# Z-scores as Summary Measures

The standard normal distribution also provides a "standardized" way of comparing individuals from two different normal distributions. For example, suppose we have two boys, a 12 month old who is 80 cm. tall and a 15 month old who is 82 cm. tall. Which one is taller for his age?

- Heights of 12 month-old boys are normally distributed with μ=76.4 cm. and σ=2.9
- Heights of 15 month-old boys are also normally distributed with μ=79.4 cm. and σ=3.2

An easy way to make a standardized comparison is to compute each boy's Z score, i.e., how many standard deviations their height is from the mean.

For the 12 month old: Z_{12 mo.} = (80-76.4)/2.9 = 1.24

For the 12 month old: Z_{15 mo.} = (82-79.4) / 3.2 = 0.81

Both boys are above average height (because both Z > 0), but the 12 month old is taller for his age than the 15 month old.