Statistics / Unit 3: The Normal Distribution
3.6 Comparing Values with Z-Scores
Notes
Z-scores put values from different distributions on the same scale, so they can be compared fairly.
- The value with the larger z-score is higher relative to its own group.
- A z-score far from 0 (beyond about $\pm 2$) is unusual.
- Standardizing every value in a data set gives a distribution with mean $0$ and standard deviation $1$.
- Example: Ana scores 84 on a test with $\mu = 75$ and $\sigma = 6$ ($z = 1.5$). Ben scores 90 with $\mu = 80$ and $\sigma = 10$ ($z = 1$). Ana did better relative to her class.
Practice
Maya scored 92 in math (class mean 80, standard deviation 6) and 88 in English (class mean 76, standard deviation 4). Relative to her classes, she did better in:
Math: $\displaystyle \frac{92 - 80}{6} = 2$. English: $\displaystyle \frac{88 - 76}{4} = 3$.
Which value is more unusual?
$-2.2$ is farther from 0 than $1.9$, whether it is above or below the mean.
If every value in a data set is converted to a z-score, the new data set has:
Standardizing centers the data at 0 and measures spread in standard deviations.
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