Algebra 2 / Unit 10: Statistics and Data Analysis
10.3 z-Scores
Notes
A z-score tells how many standard deviations a value is from the mean:
$$z = \frac{x - \mu}{\sigma}$$
- Positive $z$ is above the mean; negative $z$ is below.
- z-scores let you compare values from different distributions.
- Solve for the value: $x = \mu + z\sigma$.
- A value with $|z| > 2$ is often called unusual.
- With a standard normal table, $P(z < 1) \approx 0.84$ and $P(z < -1) \approx 0.16$.
Practice
A test has mean 75 and standard deviation 8. What is the z-score of 91?
$\displaystyle \frac{91 - 75}{8} = 2$.
Ana scored 85 on a test with mean 75 and standard deviation 5. Ben scored 90 on a test with mean 80 and standard deviation 10. Who did better compared with their class?
Ana: $\displaystyle \frac{10}{5} = 2$. Ben: $\displaystyle \frac{10}{10} = 1$.
A distribution has mean 200 and standard deviation 20. Which value has a z-score of $-1.5$?
$x = 200 + (-1.5)(20) = 170$.
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