Chapter 3: Averages

1. Chapter 3: Averages Averages: measures of central tendency, include mode, median, and mean. Mode: the most frequently occurring score on a variable. In US, the mode for gender is female because there is more female than male. The mode for education is 12 years for American because more people with 12 year-education than any other year of education.

2. Mode • Look at this graph (GSS 2000), determine which one is the mode for the variable of spanking

3. Mode 2 • When a variable has only one score that is most common, we call the variable is unimodal, such as the spanking example. • Sometimes a variable may have two scores, both of which are very common. We can this variable bimodal variable. Bimodal variables may have two humps in their bar charts, such as the one on page 63. The two humps are not identical but very close in their numbers.

4. Mode 3 • Although mode can be used to indicate nominal, ordinal, interval/ratio variables, most often mode is useful to indicate the central tendency for only nominal variables.

5. Median • Median (MD) is the value that divides an ordered set of scores in half. If the score has odd number, finding its Md involves (1), put the scores in order from lowest to highest; and (2) find the middle score Original scoresOrdered scores 4 1 3 3 9 4 12 9 1 12 78 19 19 78

6. Median 2 • If the score has an even number, finding its md involves (1), put the scores in order from lowest to highest; (2), find the two middle scores; an (3) average the two middle scores by adding them and the dividing them by 2. Original scoresOrdered scores 4 1 3 3 9 4 1 9 78 19 19 78

7. Median 3 • GSS 2000 “Will immigrants lead to economic growth?” valuefF 1 (very likely) 155 155 2 (likely) 489 644 3 (unlikely) 514 1158 4 (very unlikely) 145 1303

8. Median 4 • Extreme low or high scores do not affect the median. Set ASet BSet C 51 1 51 52 52 52 53 53 53 54 54 54 55 55 198

9. Mean • Mean: arithmetical average by dividing the sum by the number of score E(X) = Xi/N • Deviation: the difference between a score and the mean • Sum of squares: the sum of squared deviations from the mean

10. Mean 2 • XXi – E(X)(Xi - E(X))2 4 -4 16 8 0 0 10 2 4 11 3 9 9 1 1 6 -24 E(X) = 8 0 34

11. Mean 3 • Be careful of outliers in calculation of means. In contrast to median, mean is dependent on extreme values. Try to compute the two means for the two sample salaries in thousand. Sample salary oneSample salary two 10 10 20 20 30 720

12. Mean 4 • Theoretically mean is used only for interval/ratio variables. But many times social scientists use mean to indicate central tendency of ordinal variables such as social classes, attitudes from strongly agree to strongly disagree etc. • The book argued that mean can also applied to dichotomous nominal variables.

13. Which one to use Nominal Ordinal Interval/ratio Mean rare sometimes OK Median rare OK OK Mode OK OK sometimes

14. Skewness • Positive skewed/right skewed: when sample mean is larger than sample median. Typical situation would be salary survey including most common people making 50 k or less but a few Hollywood stars making millions of dollars • Negative/left skewed: When sample mean mean is smaller than median. Typical situation is a survey of highly educated persons with a few illiterates.

15. Exercise • GSS Data Case # workingprestigemaritalsibsage 1 5 32 1 2 79 2 1 30 5 1 32 3 4 46 3 7 55 4 2 28 4 4 29 5 7 52 1 3 48

16. Exercise continues • Working: What did you do last week? (1) Working full time; (2) working part time; (3) on leave; (4) unemployed; (5) retired; (6) in school; (7) keeping house • Prestige: respondents’ occupational prestige score from 17 to 86 • Marital: respondent marital status (1) married; (2) widowed; (3) divorced; (4) separated; (5) never married • Sibs: How many brothers and sisters do you have? • Age: respondents’ age in years

17. Exercise continues • Determine which measure mode, median, mean) can be used to indicate the central tendency for those variables. • Calculate those measures.