Z-Score Review

Can you answer the following question: A cumulative normal distribution table will tell the probability that a random event from that distribution will be less than a target event. In the case of our problem, it would be helpful to know the probability that a random sales figure will be less than 120.  If we know the probability that it will be less than 120, we can subtract that number from 1 to determine the probability that it will be at least 120.

Using a text book, we would look up the probability values in a cumulative normal distribution table. Or, you could use a z-score calculator.

The problem is that there are an infinite number of normal distributions defined by different means and different standard deviations. The following are all different normal distributions:

Most textbooks only include one cumulative normal distribution table -- the cumulative unit normal distribution table. That is the cumulative normal distribution table with a mean of 0 and a standard deviation of 1.

But, in our sales problem we have a mean of 130 and a standard deviation of 20.  How can we use the cumulative unit normal distribution table?

This is actually quite a simple problem if you just understand how to use z scores. Any normal distribution can be converted to a cumulative normal distribution by normalizing the target value. That is, we need to determine how many standard deviations the target value is from the given mean.

First, we subtract the distribution mean from the target value: 120-130 = -10. That implies that the target value is 20 units below the mean. (To break even, we do not need to sell the mean of 130, but can sell just 10 below the mean.)

Next, we know that a standard deviation is 20 for our distribution. Since our target is 10 below the mean (-10), our target is one-half of a standard deviation below the mean (-10/20= -0.5). That is a z-score. The z score tells how many standard deviations the target is above (or below if negative) the mean.

Using the text book's cumulative unit normal distribution table or a  z-score calculator, we look up z=-.5 and find that the probability of selling 120 or fewer widgets is 0.3085375, or 30.8 percent. Thus, the probability of selling more than 120 is one minus that amount or 0.6914625 (69.1 percent), which is the probability that we will break even.

(Note that the normal distribution is a "continuous" distribution, which implies a continuous range of sales values.)

Inverse Cumulative Normal Distribution

What about the following question: Of course, that problem is trivial given the above results. It is simply asking for the target value that corresponds with a given cumulative normal distribution probability. The answer is: 120!

If we have a target probability and are looking for a target value, we simple do the process in reverse (it is the inverse).

First, look up the z-score in the text table. Z-score tables show the probability of obtaining less than a given value, which is one minus the probability of obtaining greater than that given value.  The z score for 0.31 (i.e., 1-0.69) is about -0.5. That means that the target value is one half of a standard deviation below the mean.

To convert the z score into the target value on our Normal(mean=130,std=20) distribution, we simply multiply the z-score by the standard deviation of 20 and add the mean of 130: -0.5*20+130 = 120! There we have it.

Inverse Cumulative Normal Calculator

Here is a simple calculator that calculates that inverse cumulative normal distribution value:
Enter a cumulative probability:
(Remember that the cumulative probability is the probability of being less than a certain amount, which is one minus the probability of being greater than that amount.)
Click:
The corresponding z-score is . Multiply that z-score by the distribution standard deviation, then add the distribution mean to get the actual target value.

Try it on the following problems:

For a calculator that goes either z to probability or probability to z,  click here.

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