Corrected page numbers for optional readings in SAS® Languages and Procedures

Aug 30 pp. 65-75 and 77-83Sept 6 pp. 85-87Sept 13 pp. 79-81Sept 27 pp. 1-36 and 95-110Oct 2 pp. 37-53Oct 30 DeleteNov 13 DeleteNov 15 DeleteNov 22 pp. 81-83Nov 27 DeleteNov 29 pp. 55-63

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 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)

“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 193 13.2 Agree 480 32.8 Disagree 499 34.1 Strongly disagree 292 19.9

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 . . .

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

10 1 8 2 7 1 6 4 5 2

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,

1st 272nd 19

3rd 14

4th 12

5th 11

Median = 14.0

1st 272nd 193rd 144th 125th 116th 5

1st 272nd 193rd 14---------------4th 125th 116th 5

1st 272nd 19========= 14.53rd 14---------------4th 12========= 11.55th 116th 5

1st 272nd 19

========= 14.53rd 14---------------4th 12

========= 11.55th 116th 5

Median = 14.5 - (14.5 - 11.5)(1/2) = 13.0

1st 332nd 273rd 194th 14---------------5th 146th 147th 128th 11

1st 332nd 273rd 19=========4th 14---------------5th 146th 14=========7th 128th 11

1st 332nd 273rd 19

========= 14.54th 14---------------5th 146th 14

========= 13.57th 128th 11

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

= 14.5 - [.333]= 14.167

The Mean

For scores 2, 5, 4, 1, and 3:

1—2—3—4—5^

- 2 ————— +2- 1——— +1

Mean = 3.0

Formally,

N

YY

N

ii

1

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

16

16

1 i

iYY

16

766,1Y

375.110Y

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 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

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)

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?

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). 

112 98 20 18 18 17 14 10---

9 8 7 7 6 4 4 3

  

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

  

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?

N

YY

n

ii

1

3921

n

iiY

16

392Y

5.24Y