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Putting Variables on the Same Scale: Standard Normal Distribution (Z)

In econometrics, a specific version of a normally distributed random variable is the standard normal. A standard normal distribution is a normal distribution with a mean of 0 and a variance of 1. It’s useful because you can convert any normally distributed random variable to the same scale, which allows you to easily and quickly calculate and compare probabilities.

Typically, the letter Z is used to denote a standard normal, so the standard normal distribution is usually shown in shorthand as Z ~ N(0, 1).

You can obtain a standard normal random variable by applying the following linear transformation to any normally distributed random variable:

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where X is a normally distributed random variable with mean

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and standard deviation

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Suppose you’re working with population data for individuals living in retirement homes. The average age of these individuals is 70, the variance is 9, and the distribution of their age is normal; that is, X ~ N(70, 9). If you randomly select one person from this population, what are the chances that he or she is more than 75 years of age?

You can figure out this probability by using the normal probability density function and applying integral calculus, but fortunately the standard normal distribution simplifies the problem. Instead, you simply convert the X value of 75 to a Z value and use the standard normal probability table to look up the density in that part of the distribution. Using the formula for Z and the standard normal probability table, you get

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This answer tells you that you have a 4.75 percent chance of selecting somebody from the population who’s more than 75 years of age.

The other popular continuous probability distributions — chi-squared, t, and F — are based on the normal or standard normal distributions.

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