PS 366
5
Variation: Chapter 5
• How are observations distributed around the
central point?
• Is there one, more central point?
– unimodal
– bimodal
Variation
• Which is unimodal, which is bimodal:
– Mass public ideology
• V con, con, moderate, lib, v. lib
– Members of Congress ideology
– What does the mean mean?
Distribution
• How spread out are the observations?
• Single peak
– not much variation
• Flat?
– lots of variation; what does mean mean?
Variation
• Variance & Standard deviation
• Information about variation around the mean
Variation
125
92
72
126
120
99
130
100
mean = 108
Variance
= sum of squared
distances of each obsv
from mean, over # of
observations
Variance
• Mean
125
92
72
126
120
99
130
100
mean = 108
(x - mean)
125-108
92-108
72-108
126-108
120-108
99-108
130-108
100-108
Variance
• Mean
125
92
72
126
120
99
130
100
mean = 108
(x - mean) (x - mean)2
17
-16
-36
18
12
-9
22
-8
289
256
1296
324
144
81
484
64
sum sqs=2938
• THEN, 2938/8
Variance & Std. deviation
• Variance does not tell
us much
• Standard deviation =
square root of variance
• mean = 108
• variance = 2938 / (n-1)
• variance = 2938 / 7
• sd = sqrt 419.7
• = 20.4
• = 419.7
Variation
125
92
72
126
120
99
130
100
mean = 108
• Variance = 419
• SD =
20.4
Variation
• Range ( lo – hi)
• Variance (sum of
distances from mean,
squared) / n
• Standard Deviation
– Bigger # for each = more
variation
Variation
Standard Deviation
• expresses variation around the mean in
‘standardized’ units
– gives picture of distribution
• Bigger # = more variation
• Allow us to compare apples to oranges
Standard Deviation: Examples
• Total convictions for corruption in state
– mean = 178, s.d. = 199.7
• Per capita convictions (per 10,000 officials)
– mean = .357, s.d. = .197
Standard Deviation
Low s.d relative to mean
High s.d. relative to mean;
and/or lots of skew
Standard Deviation
Distribution of total convictions: mean 187; s.d. 199
Standard Deviation
Mean .357, s.d. .197
Standard Deviation
Turnout by state: mean = .62 ; s.d. = .07
Standard Deviation
• Tells even more if distribution ‘normal’
• If data interval
• What about a state that has 75% turnout, and
.5 corruption convictions per 10,000?
• Where are they in each distribution?
Standard Deviation
X
Mean .357, s.d. .197
Standard Deviation
X
Turnout by state: mean = .62 ; s.d. = .07
Time speaking, in seconds ranked
Frequency Distribution: Mean = 729
Trumped
• Trump
• Bush
• Mean
+2.17 sd beyond mean
+1.19 sd
0.0 sd
• Huckabee -0.92 sd
• Walker
-1.20 sd below mean
Standard Deviation & z-scores
• State w/ 75% voter turnout = z
z= (score – mean) / s .d.
= (.75 - .61) / .07 =
= .14 / .07 = +2.00
2.00 standard deviations above mean on turnout
Standard Deviation & z-scores
• State w/ .5 corruption convictions = z
z= (score – mean) / s .d.
– = (.50 - .35) / .19 =
– = +.15 / .19 = + 0.78
0.78 standard deviations above mean on corruption
Std Dev & Normal Curve
Std Dev & Normal Curve
Std Dev & Normal Curve
Std Dev & Normal Curve
Chapter 5, review
• 2010 GSS: Political views
Chapter 5, review: Explore, p. 165-6
GSS Occupational prestige, Gender
GSS occupational prestige, gender
• Male
–
–
–
–
cases
mean
variance
std. deviation
• How calculate variance?
626
44.94
196.8
14.03
• Female
–
–
–
–
cases
mean
variance
std. deviation
794
43.65
188.12
13.72
• How calculate std.
deviation?
• Which group has more
variation around mean?
• Is difference significant?
GSS occupational prestige, gender
• Male
–
–
–
–
cases
mean
variance
std. deviation
32
44.94
196.8
14.03
• Female
–
–
–
–
cases
mean
variance
std. deviation
45
43.65
188.12
13.72
• Fewer cases
• Is difference significant?
GSS occupational prestige, gender
• Male
–
–
–
–
cases
mean
variance
std. deviation
6200
44.94
196.8
14.03
• Female
–
–
–
–
cases
mean
variance
std. deviation
7900
43.65
188.12
13.72
• More cases
• Is difference significant?
