Measures of Variation

1
Measures of Variation
2
For discrete variables,the Index of
Qualitative Variation
3
Religious Preference Percent
Protestant 65.6 Catholic
24.2 Jewish 2.3 Other
1.2 None 6.1 No Answer 0.5
4
where p proportion of casesand K of
categories
5
Religious Preference Percent
Proportion Protestant 65.6 0.656 Catho
lic 24.2 0.242 Jewish
2.3 0.023 Other 1.2 0.012 None
6.1 0.061 No Answer 0.5 0.005
6
Religious Preference Percent Proportion
Proportion2 Protestant
65.6 0.656 0.430 Catholic
24.2 0.242 0.059 Jewish
2.3 0.023 0.001 Other
1.2 0.012 0.000 None
6.1 0.061 0.004 No Answer
0.5 0.005 0.000
7
Religious Preference Percent Proportion
Proportion2 Protestant
65.6 0.656 0.430 Catholic
24.2 0.242 0.059 Jewish
2.3 0.023 0.001 Other
1.2 0.012 0.000 None
6.1 0.061 0.004 No Answer
0.5 0.005 0.000
8

9
IQV (1 - 0.494) / (6 - 1) / 6 (0.506) / (5
/ 6) (0.506) / (0.833) 0.61
10
What does this mean?When there is perfect
dispersion, IQV 1.00 When there is no
dispersion, IQV 0.00
11
Religious Preference Percent
Proportion Proportion2
Protestant 16.67 0.1667 0.0279
Catholic 16.67 0.1667 0.0279
Jewish 16.67 0.1667 0.0279
Other 16.67 0.1667 0.0279
None 16.67 0.1667 0.0279 No
Answer 16.67 0.1667 0.0279 IQV
(1 - 0.1674) / (6 - 1) / 6 (0.833) / (5 / 6)
(0.833) / (0.833) 1.00
12
Religious Preference Percent
Proportion Proportion2 Protestant
100.00 1.00 1.00
Catholic 0.00 0.00
0.00 Jewish 0.00 0.00
0.00 Other 0.00
0.00 0.00 None
0.00 0.00 0.00 No Answer
0.00 0.00 0.00
IQV (1 - 1.000) / (6 - 1) / 6
(0.000) / (5 / 6) (0.000) /
(0.833) 0.00
13
For continuous variables, 1. range
2. interquartile range 3. standard
deviation 4. variance
14
The RangeThe distance across 100 of
scoresRange H L 1
15
For example, take the following 12 values (N
12) 5, 2, 27, 32, 3, 5, 35, 7, 31, 42, 37,
39To determine any of the so-called quantile
statistics such as the range, the scores first
must be ranked or ordered, here in descending
order  1st 42 39 37 35
32 31 27 15 7 5
3 12th 2
16
42.5 1st 42 39 37
35 32 31 27 15
7 5 3 12th 2
1.5 Range 42 2 1 41.0
17
The Interquartile RangeThe distance across the
middle 50 of scoresIQR Q3 Q1
18
1st 42 2nd 39 3rd 37
4th 35 5th 32 6th 31
7th 27 8th 15 9th 7 10th 5
11th 3 12th 2
19
Univariate and EDA Statistics 
PPD 404   Stem
Leaf
Boxplot 7 9
1 7
6 6 5 5 4
4 3 3 4
1
2 8 1
2 1 59
2
0 1 2
1 0
555556666777778889 18
---- 0 11111111111111111111111
1222222333344444 39 -----
--------------------------------
Multiply Stem.Leaf by 103
20
1st 42 2nd 39
3rd 37 -------------------------------
4th 35 5th 32 6th 31
7th 27 8th 15 9th
7 ------------------------------- 10th 5
11th 3 12th 2
21
1st 42 2nd 39
3rd 37 ------------------------------- Q3
4th 35 5th 32 6th 31
7th 27 8th 15 9th
7 ------------------------------- Q1
10th 5 11th 3 12th 2
22
1st 42 2nd 39
3rd 37 ------------------------------- Q3
(37.5 34.5)/2 36.0 4th 35
5th 32 6th 31 7th 27
8th 15 9th 7 -------------------------
------ Q1 (7.5 4.5)/2 6.00 10th 5
11th 3 12th 2
23
1st 42 2nd 39
3rd 37 ------------------------------- Q3
(37.5 34.5)/2 36.0 4th 35
5th 32 6th 31 7th 27
8th 15 9th 7 -------------------------
------ Q1 (7.5 4.5)/2 6.00 10th 5
11th 3 12th 2 IQR Q3 Q1
36.0 6.0 30.0
24
The Standard Deviation
25
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26
33 19.333 13.667 27
19.333 7.667 19 19.333
-0.333 14 19.333 -5.333 12
19.333 -7.333 11 19.333
-8.333 ? ? 0.000 (0.003)
27
The sum of the deviationswill always be
zero(except for rounding error)
28
The Sum of the Deviations

