STA 200 Summer I 2011
CHAPTER 13, PART 2
Z-scores/Standard Scores
 The z-score (or standard score) of an
observation is how far above or below the
mean an observation is located (in standard
deviations).
 𝑧 − 𝑠𝑐𝑜𝑟𝑒 =
𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛−𝑚𝑒𝑎𝑛
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Z-scores (cont.)
 A z-score of 1 implies that an observation is 1
standard deviation above the mean.
 A z-score of -2 implies that an observation is 2
standard deviations below the mean.
 A z-score of 1.47 implies that an observation
is 1.47 standard deviations above the mean.
Example
 The verbal section of the SAT is approximately
normally distributed with mean 500 and standard
deviation 100.
 The verbal section of the ACT is approximately
normally distributed with mean 18 and standard
deviation 6.
 Which is better: a score of 650 on the verbal
section of the SAT, or a score of 28 on the verbal
section of the ACT?
Percentiles
 The cth percentile of a distribution is a value
such that c% of the observations lie below it.
 For example…
 the median is the 50th percentile
 the first quartile is the 25th percentile
 the third quartile is the 75th percentile
Using a Normal Table
 A z-score of -1.80 corresponds to a percentile
of:____
 A z-score of 0.47 corresponds to a percentile
of:____
More IQ Score Examples
 Returning to the IQ score scenario (mean 100,
standard deviation 15), we can now answer
additional questions:
 What proportion of IQ scores are less than 90?
 What proportion of IQ scores are less than 124?
 What proportion of IQ scores are greater than 112?
 What proportion of IQ scores are greater than 97?
 What proportion of IQ scores are between 80 and 118?
More Examples (Percentiles)
 Still with mean 100 and standard deviation 15:
 How high must a person score to be in the top 25%
of all scores?
 How high must a person score to be in the top 10%
of all scores?
More Examples, Different Data
 For women, height is approximately normally
distributed with mean 63.6” and standard
deviation 2.5”.
 What proportion of women are taller than 60”?
 What proportion of women are between 61” and
66” tall?
 What is the 80th percentile of women’s height?
 What is the 90th percentile of women’s height?