This instructional aid was prepared by the Tallahassee Community College Learning Commons.
Z-scores and the Empirical Rule
The heights of individuals within a certain population are normally distributed. That is, it will be
found that most peoples' heights are clustered around an average height, and the farther a
particular height is above or below the average, the less people we would find with that height.
Graphically, it looks like this:
A Z-score tells how many Standard Deviations a value is above or below the Mean.
Suppose that the Mean height (µ) of college basketball players is 6 feet 4 inches (chance the
feet to inches; only one measuring unit is needed 76 inches = µ) with a Standard Deviation (σ)
of 2 inches.
A height of 78" (X) is 2" above the Mean; 2" is one Standard Deviation; so the Z-score
associated with 78" is positive 1.
𝑋 − 𝜇
𝜎 = 78 −76
2 = 2
2 = 1
A height of 74" (X) is 2" below the Mean; 2" one Standard Deviation; so the Z-score
associated with 72" is negative 1.
𝑋 − 𝜇
𝜎= 74 −76
2= −2
2= −1
z=1
z=-1
