Download SOC 8311 Basic Social Statistics

Chapter 2
Describing Variables
2.7 Standardized Scores (Z Scores)
Mean and Standard Deviation
Two basic descriptive statistics for the distribution of a
continuous variable with N observations:
N
Mean (central tendency):
Yi
Y
i 1 N
Standard deviation (dispersion):
s Y  s 2Y
(Yi  Y) 2
 

N 1
i 1
N
2
d
 i
N 1
EDUC Years of Schooling
600
500
Y  13.43 s Y  3.08
Frequency
400
300
200
100
0
0
3
6
9
12
15
18
HIGHEST YEAR OF SCHOOL COMPLETED
Standardized Scores (Z Scores)
Transform each score in the frequency distribution of a
continuous Y-variable into a standard or Z score:
(1) from Yi subtract the mean (= deviation di)
(2) divide case i deviation by the standard deviation
Yi  Y
Zi 
sY
di

sY
Z scores can be positive or negative, indicating the
number of standard deviation units that Yi lies above or
below the distribution mean
Yi scores into Zi scores
Yi
Zi
0
5
-2.0
Yi  Y
Zi 
sY
10
15
20
25
-1.0
Y
30
0
= 30
35
40
45
+.67
sY
= 15
Change some EDUC Y scores into Z scores:
Y  13.43
s Y  3.08
1. What is the Z score for a person with16 years?
16  13.43  2.57 

Yi  Y
3.08 ___________
3.08 __________
Zi 
 __________
sY
2. Find the Z score for a person with 8 years:
Yi  Y
Zi 
 _______________________________
sY
Now change some EDUC Z scores into Y scores:
What years of EDUC separate two persons at the upper
& lower limits of the range Z = ± 2.2 standard deviations?
• Find upper limit of range, Z = +2.20 std. dev. units :
Yi  Y
Zi 

sY
Yi   Zi s Y  Y
Yi  (2.20)(3.08)  13.43  _____________
• Find lower limit of range, Z = -2.20 std. dev. units :
Yi  Y
Zi 

sY
Yi   Zi s Y  Y
Yi  __________________________________
Find the Z scores for these ungrouped data:
First calculate mean & standard deviation, then the Z scores
Yi
 Y  di
(d i )2
Y1: 6 - _____________________
Y2: 4 -
_____________________
Y3: 5 -
_____________________
Y4: 4 -
_____________________
Y5: 3 -
_____________________
Y6: 8 -
_____________________
ZY=3 =
6
2
(d
)
 i  _______
i 1
s 2Y 
N 1
 ______________
s Y  s 2Y  ________
Yi  Y
 ____________________
sY
ZY=4 = ________
2
(d
)
 i
ZY=6 = ________
ZY=8 = ________
Find Z scores for grouped data on Pres. Obama’s handling
of the global war on terror, where Mean = 3.30 and N = 158
Rating
Yi
fi
(di)2(fi)
Poor
1
10
___________________________
Fair
2
20
___________________________
Good
3
40
___________________________
Excellent
4
88
___________________________
K
Z1  (Y1  Y) /s Y  _________________

i 1
(d i ) 2 (f i )  _______
K
Z2  (Y2  Y) /s Y  _________________
Z3  (Y3  Y) /s Y  _________________
Z4  (Y4  Y) /s Y  _________________
s 
2
Y

i 1
(d i ) 2 (f i )
N 1
sY  _______
 _______
Calculate the mean for ungrouped data
i
Yi
1
2
2
3
3
3
4
4
5
5
6
5
7
6
8
7
9
7
10
8
N
Yi
Y
i 1 N
N  _____
N
Y
i 1
i
 ________________
Y  __________
Calculate variance & std. dev. for 10 scores
Yi
 Y  di
(d i )2
2
-
= _______
_______
3
-
= _______
_______
3
-
= _______
_______
4
-
= _______
_______
5
-
= _______
_______
5
-
= _______
_______
6
-
= _______
_______
7
-
= _______
_______
7
-
= _______
_______
8
-
= _______
_______
10
2
(d
)
 i  ____________
i 1
s 2Y   (d i ) 2 / (N  1) 
______________________
s Y  s 2Y  ____________
Find All the Z Scores
Yi  Y
Zi 
sY
(Yi
 Y) / s Y

(2
-
) / _______ = _______
(3
-
) / _______ = _______
(3
-
) / _______ = _______
(4
-
) / _______ = _______
(5
-
) / _______ = _______
(5
-
) / _______ = _______
(6
-
) / _______ = _______
(7
-
) / _______ = _______
(7
-
) / _______ = _______
(8
-
) / _______ = _______