Traditionally, to find the standard normal probability distribution, you must convert the normal random variable x to the standard normal distribution using the z-value formula, and then find the area under the standard normal distribution function below z. The normal probability distribution functions described earlier in this chapter shortens this process. The Standard Normal Probability Distribution has a mean of 0 and a standard deviation of 1.
NORMSDIST
If you already have a z value, or if you"ve used the z-value formula STANDARDIZE described below to find a z-value, you can use the NORMSDIST function to find the probability that a random variable x is below z standard deviations from the mean. The NORMSDIST function uses the following syntax:
=NORMSDIST (z-value)
For example, if you create furniture that needs to fit people of various heights and know that the average American adult is 5"8" tall, and the heights are normally distributed around this mean with a standard deviation of 4", you can find the probability that one of your customers is less than 6"2".
To use this function, you must first convert the data to the standard normal distribution as described below under "STANDARDIZE." Doing so returns a z value of 1.5, meaning that 6"2" is 1.5 standard deviations above the mean. When you enter 1.5 as the z-value parameter of the NORMSDIST function, the function returns the value 0.9331.
If you want to find the probability that a person is greater than 6"2" tall, you just subtract this value from 1. If you want to find the probability that a person is between 5"4" and 6", you must make a few calculations. The probability that a person is less than 6" tall is 0.8413. This means that the probability that a person is between the mean (5"8") and 6" is .3413 (because the probability that a person is less than 5"8" is 0.5). Likewise, the probability that a person is between 5"4" and 5"8" is .34134474. Add these together to get 0.6826.
NORMSINV
If instead of having a value and needing to find the probability that a random variable falls below it, you know the probability for the range into which a random variable must fall and need to find the value defining this range, you can use the NORMSINV function.
To use the NORMSINV function, just enter the probability (between 0 and 1, of course) and the function returns the z-value below which the probability area you entered falls. If you choose a probability less than 0.5, the function returns a negative z-value. If you enter a probability greater than 0.5, the function returns a positive z-value. The NORMSINV function uses the following syntax:
=NORMSINV (probability)
STANDARDIZE
Traditionally, to answer probability questions about a normal distribution, you first convert the distribution to the standard normal distribution. The standard normal distribution has a mean of zero and a standard deviation of 1. To convert to the standard normal distribution, you find a z value using the STANDARDIZE function in Excel. The STANDARDIZE function uses the following syntax:
=STANDARDIZE(x, mean, standard deviation)
For example, if you have a product that costs an average of $6,000 to produce and a standard deviation of $800, what percentage of the items should you expect to cost more than $6,600?
To find out, enter the function as follows:
=STANDARDIZE (6600,6000,800)
The function returns the value .75. You can then use the NORMSDIST function to find the probability or area under the curve between 0 and .75.
The z-value tells you how far (in terms of the number of standard deviations) an individual observation is from the mean. It therefore also allows you to determine whether an observation
is an outlier (unusually large or small) and therefore suspect. Z-values of less than -3 or greater than +3 are generally treated as outliers and call for closer inspection.