Concepts of the normal distribution and Z scores

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Dr. Kenneth Le Cour

6/25/2023 1:29 PM

Statistics: Z-Score Table

Hello Everyone:

The following information should provide some clarification on the concepts of the normal distribution and Z scores. First we need to consider that the sum of all probabilities equals 1 no matter what probability distribution we are using. When we use the normal distribution half of the probabilities are below the mean and half of the probabilities are above the mean. We also need to consider that the Z scores or probabilities relate back to the mean. One of the best ways to approach this is to draw a bell curve on a piece of paper and then draw the Z score on the curve. A negative Z score means the value is to the left of the mean.

Let us look at an example. If you look at the chart for the normal distribution the Z scores are in the first column and first row. The first column has Z score of #.#. The first row has Z scores of 0.0. Again, the first column and first row are the Z scores. All of the other values in the table are the probabilities of getting the Z score. If we are looking for the probability of a Z of 1.25 we would go to the first column to find 1.2 and the first row to find .05. the value in the table that corresponds to this Z score is .3944.

Let us try another example. What is the probability of a Z score of 1.64?

There will also be times when we know the probability and want to determine the Z score. For example, let us say there is a .4920 probability of a value occurring and we want to know what the Z score is for this probability. We would go to the table for the normal distribution and find .4920 ion the body of the table. The Z score in the first column and first row that corresponds to this probability would be the Z score. In this case the Z score for .4920 would be 2.4 and .01 or 2.41.

Let us try another example. What is the Z score for a probability of .3315?

Thank You,

Dr. Ken LeCour