Chapter 5: z-Scores
(a)


  82
(b)
  12
X = 76 is
slightly below
average
12
82
x = 76


  70
(c)
  12
X = 76 is
slightly above
average


  70
 3
X = 76 is far
above
average
12
70
x = 76
3
70
x = 76
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

z
-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

=4


60
µ
64
68
X
66
Example 5.5
• A distribution has a mean of µ = 40 and a
standard deviation of  = 6.
To get the raw score from the
z-score:
x    z
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
80
-2
90
100

-1

0


110
120

z
+1
+2
A small population
0, 6, 5, 2, 3, 2
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
 
N


 ( x   )  SS
 24

2

N=6
N
18

6

 3
 (x  )
6




24
6



SS
2
4
2
frequency
(a)
2

1
0
1
2
3
µ
4
5
6
X
frequency
(b)
2

1
-1.5
-1.0
-0.5
0
µ
+0.5 +1.0 +1.5
z
Let’s transform every raw score
x
z
into a z-score using:

x=0
z
x=6
z
x=5
x=2
x=3
x=2
03
= -1.5
2
63
2

z
53

2

z
23
2

z
3 3
2

z
23
2

= +1.5
= +1.0
= -0.5
=0
= -0.5
Mean of z-score
z
distribution :
z


Standard deviation:


z
 
SS
N

6
0


N
(1.5 )  (1.5 )  (1.0 )  (0.5 )  (0 )  (0.5 )
z 
SS z
N

 (x  z )
2
N
z - µz
(z - µz)2
-1.5 - 0 = -1.5
2.25
+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
-1.5
+1.5
6
6


11
6.00   ( z   z )
2
Psychology
Biology
10
4
X
50
µ
X
48
µ
X = 60
52
X = 56
Converting Distributions of
Scores
Original Distribution
Standardized Distribution
14
10
X
X
57
µ
Maria
X = 43
z = -1.00
Joe
X = 64
z = +0.50
50
µ
Maria
X = 40
z = -1.00
Joe
X = 55
z = +0.50
Correlating Two Distributions
of Scores with z-scores
Distribution of
adult heights
(in inches)
=4
µ = 68
Person A
Height = 72 inches
Distribution of
adult weights
(in pounds)
 = 16
µ = 140
Person B
Weight = 156 pounds