GSS occupational prestige, gender
• Male
–
–
–
–
cases
mean
variance
std. deviation
624
54.94
196.8
14.03
• Female
–
–
–
–
cases
mean
variance
std. deviation
794
33.65
188.12
13.72
• same cases, bigger
difference?
• Is difference significant?
GSS occupational prestige, gender
• Male
– cases
– mean
– std. deviation
624
44.94
1.03
• Female
– cases
– mean
– std. deviation
• same cases, less
variation?
• Is difference significant?
794
43.65
1.72
GSS occupational prestige, gender
• Male
–
–
–
–
cases
mean
variance
std. deviation
626
44.94
196.8
14.03
• Female
–
–
–
–
cases
mean
variance
std. deviation
794
43.65
188.12
13.72
• Is difference significant?
Problem 7, p. 171-171: Divorce rates
• Calculate mean,
•
•
•
•
•
•
•
•
•
•
AK
FL
ID
ME
MD
NV
NJ
TX
VT
WI
4.3
4.7
4.9
4.5
3.1
6.5
3.0
3.3
3.8
2.9
– sum /n
• Calculate standard
deviation
– square root of variance
– variance = sum of squared
distances from mean / (n1)
Problem 7, p. 171-171: Divorce rates
•
•
•
•
•
•
•
•
•
•
AK
FL
ID
ME
MD
NV
NJ
TX
VT
WI
4.3
4.7
4.9
4.5
3.1
6.5
3.0
3.3
3.8
2.9
• Sum = 41
• N= 10
• mean = 4.1
Problem 7, p. 171-171: Divorce rates
•
•
•
•
•
•
•
•
•
•
AK
FL
ID
ME
MD
NV
NJ
TX
VT
WI
4.3
4.7
4.9
4.5
3.1
6.5
3.0
3.3
3.8
2.9
distance from mean
4.3-4.1 =0.2
4.7-4.1 =0.6
4.9-4.1=0.8
4.5-4.1 =0.4
3.1-4.1=-1.0
6.5-4.1=2.4
3.0-4.1= -1.1
3.3-4.1= -0.8
3.8-4.1=-0.3
2.9-4.1=1.2
squared
.04
.36
.64
.16
1.0
5.76
1.21
0.64
.09
1.44 Sum = 11.34
Problem 7, p. 171-171: Divorce rates
• Variance = 11.34 / (n-1)
11.34 / 9 = 1.21
• Std. Deviation = 1.1
– 68% of states +/- 1.1 units above mean (4.1)
• Why variation in divorce rates across states?
Normal Curve
• Coin flip example
http://www.fourmilab.ch/rpkp/experiments/stat
istics.html
Normal Curve
• Any one sample has random variation
• More random samples = less variation around
central point
• More random samples = approach ‘normal’
distribution
Flip 2 coins:
what probability
of 1 head, 2
heads, 3, 4?
Flip 2 coins 8
times: what
probability of 1
head, 2 heads, 3,
4?
If observation by pure chance:
Start flipping coins
• 2 coins, 8 times:
–
–
–
–
1 head:
2 heads:
3 heads:
4 heads:
• 2 coins, 8 times
–
–
–
–
1 head:
2 heads:
3 heads:
4 heads:
Start flipping coins
• 2 coins, 16 times:
–
–
–
–
1 head:
2 heads:
3 heads:
4 heads:
Project Topics
• Local area
– Presidential nominations
• Dems
• GOP
–
–
–
–
–
Coal
Planned Parenthood
Immigration
Syria / refugees
?????
• WWU
– Tuition policy
– time to degree
– satisfaction with
education
– job prospects
– Debt
– ?????
Standard Deviation & z-scores
• Apples: Turnout + 1.84
• Oranges: Corruption -1.28
• Z = 0 is mean
• Z = 3 is 3 very rare
Z scores and Normal Curve
• How many states between mean & +1.84
• How many above 1.84
• See Appendix in text
– below mean = 50%
– between mean and z=1.84 = 46.7%
– beyond mean = 3.3% [1.5 states if normal]
Z scores and Normal Curve
• How many states between mean & -1.28
• How many below z= - 1.28
• See Appendix C in text
– above mean = 50%
– between mean and z= -1.28 = 39.9%
– beyond mean = 10.3% [1.5 states if normal]