• 12345 Mean 3.0- 2 2- 1
  10Sum (-2) (2) (-1) (1) (0) 0.0

29
33 19.333 13.667 186.787 27
19.333 7.667 58.783 19
19.333 -0.333 0.111 14 19.333
-5.333 28.441 12 19.333 -7.333
53.773 11 19.333 -8.333
69.439 ? 0.000 ?
397.334
30
The Variance
31
33 19.333 13.667 186.787 27
19.333 7.667 58.783 19
19.333 -0.333 0.111 14 19.333
-5.333 28.441 12 19.333 -7.333
53.773 11 19.333 -8.333
69.439 ? 0.000 ?
397.334 sy2 397.334 / (6 - 1) 79.467
32
The standard deviation

• Simply the square root of the variance
• sY 8.914

33
Z-scorespure numbers with mean of 0.0 and
standard deviation of 1.00

• z1 (68 - 70.0) / 6.45 (-2.00) / 6.45 - 0.31
• z1 (68 - 70.0) / 12.88 (-2.00) / 12.88 -
  0.16

34
Using SAS to Produce Z-Scores  libname
old 'a\'libname library 'a\' options ps66
nodate nonumber data temp1set
old.citiespopstdpopulatrun proc standard
datatemp1 mean0.0 std1.0 outtemp2var
popstdrun proc print datatemp2id
populatvar popstdtitle1 'Z-Scores Produced by
PROC STANDARD'title2title3 'PPD 404'run
35
Z-Scores Produced by PROC
STANDARD  PPD
404  POPULAT
POPSTD  275
-0.28030 116
-0.42296 127
-0.41309 497
-0.08112 117
-0.42206 301
-0.25698 82
-0.45347 641
0.04808 453
-0.12060 100
-0.43732 241
-0.31081 82
-0.45347 101
-0.43642 72
-0.46244 393
-0.17443 86
-0.44988 175
-0.37002 68
-0.46603 108
-0.43014
36
libname mydata 'a\'libname library
'a\' options ps66 nodate nonumberproc
univariate datamydata.citiesvar
populattitle1 'Univariate Statistics'run
37
Univariate Statistics 
PPD 404 
Univariate Procedure VariablePOPULAT
NUMBER OF RESIDENTS, IN 1,000S 
Moments  N
63 Sum Wgts 63 Mean
587.4127 Sum 37007 Std Dev
1114.554 Variance 1242231
Skewness 5.090201 Kurtosis 30.74326
USS 98756687 CSS 77018305
CV 189.7395 Std Mean 140.4206
TMean0 4.183237 PrgtT
0.0001 Num 0 63 Num gt 0
63 M(Sign) 31.5
PrgtM 0.0001 Sgn Rank
1008 PrgtS 0.0001 WNormal
0.468356 PrltW 0.0001
38
Quantiles(Def5) 
100 Max 7896 99 7896
75 Q3 641 95 1949
50 Med 278 90 906
25 Q1 100 10
72 0 Min 56 5
60 1
56 Range 7840
Q3-Q1 541 Mode
56   Extremes 
Lowest Obs Highest Obs
56( 30) 1511( 56)
56( 24) 1949( 55)
58( 46) 2816( 54)
60( 21) 3367(
53) 65( 51) 7896(
52)
39
Calculate the INDEX OF QUALITATIVE VARIATION for
the data in the following table.  
 
