Integrated Math 3 / Unit 8: Statistics and Inference
8.2 z-Scores
Notes
A z-score tells how many standard deviations a value is from the mean: $\displaystyle z = \frac{x - \mu}{\sigma}$.
- Positive $z$ is above the mean, negative $z$ is below, and $z = 0$ is exactly at the mean.
- Solve for a value with $x = \mu + z\sigma$.
- Use z-scores to compare values from different distributions.
- Example: 85 on a test with $\mu = 70$ and $\sigma = 10$ has $\displaystyle z = \frac{85 - 70}{10} = 1.5$.
Practice
A test has mean 70 and standard deviation 10. What is the z-score of a score of 85?
$\displaystyle z = \frac{85 - 70}{10} = 1.5$.
A distribution has mean 50 and standard deviation 4. What value has a z-score of $-2$?
$x = 50 + (-2)(4) = 42$.
Maria scored 78 on a test with mean 70 and standard deviation 4. Jo scored 85 on a test with mean 75 and standard deviation 10. Who did better relative to their class?
Maria: $\displaystyle \frac{78 - 70}{4} = 2$. Jo: $\displaystyle \frac{85 - 75}{10} = 1$.
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