Download z-Scores - mrseehafer

Chapter 3
Lesson
Vocabulary
z-Scores
3-9
z-score
raw data
standardized data
BIG IDEA
z-scores enable a score to be compared to other
scores in the same data set.
What is a z-Score?
Sometimes a person wants to know how his or her score or salary
compares to a group as a whole. One way to do this is to compute a
percentile. Another way is to analyze how many standard deviations the
score or salary is above or below the mean.
The U.S. Bureau of Labor Statistics tracks the median hourly wages for
22 general occupational categories. The 2007 median wages for certain
occupations are listed in the table below. The mean of the 7 median
_
hourly wages is x = $22.90 and the standard deviation is s = $10.70.
Food
Production
Protective Construction Education
Services
$8.24
$16.11
$17.57
$20.47
Healthcare Architecture/
Engineering Management
$26.17
$31.14
$40.60
In general,
the transformation that maps each wage x to the score
_
x-x
x - 22.90
_____
________
s = 10.70 tells you how many standard deviations that wage
is above or below the mean. For example, the median wage for
40.60 - 22.90
Management is ___________
≈ 1.7 standard deviations above the mean,
10.70
8.24 - 22.90
while the median wage for Food Production is __________
≈ –1.4
10.70
standard deviations above, or 1.4 standard deviations below, the mean.
Mental Math
Suppose the mean of a
set of scores is 486 and
the standard deviation is
37. What number is:
a. 1 standard deviation
above the mean?
(remember – this is mental
math!)
b. 1 standard deviation
below the mean?
c. 2 standard deviations
above the mean?
d. 2 standard deviations
below the mean?
In the table below, row L1 is the original set of wages, L2 is the image of
_
L1 under the translation T where T(x) = x - x = x - 22.90, and L3 is the
x
image of L2 under the scale change S where S(x) = _xs = _____
.
10.70
L1
8.24
16.11
17.57
20.47
26.17
31.14
40.60
L2
–14.66
–6.79
–5.31
–2.43
3.27
8.24
17.70
L3
–1.4
–0.6
–0.5
–0.2
0.3
0.8
1.7
The transformation that maps the original data set L1 onto L3 is the
x - 22.90
composite S T, where S T(x) = S(T(x)) = S(x - 22.90) = ________
.
10.70
S T(x) is called the z-score for the value x. The z-score for Protective
Services is –0.6; this means that $16.11 is 0.6 standard deviations below
the mean. In the same way, the 0.8 z-score for Architecture/Engineering
means that $31.14 is 0.8 standard deviations above the mean.
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Lesson 3-9
The preceding discussion can be generalized in the following definition.
Definition of z-Score
_
Suppose a data set has mean x and standard deviation s. The
z-score for a member x of this data set is
_
deviation
x-x
_____
=
z = _______________
s .
standard deviation
A positive z-score tells how many standard deviations the score is above
the mean. A negative z-score tells how far below the mean the score is.
QY
QY
Find the z-score for
$20.47 in the table and
describe what it means.
Example 1
Nancy scored 87 on a math quiz on which the mean score was 70 and the
standard deviation of the scores was 8. Find her z-score and tell how far her
score was from the mean.
Solution Her z-score is
87 - 70
z = _______
≈ 2.1,
8
so her score was 2.1 standard deviations above the mean.
Sometimes the original data are called raw data or raw scores, and
the results of the transformation are called standardized data or
standardized scores. In Example 1, a raw score of 87 corresponds to a
standardized score of 2.1.
Properties of z-Scores
What happens to the mean and standard deviation of a data set if each
score is converted to a z-score? Refer again to the median wages shown
in row L1 of the table on page 206. Because adding (or subtracting)
the number h to every number in a data set adds (or subtracts) h to
the mean, the mean of the data set in L2 is 22.90 - 22.90 = 0. Under a
translation of a data set, the standard deviation is invariant. Thus, the
standard deviation of the data set in L2 is still 10.70. Because the scale
change S with S(x) = ax multiplies both the mean and standard deviation
1
by a, the mean of the data set in L3 is _____
· 0 = 0 and the standard
10.70
1
_____
deviation is 10.70 · 10.70 = 1. Thus, the z-scores in L3 have mean 0 and
standard deviation 1. In general, we have the following theorem.
Theorem (Mean and Standard Deviation of z-Scores)
_
If a data set has mean x and standard deviation s, the mean of
its z-scores will be 0, and the standard deviation of its z-scores
will be 1.
z-Scores
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Chapter 3
Using z-Scores to Make Comparisons
Standardized scores, or z-scores, make it easier to compare different
sets of numbers.
GUIDED
Example 2
Mark scored 78 on a history test on which the mean was 71 and the standard
deviation was 10. He scored 68 on a chemistry test on which the mean was
62 and the standard deviation was 6. Use z-scores to determine on which test
he performed better compared to his classmates.
Solution His z-scores are:
history:
chemistry:
? - 71
______
=
10
68 - ?
______
=
?
?
? .
?
Because his z-score on the
test is higher, Mark
performed better on that test compared to his classmates.
In Example 2 you should notice that Mark scored above the mean on
both tests, but the z-scores provide more information. The z-scores are
sensitive to the fact that the scores on the history test are more spread
out than those on the chemistry test.
Questions
COVERING THE IDEAS
In 1–3, refer to the data sets in rows L2 and L3 on page 206.
