SOC 8311 Basic Social Statistics
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Transcript 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 sY 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
sY 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 sY 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 sY Y
Yi __________
__________
__________
____
Find the Z scores for these ungrouped data:
First calculate mean & standard deviation, then the Z scores
Yi
Y di
(di )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 __________
_______
(d
i 1
i
) 2 (fi ) _______
K
Z2 (Y2 Y) /sY __________
_______
Z3 (Y3 Y) /sY __________
_______
Z4 (Y4 Y) /sY __________
_______
s
2
Y
i 1
(d i ) 2 (fi )
N 1
s Y _______
_______
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
(di )2
2
-
= _______
_______
3
-
= _______
_______
3
-
= _______
_______
4
-
= _______
_______
5
-
= _______
_______
5
-
= _______
_______
6
-
= _______
_______
7
-
= _______
_______
7
-
= _______
_______
8
-
= _______
_______
10
2
(d
)
i ____________
i 1
s 2Y (di ) 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
-
) / _______ = _______