Section 3.4 Measures of Position and Outliers
ΒΆDefinition 3.4.1. Z-score.
A z-score of a data point is the number of standard deviations the point is from the mean.
- population z-score
\(z = \dfrac{x-\mu}{\sigma}\)
- sample z-score
\(z = \dfrac{x-\bar{x}}{s}\)
Characteristics of z-scores:
- unitless
- mean of z-scores = 0
- std. deviation of z-scores=1
- If a datum is larger than the mean its z-score is positive.
- If a datum is smaller than the mean its z-score is negative.
- Z-scores can be used to compare measurements on different scales.
Example 3.4.3. Comparing Final Exam Scores.
Final exam scores for a Biology course have a mean of 79 and a standard deviation of 5. Final exam scores for a Statistics course have a mean of 82 and a standard deviation of 7. You scored 86 on the Biology exam and 91 on the Statistics exam. How many standard deviations above the mean were you in each course? Relative to the rest of the students, on which exam did you do better?
statistic | symbol | Biology | Statistics |
test score | \(x\) | 86 | 91 |
mean | \(\mu\) | 79 | 82 |
std. dev. | \(\sigma\) | 5 | 7 |
z-score | \(z\) | \(\dfrac{86-79}{5}\) | \(\dfrac{91-82}{7}\) |
\(z\) | 1.4 | \(\approx\) 1.3 |
Thus since, 1.4 > 1.3, you did slightly better on the Biology exam than you did on your Statistics exam, relative to your fellow students. This is despite the fact that in absolute terms, your score in Statistics was higher.
This example demonstrates that it is meaningless to compare absolute measures which come from different distributions, in this case, the test scores. However, by scaling each test score relative to the mean and the standard deviation of each distribution i.e. creating z-scores, we can meaningfully compare data from different distributions.
Example 3.4.5. Triathlon Times.
Roberto finishes a triathlon (750 m swim, 5 km run, and 20 km bicycle) in 63.2 minutes. Among all men in the race, the mean finishing time was 69.4 minutes with a standard deviation of 8.0 minutes. Zandra finishes the same triathlon in 79.3 minutes. Among all women in the race, the mean finishing time was 84.7 minutes with a standard deviation of 7.4 minutes. Who did better relative to their gender?
We compute a z-score for each athlete:
statistic | Roberto | Zandra |
z-score | \(\dfrac{63.2-69.4}{8.0}\) | \(\dfrac{79.3-84.7}{7.4}\) |
z-score | -0.775 | \(\approx\) -0.730 |
In the case of a race, it is better to have a time below the mean time of all the racers, thus the more negative the z-score the better. Since Roberto has the lower z-score, Roberto did better relative to his competitors than Zandra did relative to hers.