I am two standard deviations above the average? Then, if you know that the number of books owned by consultants is normally distributed, you could calculate exactly what percentage of my colleagues have more books than I do.

To describe my library better in this way, you can calculate what is called a standard score. It describes the location of a particular case in a distribution: whether it is above average or below average and how much above or below. The computation is simple:

1. Take the value and subtract the mean from it. If the difference is positive, you know the case is above the mean. If it is negative, the case is below the mean.

2. Divide the difference by the standard deviation. This tells you how many standard deviation units a score is above or below the average.

For example, if book ownership among consultants is normally distributed with a mean of 150 and a standard deviation of 50, you can calculate the standard score for the 250 books I own in this way:

Step 1: 250 (my books) - 150 (average number of books) = 100 (I own 100 books more than the average consultant does.)

Step 2: 100 (difference from step 1)/50 (standard deviation of books) = 2 (standard score)

My standard score is two. Since its sign is positive, it indicates that I have more books than average. The number two indicates that I am two standard deviation units above the mean. In a normal distribution, 95% of all cases are within two standard deviations of the mean. Therefore, you know that my library is remarkable .

In a sample, the average of the standard scores for a variable is always zero, and the standard deviation is always one. Suppose you ask 15 people on the street how many hamburgers they consume in a week. If you calculate the mean and standard deviation for the number of hamburgers eaten by these 15 people and then compute a standard score for each person, you will get 15 standard scores. The average of the scores will be zero, and their standard deviation will be one.

When you use standard scores, you can compare values for a case on different variables . If you have standard scores of 2.9 for number of books, -1.2 for metabolic rate, and 0.0 for weight, then you know:

• You have many more books than average.

• You have a slower metabolism than average.

• Your weight is exactly the average.

You could not meaningfully compare the original numbers since they all have different means and standard deviations. Owning 20 cars is much more extraordinary than owning 20 shirts.

It is important to recognize that only for this example we focused on two standard deviations. Nothing prevents us from going even further. For example when we study Statistical Process Control charting we will be talking about three standard deviations, and certainly when we talk about six sigma we are indeed talking about six standard deviations.