Chapter 5: z-Scores (a) (b) (c) = 82 = 70 = 70 = 12 = 12 = 3 - - PDF document

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Chapter 5: z-Scores

12 82 x = 76

(a)

µ = 82 σ =12 X = 76 is slightly below average 12 70 x = 76

(b)

µ = 70 σ =12 X = 76 is slightly above average 3 70 x = 76

(c)

µ = 70 σ = 3 X = 76 is far above average

Definition of z-score

• A z-score specifies the precise location of

each x-value within a distribution. The sign

• f the z-score (+ or - ) signifies whether the

score is above the mean (positive) or below the mean (negative). The numerical value

• f the z-score specifies the distance from

the mean by counting the number of standard deviations between X and µ.

2

X

σ µ

z

• 2
• 1

+1 +2 X

σ

100

z

• 2
• 1

+1 +2

116 132 84 68

Intelligence Quota - IQ Scores

µ = 100

σ = 16

X

σ

40

z

• 2
• 1

+1 +2

47 54 33 26 µ = 40 σ = 7

3

X

σ

500

z

• 2
• 1

+1 +2

600 700 400 300

µ = 500 σ = 100

SAT Scores X

σ

70

z

• 2
• 1

+1 +2

72.7 75.4 67.3 64.6

Average male height (in inches) for American Males (ages 20-29) – taken from Wikipedia 2014

µ = 70 σ = 2.7

X

σ

64.5

z

• 2
• 1

+1 +2

66.9 69.3 62.1 59.7

Average female height (in inches) for American Females (ages 20-29) – taken from Wikipedia 2014

µ = 64.5 σ = 2.4

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Example 5.2

• A distribution of exam scores has a mean

(µ) of 50 and a standard deviation (σ) of 8.

z =

x − µ σ

s s s = 4 60 64 68 µ X 66

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Example 5.5

• A distribution has a mean of µ = 40 and a

standard deviation of s = 6.

To get the raw score from the z-score:

x = µ + zσ

If we transform every score in a distribution by assigning a z-score, new distribution:

• 1. Same shape as original distribution
• 2. Mean for the new distribution will be zero
• 3. The standard deviation will be equal to 1

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X

σ µ

z

• 2
• 1

+1 +2 100 110 120 90 80

µ

A small population

0, 6, 5, 2, 3, 2 N=6

x x - µ (x - µ)2 0 - 3 = -3 9 6 6 - 3 = +3 9 5 5 - 3 = +2 4 2 2 - 3 = -1 1 3 3 - 3 = 0 2 2 - 3 = -1 1 (x −µ)2 = SS ∑ = 24 µ = x ∑ N = 18 6 = 3 σ = SS N = (x − µ)2 ∑ 6 = 24 6 = 4 = 2 2 1 1 2 3 4 5 6 µ s X

frequency (a)

2 1

• 1.5
• 1.0
• 0.5

+0.5 +1.0 +1.5 µ s z

frequency (b)

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Let’s transform every raw score into a z-score using:

z = x − µ σ

x = 0 x = 6 x = 5 x = 2 x = 3 x = 2 z = 0 − 3 2 z = 6 − 3 2 z = 5 − 3 2 z = 2 − 3 2 z = 3− 3 2 = -1.5 = +1.5 = +1.0 = -0.5 = 0 = -0.5 z = 2− 3 2

Mean of z-score distribution :

µz = z ∑ N = (−1.5)+(1.5)+(1.0)+(−0.5)+(0)+(−0.5) 6 = 0

Standard deviation: σ z =

SSz N = (x − µz)2 ∑ N

z z - µz (z - µz)2

• 1.5
• 1.5 - 0 = -1.5

2.25 +1.5 +1.5 - 0 = +1.5 2.25 +1.0 +1.0 - 0 = +1.0 1.00

• 0.5
• 0.5 - 0 = -0.5

0.25 0 - 0 = 0

• 0.5
• 0.5 - 0 = -0.5

0.25 σ = SS N = 6 6 = 1 =1 6.00 = (z − µz)2 ∑

What do we use z-scores for?

• Can compare performance on two different

scales (e.g. compare your score on the ACT to your score on SAT) by converting scores to z scores and comparing z scores

• Can convert a distribution of scores with a

specific mean and standard deviation to completely new distribution with a new mean and standard deviation

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Comparing test scores on two different scales Meghan’s Semester Test Results

• Psychology Exam: score of 60
• Biology Exam:

score of 56

• Which was her better score, relative to the
• thers in each class?

To answer the question convert her scores to standard scores and compare—explain your answer fully in terms of z scores and standard deviations

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10 50 µ X = 60 X

Psychology

4 48 µ X = 56 X

Biology

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Converting Distributions of Scores

14 57 µ Joe X = 64 z = +0.50 X

Original Distribution

10 50 µ X

Standardized Distribution

Maria X = 43 z = -1.00 Maria X = 40 z = -1.00 Joe X = 55 z = +0.50

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Correlating Two Distributions

• f Scores with z-scores

s = 4 µ = 68 Person A Height = 72 inches

Distribution of adult heights (in inches)

s = 16 µ = 140 Person B Weight = 156 pounds

Distribution of adult weights (in pounds)