1. Introduction to Descriptive Statistics
2/25/03

2. Population vs. Sample Notation
Greeks
Romans
, , 
s, b

3. Types of Variables

4. Describing data Moment Non-mean based measure Center Mean Mode, median
Spread
Variance (standard deviation)
Range,
Interquartile range
Skew
Skewness --
Peaked
Kurtosis

5. Mean

6. Variance, Standard Deviation

7. Variance, S.D. of a Sample

8. Coefficient of variation

9. Skewness Symmetrical distributionIQ
SAT

10. Skewness Asymmetrical 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 The instructor and/or section leader: Mean s.d. Skew Kurt
Graph
Gives well-prepared, relevant presentations
6.0
0.69
-1.7
8.5
Explains clearly and answers questions well
5.9
0.68
-1.0
4.8
Uses visual aids well
5.6
0.85
-1.8
8.9
Uses information technology effectively
5.5
0.91
-1.1
5.0
Speaks well
6.1
-1.5
6.8
Encourages questions & class participation
0.66
-0.88
3.7
Stimulates interest in the subject
0.76
4.7
Is available outside of class for questions
-1.3
6.3
Overall rating of teaching
0.67
-1.2

18. Graph some SEG variables
The instructor and/or section leader:
Mean
s.d.
Skew
Kurt
Graph
Uses visual aids well
5.6
0.85
-1.8
8.9
Encourages questions & class participation
6.1
0.66
-0.88
3.7

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
14.02D
21W
21M

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
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
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)

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
months
months
years
years
6 5 years or longer

32. Simple graph

33. Solution, Step 1 Map 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
months  3.5/12 = 0.29
months  9/12 = 0.75
years  1.5
years  3.5
6 5 years or longer  10 (arbitrary)

34. Graph of recoded data

35. Density plot of data