# 2.6 Z-Score as a Measure of Relative Standing

A college admissions officer is looking at the files of two international candidates, one with a computer-based TOEFL score of 210 and the other with an IELTS score of 7.5. Which score is better? How can we compare measurements with different scales? The solution is the z-score, which gives a relative standing among the population.

Suppose variable [latex]X[/latex] follows a distribution with a mean [latex]\mu[/latex] and a standard deviation [latex]\sigma[/latex], the corresponding standardized variable is defined as

[latex]Z = \frac{X - \mu}{\sigma}.[/latex]

It can be shown that the standardized variable [latex]Z[/latex] has a mean 0 and a standard deviation 1.

For a given value of [latex]X[/latex], denoted as the corresponding lower-case [latex]x[/latex], the value of the standardized variable is called the z-score of [latex]x[/latex], which is given by

[latex]z = \frac{x-\mu}{\sigma}.[/latex]

If the population parameters [latex]\mu[/latex] and [latex]\sigma[/latex] are unknown, use the sample mean and standard deviation [latex]\bar{x}[/latex] and [latex]s[/latex] to estimate them, then the z-score becomes

[latex]z = \frac{x - \bar{x}}{s}.[/latex]

Properties of the z-score:

- It measures how far an individual is away from the mean using standard deviation as the unit (ruler).
- It represents a relative standing of an observation.
- A z-score > 0 means the observation x is above the mean; z-score < 0 means the observation x is below the mean; z-score = 0 means the observation x is equal to the mean.
- A z-score has no unit.

Example: Z-score

Suppose that the TOEFL score for admission at MacEwan has a mean 200 and a standard deviation 10, and the IELTS score has a mean 6 and a standard deviation 1. Two international candidates, one with a TOEFL score of 210 and the other with an IELTS score of 7.5. Which score is relatively better?

Calculate the z-score for the student who took the TOEFL exam:

[latex]z_T = \frac{x - \mu}{\sigma} = \frac{210 - 200}{10} = 1.[/latex]

This z-score means that the student who took the TOEFL exam is one standard deviation **above the average**.

The z-score for the student who took the IELTS exam:

[latex]z_I = \frac{x - \mu}{\sigma} = \frac{7.5 - 6}{1} = 1.5.[/latex]

This z-score means that the student who took the IELTS exam is 1.5 standard deviation **above the average**. Therefore, the student who took the IELTS exam is better because the z-score is larger. This tells us that the IELTS student score is relatively further above the mean than the TOEFL student score is above its mean.

Exercise: Women Heptathlon Champion

## Show/Hide Answer

The events are measured in different units, some are in metres and some are in seconds. Moreover, for those runs and high hurdles, results are the smaller the better; for jumps, shot put, and javelin, the larger the better. In order to combine the results of the seven events to a single score to determine the winner, one possible solution is the z-score. The idea is as follows:

For each athlete, calculate her z-score in each of those seven events:

Put a negative sign in front of those z-scores whose results are the smaller the better, i.e., for 200-m and 800-m runs, 100-m high hurdles. Note that a positive z-score in 100-m run indicates the athlete is below average, since she spends more time than the mean to finish the run. The athlete has the smallest z-score, which is a negative z-score, is the winner of this event. Putting a negative sign in front of the smallest z-score will make it positive and the largest z-score for this event. This makes sense in the way that the larger the z-score the better performance.

Add the seven z-scores together and get a single value. The one has the largest value is the winner.