### Chapter 1. Descriptive Statistics: Measures of Location II: z-Scores

### Z-scores

Earlier, percentiles and quartiles were introduced as a way to express the location of a score relative to *all other scores *within a distribution. This section will introduce another way to determine the location of a score within a distribution, but this time the location of the score is expressed relative to the *mean of the distribution*.

At first glance, deviation scores might be able to serve this purpose. One of the shortcomings of deviation scores, however, is that they do a relatively poor job of expressing the *exact* location of a score within a distribution. Although the sign of a deviation score indicates on *which side* of the mean the score is located, the value of a deviation score is not a good indication of *how far* from the mean the score is located. This is because the value of a deviation score is an absolute measure of the distance from the mean, which makes it heavily dependent on the magnitude of the scores that make up the distribution.

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For example, a score that is located 5 units above the mean could either represent a large or a small distance from the mean, depending on the magnitude of the scores:

- If a distribution has a standard deviation of #\sigma=10#, then a deviation score of 5 would indicate that the score lies relatively close to the mean of the distribution.
- If a distribution has a standard deviation of #\sigma=2#, however, then a deviation score of 5 would indicate that the score lies somewhere near the edge of the distribution.

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In order to transform a deviation score into a useful measure of location, standardize the deviation score by dividing it by the standard deviation of the distribution. The resulting measure is called a *z-score*.

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Z-scores

**Definition**

A **z-score** or **standard score** is the standardized form of a *deviation score. *

Specifically, a z-score expresses a score's deviation from the mean in terms of standard deviation units.

**Formula**

\[z = \dfrac{X - \mu}{\sigma}\]

The *sign *of a z-score conveys the same information as the sign of a deviation score, namely on which side of the mean the score is located:

- Scores with a #\blue{\text{positive}}# z-score are located #\blue{\text{above}}# the mean.
- Scores with a #\orange{\text{negative}}# z-score are located #\orange{\text{below}}# the mean.
- A z-score of #\purple{\text{zero}}# means the score is #\purple{\text{equal}}# to the mean.

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Contrary to the value of a deviation score, the *value* of a z-score is a good indication of the exact location of a score because z-scores are *independent* of the measurement units used. This makes z-scores particularly well-suited for comparing scores that originate from different distributions. If two observations from two different distributions have the same z-score, this means that both scores are located at the same relative position within their respective distributions.

The formula for transforming sample scores into their corresponding z-scores is the following:

\[z=\dfrac{X-\bar{X}}{s}=\dfrac{45-40}{10}=0.50\]

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