Statistics / Unit 3: The Normal Distribution
3.1 Z-Scores
Notes
A z-score tells how many standard deviations a value is from the mean.
- $\displaystyle z = \frac{x - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation.
- Positive $z$ means above the mean, negative $z$ means below, and $z = 0$ means exactly at the mean.
- To go backward, use $x = \mu + z\sigma$.
- Example: A score of 85 on a test with $\mu = 70$ and $\sigma = 10$ has $\displaystyle z = \frac{85 - 70}{10} = 1.5$.
Practice
A value of 64 comes from a distribution with $\mu = 80$ and $\sigma = 8$. Its z-score is:
$\displaystyle z = \frac{64 - 80}{8} = -2$.
IQ scores have $\mu = 100$ and $\sigma = 20$ in a certain test. What score has a z-score of $1.5$?
$x = 100 + 1.5(20) = 130$.
A student's z-score on a quiz is $-0.5$. This means the student's score is:
The sign shows below or above the mean, and the size is measured in standard deviations.
Ready for more? Take the Unit 3 test.