Is 82 a good score? It depends on the test. A z-score answers by measuring a value against its own distribution, using the mean and the standard deviation as a ruler.
Key idea
The z-score of a value $x$ is
$$z = \dfrac{x - \mu}{\sigma},$$
the number of standard deviations $x$ is from the mean. Positive $z$: above the mean; negative $z$: below. Working backward, $x = \mu + z\sigma$.
Why it works
Subtracting the mean measures the distance from the center (with a sign that says which side). Dividing by the standard deviation rewrites that distance in "standard deviation units," which have no ordinary units at all. That is what lets you compare a history score with a science score, or a height in inches with a weight in pounds: each becomes "how unusual is this, here?"
Worked example: Computing z
A test has mean $\mu = 70$ and standard deviation $\sigma = 8$. Find the z-scores of 82 and 62.
- $z = \dfrac{82 - 70}{8} = \dfrac{12}{8} = 1.5$: one and a half standard deviations above the mean.
- $z = \dfrac{62 - 70}{8} = \dfrac{-8}{8} = -1$: one standard deviation below.
Answer: $z = 1.5$ and $z = -1$.
Worked example: From z back to x
On the same test, what score has $z = -0.5$?
- $x = \mu + z\sigma = 70 + (-0.5)(8)$.
- $= 70 - 4 = 66$.
Answer: 66
Worked example: Comparing across scales
Ana scored 82 on a test with mean 70 and SD 8. Ben scored 650 on a test with mean 500 and SD 120. Who did better relative to their own test?
- Ana: $z = (82 - 70)/8 = 1.5$.
- Ben: $$z = (650 - 500)/120 = 150/120 = 1.25$$
- Ana is farther above her mean, in standard deviation units.
Answer: Ana, with $z = 1.5$ versus $1.25$.
Common mistake
Subtracting in the wrong order, $\mu - x$, which flips the sign: a score below the mean would get a positive z. Always compute value minus mean. Check: if $x$ is below the mean, z must be negative.
For bell-shaped data, a z-score also gives a percentile: $z = 1.5$ is above about 93% of the data. You will do this in the normal curve lessons.