Corrected page numbers for optional readings in SAS

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Corrected page numbers for optional readings in
SAS Languages and Procedures Aug 30 pp. 65-75
and 77-83 Sept 6 pp. 85-87 Sept 13 pp.
79-81 Sept 27 pp. 1-36 and 95-110 Oct 2 pp.
37-53 Oct 30 Delete Nov 13 Delete Nov
15 Delete Nov 22 pp. 81-83 Nov
27 Delete Nov 29 pp. 55-63
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Also, will NOT be placing materials for Sept 11,
Sept 13, and Nov 27 on reserve Nor will there
be a Handout for Nov 29
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Measures of Central Tendency
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For Discrete Variables

• These are variables having categoriesa. which
  are mutually exclusiveb. and exhaustive
• Identify the category having the highest
  frequency
• Is known as the modal category(after another
  statistic, the mode)

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If social welfare benefits such as disability,
unemployment compensation, and early retirement
pensions are as high as they are now, it only
makes people not want to work anymore. Do you
agree or disagree with this statement? GSS
1984
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Answer Category Frequency Percent
Strongly agree 193 13.2 Agree
480 32.8 Disagree 499 34.1
Strongly disagree 292 19.9
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Answer Category Frequency Percent
Strongly agree 193 13.2 Agree
480 32.8 Disagree 499 34.1
Strongly disagree 292 19.9 Model
Category Disagree (499), but . . .
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For Continuous variables

• Three statistics
• the mode
• the median
• and the mean

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The ModeThe score value that occurs most
frequently in a set of score values6, 5, 7, 8,
10, 6, 6, 8, 6, 5
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Score Value Frequency 10 1
8 2 7 1 6 4 5 2 Mode
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The Median

• Divides an ordered set of scores into two equal
  halvesNOTE The scores must first be ranked,
  usually in descending order

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12, 27, 19, 11, 14

• After ranking,
• 1st 27 2nd 19
• 3rd 14
• 4th 12
• 5th 11 Median 14.0

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1st 272nd 193rd 144th 125th 116th
5
14
1st 272nd 193rd 14---------------4th 125
th 116th 5
15
1st 272nd 19 14.5 3rd 14---------
------4th 12 11.55th 116th 5
16
1st 272nd 19
14.53rd 14---------------4th 12
11.55th 116th 5Median 14.5
- (14.5 - 11.5)(1/2) 13.0
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1st 332nd 273rd 194th 14---------------
5th 146th 147th 128th 11
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1st 332nd 273rd 194th 14-----
----------5th 146th 147th 128th
11
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1st 332nd 273rd 19
14.54th 14---------------5th 146th 14
13.57th 128th 11
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Median 14.5 (14.5 13.5)(1/3) 14.5 -
(1)(1/3) 14.5 - .333 14.167
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The Mean

• For scores 2, 5, 4, 1, and 3 12345- 2
  2- 1 1 Mean 3.0

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Formally,
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For example, the 1980 homicide rates per 1,000 in
16 (N) Southern states were51, 69, 71, 86,
88, 92, 95, 106, 108, 114, 132, 138, 145, 145,
157, 169
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libname mydata 'a\'libname library
'a\' options ps66 nodate nonumberproc
univariate datamydata.citiesvar
populattitle1 'Univariate Statistics'run
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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
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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)
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Here are data 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 19Netherlands
18 13 18Norway 7 24 18Portugal
4 48 23Spain 18 36 21Sweden
20 18 16Switzerland 14 15 19-------------
--------------------------------------------------
---------------  What is the MODE for the crude
birth rate? What is the MEDIAN for Gross
Domestic Product (GDP)? What is the MEAN for
percent in agriculture?
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What is the MODE for the crude birth
rate? First, determine the frequency of each of
the 16 values.  18,18,18,18,18,18,18 (7) 16,16
(2) 17,17 (2) 22 (1) 19,19
(2) 23 (1) 21 (1)
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What is the MODE for the crude birth
rate? First, determine the frequency of each of
the 16 values.  18,18,18,18,18,18,18 (7) 16,16
(2) 17,17 (2) 22 (1) 19,19
(2) 23 (1) 21 (1) Mode 18 
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What is the MEDIAN for gross domestic product
(GDP)? First, rank the birth rate values
(typically in descending order), then determine
the value at the location dividing the top half
(8 values) from the bottom half (8
values).  112 98 20 18
18 17 14 10 --- 9
8 7 7 6 4 4
3  
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What is the MEDIAN for gross domestic product
(GDP)? First, rank the birth rate values
(typically in descending order), then determine
the value at the location dividing the top half
(8 values) from the bottom half (8
values).  112 98 20 18
18 17 14 10 --- Median
(10.5 8.5)/2 9.5 9 8 7
7 6 4 4 3  
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Here are data 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 19Netherlands
18 13 18Norway 7 24 18Portugal
4 48 23Spain 18 36 21Sweden
20 18 16Switzerland 14 15 19-------------
--------------------------------------------------
---------------  What is the MEAN for percent
in agriculture?
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