1. Corrected page numbers for optional readings in SAS ® Languages and Procedures Aug 30pp. 65-75 and 77-83 Sept 6pp. 85-87 Sept 13pp. 79-81 Sept 27pp. 1-36 and 95-110 Oct 2pp. 37-53 Oct 30Delete Nov 13Delete Nov 15Delete Nov 22pp. 81-83 Nov 27Delete Nov 29pp. 55-63

2. 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

3. Measures of Central Tendency

4. 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)

5. “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]

6. Answer Category Frequency Percent Strongly agree 193 13.2 Agree 480 32.8 Disagree 499 34.1 Strongly disagree 292 19.9

7. 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...

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

9. 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

10. Score Value Frequency 101 82 71 64 52 Mode = 6

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

12. 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

13. 1 st 27 2 nd 19 3 rd 14 4 th 12 5 th 11 6 th 5

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

15. 1 st 27 2 nd 19 ========= 14.5 3 rd 14 --------------- 4 th 12 ========= 11.5 5 th 11 6 th 5

16. 1 st 27 2 nd 19 ========= 14.5 3 rd 14 --------------- 4 th 12 ========= 11.5 5 th 11 6 th 5 Median = 14.5 - (14.5 - 11.5)(1/2) = 13.0

17. 1 st 33 2 nd 27 3 rd 19 4 th 14 --------------- 5 th 14 6 th 14 7 th 12 8 th 11

18. 1 st 33 2 nd 27 3 rd 19 ========= 4 th 14 --------------- 5 th 14 6 th 14 ========= 7 th 12 8 th 11

19. 1 st 33 2 nd 27 3 rd 19 ========= 14.5 4 th 14 --------------- 5 th 14 6 th 14 ========= 13.5 7 th 12 8 th 11

20. Median = 14.5 – [(14.5 – 13.5)(1/3)] = 14.5 - [(1)(1/3)] = 14.5 - [.333] = 14.167

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

22. Formally,

23. 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

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

29. Univariate Statistics PPD 404 Univariate Procedure Variable=POPULAT 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 T:Mean=0 4.183237 Pr>|T| 0.0001 Num ^= 0 63 Num > 0 63 M(Sign) 31.5 Pr>=|M| 0.0001 Sgn Rank 1008 Pr>=|S| 0.0001 W:Normal 0.468356 Pr<W 0.0001

30. Quantiles(Def=5) 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)

31. Here are data from 16 European countries. ============================================================================== Gross Domestic Percent in Crude Birth Nation Product (GDP) Agriculture Rate per (in billion$) 1,000 ------------------------------------------------------------------------------ Austria 31818 Belgium 4 716 Denmark 62318 Finland 73817 France 82518 Germany 112 817 Great Britain 98 518 Greece 94818 Ireland 104222 Italy 172419 Netherlands 181318 Norway 72418 Portugal 44823 Spain 183621 Sweden 201816 Switzerland 141519 ------------------------------------------------------------------------------ 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?

32. 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)

33. 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

34. 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

35. 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

36. Here are data from 16 European countries. ============================================================================== Gross Domestic Percent in Crude Birth Nation Product (GDP) Agriculture Rate per (in billion$) 1,000 ------------------------------------------------------------------------------ Austria 31818 Belgium 4 716 Denmark 62318 Finland 73817 France 82518 Germany 112 817 Great Britain 98 518 Greece 94818 Ireland 104222 Italy 172419 Netherlands 181318 Norway 72418 Portugal 44823 Spain 183621 Sweden 201816 Switzerland 141519 ------------------------------------------------------------------------------ What is the MEAN for percent in agriculture?