Service Branch Frequency P P2 -------------
--------------------------------------------------
 Air Force 56Army 166Marine Corps
14Merchant Marines 1Navy 70
-------Total 307 --------------------------
-------------------------------------
40

Service Branch Frequency
P P2 ----------------------------------------
----------------------- Air Force
56 0.182 0.033Army 166 0.541 0.292M
arine Corps 14 0.046 0.002Merchant Marines
1 0.003 0.000Navy 70 0.228 0.052
--- ------Total
307 0.379 -----------------------------------
----------------------------   INDEX OF
QUALITATIVE VARIATION 0.776 
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42
Here are data once again from 16 European
countries. 

Gross Domestic Percent in Crude
Birth Nation Product (GDP)
Agriculture Rate per (in
billion) 1,000 ---------------------
--------------------------------------------------
------- Austria 3 18 18Belgium 4
7 16Denmark 6 23 18Finland
7 38 17France 8 25 18Germany
112 8 17Great Britain 98
5 18Greece 9 48 18Ireland
10 42 22Italy 17 24 19Netherlan
ds 18 13 18Norway
7 24 18Portugal 4 48 23Spain
18 36 21Sweden 20 18 16Switzerland
14 15 19---------------------------------------
---------------------------------------  What
is the RANGE for the PERCENT IN
AGRICULTURE? What is the INTERQUARTILE RANGE
for the PERCENT IN AGRICULTURE?
43
First, rank the values in descending order. Find
the difference between the HIGHEST and LOWEST
values (and add 1).  48 48 42 38
36 25 24 24 23 18 18
15 13 8 7 5  
RANGE H L 1 48 5 1 44.0
44
Having ranked the values in descending order,
determine the value at the location dividing the
upper 4 values from the lower 12 values. Then
determine the value at the location dividing the
upper 12 values from the lower 4 values. Find
the difference between these two
values.  48 48 42 38 -- Q3
(38.5 35.5) / 2 37.0 36 25 24
24 23 18 18 15 -- Q1
(15.5 12.5) / 2 14.0 13 8
7 5   IQR Q3 Q1 37.0 14.0
23.0
45
 

Gross Domestic Percent in Crude Birth
Nation Product (GDP) Agriculture
Rate per (in billion)
1,000 ----------------------------------------
-------------------------------------- Austria
3 18 18Belgium 4 7 16Denmark
6 23 18Finland 7 38 17France
8 25 18Germany 112 8 17Great
Britain 98 5 18Greece
9 48 18Ireland 10 42 22Italy
17 24 19Netherlands
18 13 18Norway 7 24 18Portugal
4 48 23Spain 18 36 21Sweden
20 18 16Switzerland 14 15 19-------------
--------------------------------------------------
---------------  What is the STANDARD
DEVIATION for GDP? What is Germanys Z-SCORE
for GDP?
46
First, determine the value of the mean.
47
Next, determine the deviations and squared
deviations for each value.  3
19.1875 368.1602 4 -18.1875 330.7852 6 -16.1
875 262.0352 7 -15.1875 230.6602 8 -14.1875
201.2852 112 89.8125 8066.285
98 75.8125 5747.535 9 -13.1875 173.9102
10 -12.1875 148.5352 17
-5.1875 26.91016 18 -4.1875 17.53516 7
-15.1875 230.6602 4 -18.1875 330.7852
18 -4.1875 17.53516 20
-2.1875 4.785156 14 -8.1875 67.03516 
355 16224.44
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49
What is Germanys Z-SCORE for GDP?Germanys GDP
112Mean GDP 22.188