Descriptive Statistics

Chapter 2

§ 2.5

Measures of Position

Larson & Farber, Elementary Statistics: Picturing the World, 3e 3

Standard Scores

The standard score or z-score, represents the number of standard deviations that a data value, x, falls from the mean, μ.

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.

value mean standard deviation

xz

Continued.

Larson & Farber, Elementary Statistics: Picturing the World, 3e 4

Standard Scores

xz

Example continued:

a.) μ = 78, σ = 7, x = 85

85 787 1.0 This score is 1 standard

deviation higher than the mean.

xz

b.) μ = 78, σ = 7, x = 70

70 787 1.14 This score is 1.14 standard

deviations lower than the mean.

xz

c.) μ = 78, σ = 7, x = 78

78 787 0 This score is the same as the

mean.

Larson & Farber, Elementary Statistics: Picturing the World, 3e 5

Relative Z-ScoresExample: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-scorexz

75 73.2

4.5

0.4

xz 68.6 65

3.9

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 6

Exercises

Pg 100-104 # 10, 21, 25-31, 35-38

Larson & Farber, Elementary Statistics: Picturing the World, 3e 7

Exercises # 10, 21, 25-31, 35-38

10 false, negative z-scores simply indicate that the data value is below the mean21 2.8, 3.2, 3.65, 3.9, 4.625A z=-1.43, B z=0, C z=2.1426 A z=-1.54, B z=0.77 C z=1.54 Stats Bio Better27 1.43 .77 Stats 28 -.43 -.77 Stats29 2.14 1.54 Stats30 0 0 the same

Larson & Farber, Elementary Statistics: Picturing the World, 3e 8

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 unusual36 0.28, -1.10, -0.07 none are particularly unusual37 .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 9

Handout solutions

1a 30 b 2 c 2 d the same2a 2.5 b 1 c -1 d opposite3a 5.71 b -1.79 c 0.264a 2.90 b -2 c 05 2.56 unusual 6 2.67 unusual7 4.52 very unusual 8 1.99 not unusual9 -.42 -.38 -.38 psychology is better10 A .47 b .22 c .6 best37 Units of measure do not matter for z scores38 60.475 inches

Larson & Farber, Elementary Statistics: Picturing the World, 3e 10

Handout solutions (continued)

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Handout solutions (continued)

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Handout solutions (continued)