1. By computing directly, find the mean and standard deviation of
each data set.
2. What does the z-score of –0.2 mean?
3. Which median wage is 0.5 standard deviations below the mean?
4. Find the z-score for a test score of 84 for each situation.
a. mean = 89; standard deviation = 6.1
b. mean = 67; standard deviation = 8.8
5. In which situation, 4a or 4b, is the test score better compared to
others who took the test?
6. What is a standardized score?
7. A data set has a mean of 19 and a standard deviation of 4.3. How
can the data set be transformed so that the mean is 0 and the
standard deviation is 1?
8. Refer to Example 2. Mary scored 54 on the history test and 47 on
the chemistry test. Use z-scores to determine on which test she did
better compared to her classmates.
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Lesson 3-9
9. In 2007, Colorado high school students’ average ACT scores were:
English, 19.7; Math, 20.1; Reading, 20.8; and Science, 20.4. Jennie’s
scores were: English, 24; Math, 21; Reading, 18; and Science, 20.
Find Jennie’s z-score for each section assuming that each section
had a standard deviation of 1.8.
APPLYING THE MATHEMATICS
In 10 and 11, the mean boys’ time for a one-mile race was 6 minutes 6
seconds with a standard deviation of 16 seconds. The mean girls’ time
was 8 minutes 15 seconds with a standard deviation of 24 seconds.
10. Who is faster relative to others of his or her gender, a boy who runs
a mile in 5 minutes 34 seconds, or a girl who runs a mile in
7 minutes 35 seconds?
11. Suppose a girl runs the race in 8 minutes 30 seconds. What boy’s
time would have the same z-score?
12. A student got a z-score of 1.33 on a test with a mean of 73 and a
standard deviation of 9. What was the student’s raw score?
13. The graph at the right shows how much
10
Frequency
customers spend at a grocery store. The costs
have mean $125 and standard deviation $65.
Copy the horizontal axis as shown.
a. Change each value labeled on the axis to a
z-score. Values for 100 and 250 are shown.
b. Locate the points on the axis that correspond
to z-scores of 0, 1, and –1.
8
6
4
2
14. A teacher tells a class that the mean raw score on
a test was 58. Alex has a raw score of 73 and a
z-score of 1.25. What was the standard deviation
on the test?
15. Considering the data sets in L1 and L3 at the _start
0
50 100 150 200 250 300 350
Grocery Costs
-0.4
0
1.9
z-score
50 100 150 200 250 300 350 cost
of the lesson, use the translation T(x) = x - x
and the scale change S(x) = _xs .
a. Find (T S)(x).
b. Apply T S to the data set in L1. What are the mean and
standard deviation of this new data set?
c. Are these the same mean and standard deviation as for the data
set in L3? Explain why or why not.
In 16–19, use the table at the right with the scores from
two tests.
16. How many standard deviations above the mean is
Fiona’s score on the physics test?
17. On which test did Raj do better compared to others who
took the test?
Physics
Mathematics
Fiona
84
64
Raj
70
76
mean: 60
standard
deviation: 6.9
mean: 70
standard
deviation: 10.7
Test
Statistics
z-Scores
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Chapter 3
18. Simon had a 58 on the physics test. He scored equally well (in
terms of z-score) on the mathematics test. What was his raw score
on the mathematics test?
19. Andrea had the same raw score on each test, and she also had the
same z-score on each test. What raw score did she get?
REVIEW
20. a. Write an equation for the inverse of g where g(t) = √
t + 4.
b. Is the inverse of g a function? (Lesson 3-8)
21. True or False Let f(x) = x3 and g(x) = x –3. (Lessons 3-8, 3-7)
a. For all x, f(g(x)) = g(f(x)).
b. f and g are inverses.
In 22 and 23, tell whether the statement is true or false. If it is
false, give a counterexample. (Lessons 3-8, 3-4)
22. The inverse of an even function is a function.
23. The inverse of an odd function is a function.
24. Consider a 15% discount function D where D(x) = 0.85x and
a 7% total-with-tax function T where T(x) = 1.07x.
(Lesson 3-7)
a. If you buy an item with a list price of x dollars, what will it cost
you after this discount and tax?
b. Which is better to take first, the discount or the tax? Explain.
25. Multiple Choice A transformed set of data has a variance three
times that of the original set. How were the data transformed?
(Lesson 3-6)
A translated by 3
B multiplied by √
3
C multiplied by 3
D multiplied by 9
26. Multiple Choice The graph of which relation has point
symmetry? (Lesson 3-4)
A y = ⎪x⎥
C y = x3
B y = x2
1
D y = __
2
x
EXPLORATION
27. In each of the 22 general occupation categories tracked by the
U.S. Bureau of Labor Statistics (BLS), there are a number of more
specific occupation descriptions. Visit the BLS website and identify
a specific occupation description that interests you.
a. Calculate a mean and standard deviation for the median wages
of the specific occupations within the general category.
b. Calculate the z-scores for the highest and lowest median wages
in the general occupation category. What do these z-scores tell
you about the range and distribution of wages within this group?
c. Calculate the z-score for the occupation you have chosen.
Describe how the wage for your occupation compares to the
other wages within the category.
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Transformations of Graphs and Data
QY ANSWER
The z-score is -0.2, so the
median wage for Education is
0.2 standard deviations below
the average median wage for
the seven occupations.