Corrected page numbers for optional readings in SAS ® Languages and Procedures Aug 30pp and Sept 6pp Sept 13pp Sept 27pp and Oct 2pp Oct 30Delete Nov 13Delete Nov 15Delete Nov 22pp Nov 27Delete Nov 29pp

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

Measures of Central Tendency

For Discrete Variables These are variables having categories a. which are mutually exclusive b. and exhaustive Identify the category having the highest frequency Is known as the modal category (after another statistic, the mode)

“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]

Answer Category Frequency Percent Strongly agree Agree Disagree Strongly disagree

Answer Category Frequency Percent Strongly agree Agree Disagree Strongly disagree Model Category = “Disagree” (499), but...

For Continuous variables Three statistics : 1.the mode 2.the median 3.and the mean

The Mode The score value that occurs most frequently in a set of score values 6, 5, 7, 8, 10, 6, 6, 8, 6, 5

Score Value Frequency Mode = 6

The Median Divides an ordered set of scores into two equal halves NOTE: The scores must first be ranked, usually in descending order

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

1 st 27 2 nd 19 3 rd 14 4 th 12 5 th 11 6 th 5

1 st 27 2 nd 19 3 rd th 12 5 th 11 6 th 5

1 st 27 2 nd 19 ========= rd th 12 ========= th 11 6 th 5

1 st 27 2 nd 19 ========= rd th 12 ========= th 11 6 th 5 Median = ( )(1/2) = 13.0

1 st 33 2 nd 27 3 rd 19 4 th th 14 6 th 14 7 th 12 8 th 11

1 st 33 2 nd 27 3 rd 19 ========= 4 th th 14 6 th 14 ========= 7 th 12 8 th 11

1 st 33 2 nd 27 3 rd 19 ========= th th 14 6 th 14 ========= th 12 8 th 11

Median = 14.5 – [(14.5 – 13.5)(1/3)] = [(1)(1/3)] = [.333] =

The Mean For scores 2, 5, 4, 1, and 3: 1—2—3—4—5 ^ - 2 ————— ——— +1 Mean = 3.0

Formally,

For example, the 1980 homicide rates per 1,000 in 16 (N) Southern states were: 51, 69, 71, 86, 88, 92, 95, 106, 108, 114, 132, 138, 145, 145, 157, 169

libname mydata 'a:\'; libname library 'a:\'; options ps=66 nodate nonumber; proc univariate data=mydata.cities; var populat; title1 'Univariate Statistics'; run;

Univariate Statistics PPD 404 Univariate Procedure Variable=POPULAT NUMBER OF RESIDENTS, IN 1,000S Moments N 63 Sum Wgts 63 Mean Sum Std Dev Variance Skewness Kurtosis USS CSS CV Std Mean T:Mean= Pr>|T| Num ^= 0 63 Num > 0 63 M(Sign) 31.5 Pr>=|M| Sgn Rank 1008 Pr>=|S| W:Normal Pr<W

Quantiles(Def=5) 100% Max % % Q % % Med % % Q % 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)

Here are data from 16 European countries. ============================================================================== Gross Domestic Percent in Crude Birth Nation Product (GDP) Agriculture Rate per (in billion$) 1, Austria Belgium Denmark Finland France Germany Great Britain Greece Ireland Italy Netherlands Norway Portugal Spain Sweden Switzerland 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?

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)

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

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)

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) Median = ( )/2 =

Here are data from 16 European countries. ============================================================================== Gross Domestic Percent in Crude Birth Nation Product (GDP) Agriculture Rate per (in billion$) 1, Austria Belgium Denmark Finland France Germany Great Britain Greece Ireland Italy Netherlands Norway Portugal Spain Sweden Switzerland What is the MEAN for percent in agriculture?