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Introduction to Descriptive Statistics

Introduction to Descriptive Statistics. 2/25/03. Population vs. Sample Notation. Types of Variables. Describing data. Mean. Variance, Standard Deviation. Variance, S.D. of a Sample. Coefficient of variation. Skewness Symmetrical distribution. IQ SAT. Skewness Asymmetrical distribution.

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Introduction to Descriptive Statistics

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  1. Introduction to Descriptive Statistics 2/25/03

  2. Population vs. Sample Notation

  3. Types of Variables

  4. Describing data

  5. Mean

  6. Variance, Standard Deviation

  7. Variance, S.D. of a Sample

  8. Coefficient of variation

  9. SkewnessSymmetrical distribution • IQ • SAT

  10. SkewnessAsymmetrical distribution • GPA of MIT students

  11. Skewness(Asymmetrical distribution) • Income • Contribution to candidates • Populations of countries • “Residual vote” rates

  12. Skewness

  13. Skewness

  14. Kurtosis

  15. A few words about the normal curve • Skewness = 0 • Kurtosis = 3

  16. More words about the normal curve

  17. SEG example

  18. Graph some SEG variables

  19. Binary data

  20. Commands in STAT for getting univariate statistics • summarize • summarize, detail • graph, bin() normal • graph, box • tabulate [NB: compare to table]

  21. Explore Q9: Overall teaching evaluation subject q9 n 3.371 6.4375 16 3.982 6.73333 15 3.14 6.46154 13 14.02D 5.66667 3 21W.803 5.66667 12 21M.480 5.69231 13 17.906 5.28571 14 2.51 5.88235 17

  22. Graph Q9 . graph q9

  23. Divide into 7 “bins” and have them span 1, 1..2, 2..3, … 6..7 . graph q9,bin(7) xscale(0,7)

  24. Add ticks at each integer score . graph q9,bin(7) xscale(0,7) xlabel(0,1,2,3,4,5,6,7)

  25. Add a finer grain to the bars . graph q9,bin(14) xscale(0,7) xlabel(0,1,2,3,4,5,6,7)

  26. Even finer grain • . graph q9,bin(28) xscale(0,7) xlabel(0,1,2,3,4,5,6,7)

  27. Superimpose the normal curve (with the same mean and s.d. as the empirical distribution) . graph q9,bin(28) xscale(0,7) xlabel(0,1,2,3,4,5,6,7) norm

  28. Do the previous graph with only larger classes (n > 20) . graph q9 if n>20,bin(28) xscale(0,7) xlabel(0,1,2,3,4,5,6,7)

  29. Draw the previous graph with a box plot . graph q9 if n>20,box ylabel

  30. Draw the box plots for small (0..20), medium (21..50), and large (50+) classes . gen size = 0 if n <=20 (237 missing values generated) . replace size=1 if n > 20 & n <=100 (196 real changes made) . replace size = 2 if n > 100 (41 real changes made) . sort size . graph q9 ,box ylabel by(size) . graph q9 ,box ylabel by(size)

  31. A note about histograms with unnatural categories From the Current Population Survey (2000), Voter and Registration Survey How long (have you/has name) lived at this address? -9 No Response -3 Refused -2 Don't know -1 Not in universe 1 Less than 1 month 2 1-6 months 3 7-11 months 4 1-2 years 5 3-4 years 6 5 years or longer

  32. Simple graph

  33. Solution, Step 1Map artificial category onto “natural” midpoint -9 No Response  missing -3 Refused  missing -2 Don't know  missing -1 Not in universe  missing 1 Less than 1 month  1/24 = 0.042 2 1-6 months  3.5/12 = 0.29 3 7-11 months  9/12 = 0.75 4 1-2 years  1.5 5 3-4 years  3.5 6 5 years or longer  10 (arbitrary)

  34. Graph of recoded data

  35. Density plot of data

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