Download Chapter 2: Descriptive Statistics

Chapter 2
Descriptive Statistics
§ 2.5
Measures of
Position
Standard Scores
The standard score or z-score, represents the number of
standard deviations that a data value, x, falls from the
mean, μ.
x

value  mean
z

standard deviation

Example:
The test scores for all statistics finals at Union College
have a mean of 78 and standard deviation of 7. Find the
z-score for
a.) a test score of 85,
b.) a test score of 70,
c.) a test score of 78.
Continued.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Standard Scores
Example continued:
a.) μ = 78, σ = 7, x = 85
x
  85  78
z
 1.0
  7
This score is 1 standard deviation
higher than the mean.
b.) μ = 78, σ = 7, x = 70
x
  70  78
z
  7  1.14
This score is 1.14 standard
deviations lower than the mean.
c.) μ = 78, σ = 7, x = 78
x
  78  78
z
0
  7
This score is the same as the mean.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Relative Z-Scores
Example:
John received a 75 on a test whose class mean was 73.2
with a standard deviation of 4.5. Samantha received a 68.6
on a test whose class mean was 65 with a standard
deviation of 3.9. Which student had the better test score?
John’s z-score
Samantha’s z-score
z  x    75  73.2

4.5
z  x    68.6  65

3.9
 0.4
 0.92
John’s score was 0.4 standard deviations higher than
the mean, while Samantha’s score was 0.92 standard
deviations higher than the mean. Samantha’s test
score was better than John’s.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Exercises
Pg 100-104 # 10, 21, 25-31, 35-38
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Exercises # 10, 21, 25-31, 35-38
10 false, negative z-scores simply indicate that the
data value is below the mean
21 2.8, 3.2, 3.65, 3.9, 4.6
25A z=-1.43, B z=0, C z=2.14
26 A z=-1.54, B z=0.77 C z=1.54
Stats
Bio
Better
27
1.43
.77
Stats
28
-.43
-.77
Stats
29
2.14
1.54
Stats
30
0
0
the same
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Exercises # 10, 21, 25-31, 35-38
31 a -0.44, 0.89, -1.78 none seem particularly
unusual
b 2.5th (no such thing) 84th 50th
35 1.66, -2.48, 3.72 the heights 62 and 80 inches
are a bit unusual
36 0.28, -1.10, -0.07 none are particularly
unusual
37 .66 about the 70th (from the graph)
38 -1 16th (looks closer to 11th on the graph)
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Handout solutions
1a 30
b 2
c 2
d the same
2a 2.5
b 1
c -1
d opposite
3a 5.71
b -1.79
c 0.26
4a 2.90
b -2
c 0
5 2.56 unusual
6 2.67 unusual
7 4.52 very unusual
8 1.99 not unusual
9 -.42 -.38 -.38 psychology is better
10 A .47
b .22
c .6 best
37 Units of measure do not matter for z scores
38 60.475 inches
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Handout solutions (continued)
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Handout solutions (continued)
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Handout solutions (continued)
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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