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# Distribution Summaries

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1. Distribution Summaries • Measures of central tendency • Mean • Median • Mode • Measures of spread • Standard Deviation • Interquartile Range (IQR)

2. Mean • Mean or Arithmetic Average or Average is what we normally understand as the average of a set of numbers (observations) • Add up all the numbers and divide by the number of numbers (n).

3. Example • For the observations: 21 25 32 • n = 3 • The sum is 78 • The mean is 78/3 = 26 21 22 23 24 25 26 27 28 29 30 31 32

4. Consider the formula End Begin Observation Sigma - means add ‘em up! The Mean Sample Size Subscripts Indicating (indexing) Individuals

5. Characteristics of Mean • Balance point • Sum of deviations is zero • Influenced by extreme scores

6. Balance Point 21 22 23 24 25 26 27 28 29 30 31 32

7. Sum of deviations is zero • Deviation is score minus mean 21 - 26 = -5 25 - 26 = -1 32 - 26 = 6 78 78 0

8. Influenced by Extremes Extreme 20 30 40 50 60 70 80 90 100 100 120 130 Mean 42 Most Cases, Typical Case

9. Instead of a few cases .4 Distribution of ages N = 626 Fraction .2 0 18 22 26 30 34 38 42 46 M = 25.43

10. Self-Esteem The mean is not the most common N = 624 M = 33.3

11. Self-esteem by sex .2 .1 0 10 20 30 40 10 20 30 40 M = 32.9 M = 33.6 Females Males

12. How about numerically? . tabulate sex, summ(love1) | Summary of Rubin Love Scale sex | Mean Std. Dev. Freq. ------------+------------------------------------ female | 86.153674 10.426731 313 male | 89.00639 8.7537514 313 ------------+------------------------------------ Total | 87.580032 9.7242954 626

13. Another? (Depression) . tabulate sex, summ(depres1) | Summary of ces-d scale items s112-s123 sex | Mean Std. Dev. Freq. ------------+------------------------------------ female | 13.463899 11.27072 311 male | 11.405751 8.9336186 313 ------------+------------------------------------ Total | 12.431527 10.209712 624

14. Depression by Education . tabulate ed1, summ(depres1) years of | Summary of ces-d scale items s112-s123 education | Mean Std. Dev. Freq. ------------+------------------------------------ 9 | 41 0 1 11 | 15.76 11.475626 25 12 | 14.608993 11.434858 93 13 | 13.592593 10.478033 27 14 | 13.887233 10.608088 183 15 | 12.14838 8.6328703 87 16 | 9.9244392 8.6261835 154 18 | 11.037037 12.270921 27 20 | 6.3703704 5.2925793 27 ------------+------------------------------------ Total | 12.431527 10.209712 624

15. Cohabitation by income . tabulate income1, summ( cohblnth) | Summary of # Months Cohabited income1 | Mean Std. Dev. Freq. ------------+------------------------------------ 3.5 | 9 14.455185 43 7.5 | 9.5376344 13.7173 93 12.5 | 13.5 17.588839 114 20 | 12.284768 17.36486 151 30 | 12.722772 18.828765 101 42.5 | 15.135135 23.566881 37 62.5 | 14.916667 15.529786 12 82.5 | 0 0 1 ------------+------------------------------------ Total | 12.123188 17.370189 552

16. Income by education . tabulate ed1, summ(income1) years of | Summary of income1 education | Mean Std. Dev. Freq. ------------+------------------------------------ 11 | 11.452381 7.6483736 21 12 | 17.055556 11.563953 81 13 | 17.76087 8.5464899 23 14 | 14.876543 9.0504215 162 15 | 18.65 10.46616 80 16 | 24.960145 14.16097 138 18 | 22.204545 15.181959 22 20 | 34.5 12.788341 25 ------------+------------------------------------ Total | 19.434783 12.557429 552

17. The formula (again) End Begin Observation Sigma - means add ‘em up! The Mean Sample Size Subscripts Indicating (indexing) Individuals