Wiki Page, Math 13 - Perez Sandbox : Module 12 – Introduction to Normal Probability Distributions (5 of 5)

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2026-01-17T21:15:14-08:00

Now that we know how to use a Z-score to find a probability with the Normal Distribution Calculator, we will briefly discuss what is going on behind the scenes of this process.

As we discussed previously, a Z-score is a standardized score. It uses standard deviation as a yardstick to describe the distance between an X-value and the mean. In other words, a Z-score is the number of standard deviations an X-value is from the mean.

Imagine translating every X-value in a normal probability distribution into a Z-score. What would the distribution of Z-scores look like?

Not surprisingly, the resulting distribution of Z-scores will also be a normal probability distribution, called the Standard Normal Distribution. It has a mean of 0 and a standard deviation of 1.

EXAMPLE

Here we compare the normal probability curve for the foot lengths to the standard normal curve.

Understanding these probability distributions:

Look at the values on the horizontal axes. In the top graph we have the X-values of foot length. The mean is 11 inches with a standard deviation of 1.5 inches. The bottom graph has Z-scores on the horizontal axis. The mean is 0 with a standard deviation of 1.

Both graphs are probability distributions with area under the curve equal to 1. Areas in the probability distribution for foot length equal the corresponding areas in the Z-score distribution.

Compare the two normal curves and notice the following:

The Z-score at the mean for X is 0. This is always true.
For example, foot lengths have a mean of 11 inches: Z = (11 – 11) / 1.5 = 0/1.5 = 0.

The X-value one standard deviation above the mean has a Z-score of 1 and the X-value one standard deviation below the mean has a Z-score of –1. This is always true.
For example, the foot length of 12.5 is one standard deviation above the mean and its Z-score is (12.5 – 11) / 1.5 = 1.5 / 1.5 = 1. Similarly, the foot length of 9.5 is one standard deviation below the mean and its Z-score is (9.5 – 11) / 1.5 = –1.5 / 1.5 = –1.

We can use the area under the Standard Normal Distribution to find probabilities in the normal probability distribution of X. For example, The Z-score for X=13 is 1.33. The probability associated with X-values greater than 13 is the same as the probability associated with Z-scores greater than 1.33.

Why do we standardize X-values into Z-scores and use the Standard Normal Distribution to find probabilities?

The answer is really historical. Before the advent of more sophisticated technology, statisticians used tables, instead of applets or statistical programs, to find the probabilities associated with normal distributions.

They could not produce a table for every possible normal distribution because there are an infinite number of possibilities for the mean and standard deviation, so they simplified the process by standardizing X-values into Z-scores. They then used a single table associated with the Standard Normal Curve to find probabilities. Some statistics textbooks still contain a Z-score table.

In current practice, statisticians often do not use standardized curves. They don't need to. Why? Because they have access to more sophisticated technology. For example, StatCrunch has a Normal Distribution Calculator that does not require a standardized score, as shown below. We can simply put in the mean and standard deviation of X, indicate the range of X-values associated with the desired area, and the calculator determines the probability. This allows us to skip the step of standardizing the X-value into a Z-score.

Why do we still teach the Standard Normal Curve?

Even though StatCrunch and similar technology allows us to skip the step of standardizing scores with normal distributions, we think it is important to understand the standardized Z-score. Why? Because it lays the foundation for our general approach to statistical inference.

As we get further into statistical inference, you will notice a pattern in our process. We will always translate an observed data value into a score called a test statistic. (A Z-score is an example of a test statistic.) We will then use a density curve based on the distribution of these scores to estimate probabilities. (The Standard Normal Distribution is an example of a density curve that we use to estimate probability.)

That said, you are free to use StatCrunch instead of the Standard Normal Curve and Z-score applets provided here to find probabilities.