1 / 33

Assignment

Assignment. After this lecture, start Assignment #2. It’s in the course binder. Sociology 549, Lecture 4. What is typical? Measures of center by Paul von Hippel Sociology 549. What is typical?. Measures of center Mean Trimmed mean Median Mode Sensitivity to extreme values.

liam
Download Presentation

Assignment

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Assignment • After this lecture, start Assignment #2. • It’s in the course binder.

  2. Sociology 549,Lecture 4 What is typical? Measures of center by Paul von Hippel Sociology 549

  3. What is typical? • Measures of center • Mean • Trimmed mean • Median • Mode • Sensitivity to extreme values

  4. Katie Harman (2002) Arguing from examples • “Is Miss America an Undernourished Role Model?” • Rubinstein, Sharon MHS. Caballero, Benjamin MD, PhD. Journal of the American Medical Association, 283(12):1569, March 22/29, 2000. • Yes example • Suzette Charles (1984) weighed just 100 pounds • No example (Robert Renneisen Jr., CEO Miss America Org.) • “Recent winners have had some of the highest body mass readings, • a reflection of pageant officials’ emphasizing brains over beauty.” • Source: John Curran, Associated Press • http://abcnews.go.com/sections/living/DailyNews/missamerica000322.html

  5. What’s typical? • Isolated examples are useless • Let’s get (available) winners from the past 19 years • http://www.pbs.org/wgbh/amex/missamerica/sfeature/sf_list.html BMI = 703 Weight / Height2. (Weight in pounds, height in inches.) BMI<19.1: “underweight” (NHANES) BMI<18.5: “undernourished” (WHO) BMI=22.7: average for women age 20-29

  6. Measures of center1. Mean (average) The mean “Y bar” equals the sum (S) of all the values Y, divided by the number of values N.

  7. Calculating the mean: Example aka center of mass, balance point BMI<19.1: “underweight” (NHANES) BMI<18.5: “undernourished” (WHO) BMI=22.7: average for women age 20-29

  8. Extreme value • Prediction • Miss America 2004=Ruben • Talent: popular song • height: 6’4” (76”) • weight: 350 pounds

  9. Mean is sensitive to extreme values without Ruben: 18.6 BMI<19.1: “underweight” (NHANES) BMI<18.5: “undernourished” (WHO) BMI=22.7: average for women age 20-29

  10. Measures of center2. Trimmed mean • Trim (discard) most extreme values • Calculate mean for the remaining values • Not as sensitive to extreme values • because it ignores them • not sensitive = robust

  11. Calculate the trimmed mean • Sort data by Y • Trim largest and smallest values • Get mean of remaining values Interpretation: Ignoring the 2 most extreme values, the average BMI is 18.8

  12. How much to trim? • Start with 13 values • Trim the 2 most extreme • 2/13=15% trimmed mean • Trim the 4 most extreme • 4/13=31% trimmed mean • etc. • Note • 15% trimmed • 7.5% off top + 7.5% off bottom • not 15% top, 15% bottom • Often use round numbers, then round down • e.g., 20% trimmed mean • trim 20%X13=2.6 cases •  round down to 2 cases

  13. Measures of Center3. Median (maximally trimmed mean) • For odd N, e.g. N=13 • Trim till you just can’t trim no more • 12 cases trimmed • 1 is left • its value=its mean=18.8 • This is the median

  14. Measures of center3. Median • For even N • Trim till you just can’t trim no more • 10 cases trimmed • 2 are left • their mean=18.5 • This is the median

  15. Median: Interpretation • Your book says • Half of values are larger than the median • Half are smaller • In this example, that’s correct median

  16. Median: Interpretation • But the book’s interpretationisn’t always quite right • More precisely(gray) • At most half of values are more than the median • At most half are less • or, equivalently (brackets) • At least half are the median or less • At least half are the median or more

  17. Median: Interpretation • Here far less than half of households have more than the median number of residents. • But the careful interpretation is still correct: • At least half of households (87%) have 2 residents or fewer • At least half (62%) have 2 residents or more. Distribution of adult residents across US households (from GSS)

  18. Common misconception • Median is not halfway between the largest and smallest values • That’s the midpoint • very sensitive to extremes • Median is not sensitive to extremes; • it ignores them median midpoint

  19. Summary: Sensitive vs. Robust Statistics • Mean: • affected by all values • including extreme values • efficient but sensitive • Median: • nearly all values trimmed • affected only by middle values • crude but robust • Trimmed mean: • not affected by trimmed values • compromise between mean and median

  20. Working with frequency tables

  21. crazy N=12 winners? N=7 values? still wrong Mean of a frequency table Suppose the heights of Miss Americas are summarized in a frequency table. Can you use the same formula for the mean? No!

  22. Mean formula for a frequency table In frequency tables, each line can represent more than one case. So you use a different formula! total inches total winners

  23. Comparison of mean formulas • Frequency table • Some lines represent multiple cases • Those lines get multiplied • Data set • Each line represents one case • Each line gets equal weight 3 winners, each 63 inches tall, contribute 189 inches to the total Both formulas give total inches (802) over total winners (12)

  24. Mean of a dummy variable Underweight. Y=1 if BMI<19.1. Otherwise Y=0. • …is the proportion p of cases with a value of 1 • 58% of recent Miss Americas are underweight

  25. Median from a frequency table • Usually there’s no value with cumulative % of exactly 50% • So median is first value with cum. % > 50% • Here median=68” with cum. %=66.67% • Check against definition: • At least half (66.67%) are at or below 68”, and • at least half (100-41.67=58.33%) are at or above 68” • Interpretation: • At least half of recent Miss Americas are 68” or taller, and • at least half of recent Miss Americas are 68” or shorter (real data)

  26. Median from a frequency table • If there is a value with cum. % of 50% • then median is average of that value and the next • Here 66” has c%=50% • So median is average of 66” and 68” • median is 67” • Interpretation: • Exactly half of recent Miss Americas shorter than 67” • Exactly half are taller median (not real data)

  27. Measures of center4. Mode • Most common value(s) • e.g., for quantitative variable (height) • modes or modal heights are 63” and 68” (bimodal) • Interpretation: • The most common heights for recent Miss Americas are 63” and 68”

  28. Mode • Not so easy if everyone has a slightly different value • Here mode is about $24-$29K • Interpretation: • The most common starting salaries are in the mid-to-high $20s

  29. Mode: Qualitative variable • Most common value(s) • Here mode is song • Interpretation: • The most common talent for recent Miss Americas is popular song

  30. Summary: Measures and variables Can treat dummy variables like interval

  31. Summary: Interpretation • Mean: average, center of gravity • for a dummy variable: proportion with a value of 1 • Trimmed mean: mean of non-extreme values • Median: halfway point, • (but not halfway between smallest and largest) • At least half the values are or above at the median. • At least half are at or below. • Mode: Most common value

  32. Bonus slides

  33. “Brains over beauty”?! • Kate Shindle (1998) • student at Northwestern • double major: • theater & sociology • speeches • “Miss America: Exploitation or Empowerment” • “Crowns and Condoms: Renewing Our Fight Against AIDS” • 145 pounds • 5 feet 11 inches

More Related