## Question 452:

1

When you generate a z-score from the raw data, what you are doing is making a linear transformation (subtracting by a constant and dividing by a constant). You subtract each value by the mean and divide by the standard deviation. When you do this, you convert the normal distribution of the data into a standard normal distribution. The standard normal distribution has tabled values and computations. If we didn't convert to a standard normal distribution, you'd just need to compute the area under the normal curve every time. This involves integrating (using calculus) between two points.

So why 0 and 1. It becomes apparent when we walk through any example. If we have a population, say all 10th grade students IQ scores and it is normally distributed with a mean of 100 and standard deviation of 16. If we want to know what percent of the population has IQ scores above 135 it is easier if we convert the numbers to the standard form by subtracting the mean and dividing that result by 0. Lets take 135, the z-score is (135-100)/16 = 2.187. If we now compute the z-score for the mean we have (100-100)/16 = 0 (since 0 divided by anything is 0). Since we divide everything by the standard deviation, the standard deviation becomes 16/16 = 1 (anything divided by itself =1). Statisticians could have also created a standard distribution by subtracting the mean from another constant and dividing by something other than the standard deviation, but you can see the intuitive appeal to having a simpler scale which does not depend on the raw units the data are in.

Notice that both the untransformed numbers and transformed numbers maintain the same property: 135 is 2.187 standard deviations above the mean of 100 (16*2.187 = 135) and 2.187 is 2.187 standard deviations above the mean 0 (2.187*1 = 2.187). By making the distribution mirror the standard normal we can more easily understand the properties of that distribution by quickly looking up 2.187 in a z-table or using the z-score to percentile calculator (which does the integration) to get an area of 98.56%.
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