Math for Liberal Arts

Section 10.3 Z-Scores

Using the Empirical Rule can give you a good idea of the probability of occurrence of a value that happens to be exactly one, two or three to either side of the mean, but how do you compare the probabilities of values that are in between standard deviations?

Z-scores are related to the Empirical Rule from the standpoint of being a method of evaluating how extreme a particular value is in a given set. You can think of a z-score as the number of standard deviations there are between a given value and the mean of the set. While the Empirical Rule allows you to associate the first three standard deviations with the percentage of data that each SD includes, the z-score allows you to state (as accurately as you like), just how many SDs a given value is above or below the mean.

Conceptually, the z-score calculation is just what you might expect, given that you are calculating the number of SDs between a value and the mean. You calculate the z-score by first calculating the difference between your value and the mean, and then dividing that amount by the standard deviation of the set. The formula looks like this:

where \(z\) is the z-score corresponding to a data value (\(x\)) in a normal distribution with a mean of \(\mu\) and a standard deviation of \(\sigma\text{.}\)

In this lesson, we will practice calculating the z-score for various values. In the next lesson, we will learn how to associate the z-score of a value with the probability that the value will occur.

Knowing the z-score of a given value is great, but what can you do with it? How does a z-score relate to probability? What is the probability of occurrence of a z-score less than +2.47?

Since z-scores are a measure of the number of SDs between a value and the mean, they can be used to calculate probability by comparing the location of the z-score to the area under a normal curve either to the left or right. The area can be calculated using calculus, but we will just use a table to look up the area.

In order to find the values for z-scores that aren’t integers, you can use a table like the one below. To find the value associated with a given z-score, you find the first decimal of your z-score on the left or right side and then the 2nd decimal of your z-score across the top or bottom of the table. Where they intersect you will find the decimal expression of the percentage of values that are less than your sample.

Z-score tables like the one above describe the probability that a given value, or any value less than it, will occur in a given set. This particular table assumes you are looking to find the probability associated with a positive z-score. You may have additional work to do if the z-score is negative.

• To find the percentage of values greater than a negative Z score, just look up the matching positive Z score value.

• To find the percentage of values less than a negative z-score, subtract the chart value from 1.

• To find the percentage of values greater than a positive z-score, subtract the chart value from 1.

Returning to the problem,

Knowing the z-score of a given value is great, but what can you do with it? How does a z-score relate to probability? What is the probability of occurrence of a z-score less than 2.47?

A z-score lets you calculate the probability that a randomly selected value will be greater or less than a particular value in a set.

To find the probability of a z-score below +2.47, using a reference such as the table in the lesson above:

1. Find 2.4 on the left or right side

2. Move across to 0.07 on the top or bottom.

3. The cell you arrive at says: \(0.9932\text{,}\) which means that apx 99.32% of the values in a normal distribution will occur below a z-score of 2.47.

Do z-score probabilities always need to be calculated as the chance of a value either above or below a given score? How would you calculate the probability of a z-score between two values?

To calculate the probability of getting a value with a z-score between two other z-scores, you can use a reference table (such as Table 10.3.7) to look up the value for both scores and subtract them to find the difference.

Example 10.3.13.

What is the probability associated with a z-score between 1.2 and 2.31?

Solution.

To evaluate the probability of a value occurring within a given range, you need to find the probability of both the upper and lower values in the range, and subtract to find the difference.

First find z = 1.2 on the z-score probability reference above: \(.8849\text{.}\) Remember that value represents the percentage of values below 1.2.

Next, find and record the value associated with z = 2.31: \(.9896\)

Since approximately 88.49% of all values are below z = 1.2 and approximately 98.96% of all values are below z = 2.31, there are 98.96% -88.49% = 10.47% of values between.