z-scores

In the 2012 SAT, verbal scores had mean 496 and standard deviation 114, so a score of 325 has z = (325 − 496) ÷ 114 = −1.5.

A raw score on its own says little. Is 168 cm tall? Is 10 points a good game? The answer depends on what is usual for that group and how spread out the group is. A z-score answers both at once. It tells you how many standard deviations a value is above or below the mean, which lets you compare values measured on completely different scales.

The formula

The z-score of a value x from a distribution with mean μ and standard deviation σ is z = (x − μ) ÷ σ. A positive z means x is above the mean, a negative z means below, and z = 0 means x equals the mean. Turned around, x = μ + zσ. OpenStax's first example is X ~ N(5, 6) and x = 17: z = (17 − 5) ÷ 6 = 2, so 17 is two standard deviations above the mean. For x = 1 in the same distribution, z = (1 − 5) ÷ 6 ≈ −0.67.

Comparing across scales

OpenStax compares two groups trying to gain weight over six weeks: X ~ N(5, 6) and Y ~ N(2, 1), in pounds. A gain of 17 in the first group and a gain of 4 in the second look very different, yet both have z = 2. Relative to their own groups, they are equally unusual. The same trick works for heights across decades. In 2009 to 2010 Chilean males aged 15 to 18 had N(170, 6.28) and in 1984 to 1985 N(172.36, 6.34). A height of 160.58 cm in the first period and 162.85 cm in the second both have z = −1.5.

From z to a percentile

For a normal distribution, the z-score fixes the area to the left, which is the percentile. z = 0 is the 50th percentile. z = 1 leaves about 84.13% below. z = −1.5 leaves about 6.68% below, so the SAT score of 325 was higher than only about 6.7% of scores, if the scores were normal. The 2012 SAT had 1,664,479 test takers, so about 15.87% of them, roughly 264,000 students, would have scored above 496 + 114 = 610. A Chilean male of 168 cm in 2009 to 2010 has z = (168 − 170) ÷ 6.28 ≈ −0.32, between the 37th and 38th percentile. The lab prints these areas as you move x.

z-scores for data that are not normal

A z-score can be computed for any value of any data set: it still says how many standard deviations the value sits from the mean. What changes is the link to percentiles. For the gaming hours, the classmate who played 0 hours has z ≈ −2.24, but with only 25 values and a left-skewed shape, the normal table's 1.2% below that point means little; in fact 1 of the 25 values, 4%, is at or below it. Use z to compare positions within a group freely, and use normal-curve percentiles only when the shape is close to a bell.