1. The average IQ of a group of people is 105 with a
standard deviation of 15. What is the standardized (or z)
- a) someone with an IQ of 93?
b) someone with an IQ of 135?
(93-105)/15 = -12/15 = -4/5 =
-.8 (Sign matters.)
(135 - 105)/15 = 20/15 = 4/3 = 1.333
2. Suppose a tree farm finds that the mean height of
three-year-old trees is 47 inches with a standard deviation
of 3 inches.
- a) Compute the z-score of a tree 51 inches tall.
b) Compute the z-score of a tree 49 inches tall.
c) A tree has a z-score of -3. How tall is it?
(51-47)/3 = 4/3 = 1.33
(49 - 47)/3 = 2/3 = .67
(x - 47)/3 = -3; x - 47 = -9; x = -9 + 47 =
3. On the standard IQ test, the mean is 100 and the
standard deviation is 15.
- a) Compute the z-score of a person with an IQ of
b) Compute the z-score of person with an IQ of 20.
c) A person has a z-score of +3. What is her IQ?
4. A group of people is tested to see how much weight
they can lift. The mean lift is 120 pounds with a standard
deviation of 20 pounds. Compute the z-score for someone who
- a) 125 pounds.
b) 80 pounds.
c) If someone has a z-score of +3, how much can that
5. In the not too distant past, a professor at some
prestigious school (it might have been Duke) suggested that
in an effort to combat grade inflation, the school should
issue standardized grades (or z-scores) in addition to the
Let us see how this might work. Suppose Sue was in three
classes, each with 20 students. In Introductory Alchemy, the
professor awarded an A to 15 students and an A- to 5. The
average grade point in this class was 3.90 and the standard
deviation was .15. In her Intermediate Astrology class, the
professor was a little tougher, giving five As, five Bs,
five A-s, and five B+s. The GPA of the class was 3.50 with a
standard deviation of .38. In her class of Ancient
Phoenician literature, the prof gave mostly Bs, with a
smattering of other grades, for a GPA of 2.72 and a standard
deviation of .7
Sue got an A- (3.67) in Alchemy, a B+ (3.33) in
Astrology, and a B (3.00) in Phoenician lit. Compute her
standardized grade in each class. If we judged by
standardized grades, where did she do best? Where did she do
worst? Explain carefully.
Alchemy: (3.67 - 3.9)/.15 =
-.23/15 = -1.53
Astrology: (3.33 - 3.50)/.38 = -.17/.38 = -.45
Phoenician: (3.00 - 2.72)/.7 = .28/.7 =
In terms of standardized
scores, she did best in Ancient Phoenician Literature where
she was above average in the class. Even though her best
grade was in Alchemy, she did the worst there because in
that class an A- was a poor grade.
6. A test in economics has an average grade of 12 with a
standard deviation of 3. The high score on the quiz was 20.
How many standard deviations above the mean was this
z=(20-12)/3 = 8/3 =
7. A group of students has an average height of 6 feet
with a standard deviation of .25 feet.
- a) Suppose we had measured in inches instead. Can we
tell what their mean and standard deviation would be? If
we can tell, what would they be?
- b) A student is 6.4 feet tall. How many standard
deviations away from the average is he?
- c) How tall is a student who is 2 standard deviations
below the average?
a) Multiply by 12: 72inches
and 3 inches.
b) z = (6.4 - 6.0)/.25 = .4/.25 = 1.6
c) -2 = (x - 6.0)/.25; -.5 = x - 6.0; x = 5.5 or five feet
and six inches.