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 of the z-score (+ or - ) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and µ.

X€

σ

€

μz

-2 -1 0 +1 +2

Example 5.2

• A distribution of exam scores has a mean (µ) of 50 and a standard deviation (σ) of 8.

€

z = x−μσ

σ

σ

σ = 4

60 64 68µ

X66

Example 5.5

• A distribution has a mean of µ = 40 and a standard deviation of σ = 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

X

€

σ

€

μz

-2 -1 0 +1 +2

100 110 1209080

€

μ

A small population

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

x x - µ (x - µ)2

0 0 - 3 = -3 9

6 6 - 3 = +3 9

5 5 - 3 = +2 4

2 2 - 3 = -1 1

3 3 - 3 = 0 0

2 2 - 3 = -1 1

€

(x −μ )2 = SS∑

€

=24

€

μ = x∑N

€

=186

€

=3€

σ = SSN

€

= (x −μ )2∑6

€

= 246

€

= 4

€

=2

2

1

0 1 2 3 4 5 6µ

X

freq

uenc

y

(a)

2

1

-1.5 -1.0 -0.5 0 +0.5 +1.0 +1.5µ

z

freq

uenc

y

(b)

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

€

z =x −μσ

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 =SSzN

€

=(x −μ z )

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 0

-0.5 -0.5 - 0 = -0.5 0.25

€

σ = SSN

€

= 66

€

= 1 =1

€

6.00 = (z−μ z )2∑

10

50µ

X = 60

X

Psychology

4

48µ

X = 56

X

Biology

52

Converting Distributions ofScores

14

57µ

JoeX = 64

z = +0.50

X

Original Distribution

10

50µ

X

Standardized Distribution

MariaX = 43

z = -1.00

MariaX = 40

z = -1.00

JoeX = 55

z = +0.50

Correlating Two Distributionsof Scores with z-scores

= 4

µ = 68

Person AHeight = 72 inches

Distribution of adult heights (in inches)

= 16

µ = 140

Person BWeight = 156 pounds

Distribution of adult weights (in pounds)