Introduction to Descriptive Statistics

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Introduction to Descriptive Statistics

• 2/25/03

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Population vs. Sample Notation
Population Vs Sample
Greeks Romans
?, ?, ? s, b
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Types of Variables
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Describing data
Moment Non-mean based measure
Center Mean Mode, median
Spread Variance (standard deviation) Range, Interquartile range
Skew Skewness --
Peaked Kurtosis --
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Mean
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Variance, Standard Deviation
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Variance, S.D. of a Sample
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Coefficient of variation
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SkewnessSymmetrical distribution

• IQ
• SAT

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SkewnessAsymmetrical distribution

• GPA of MIT students

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Skewness(Asymmetrical distribution)

• Income
• Contribution to candidates
• Populations of countries
• Residual vote rates

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Skewness
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Skewness
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Kurtosis
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A few words about the normal curve

• Skewness 0
• Kurtosis 3

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More words about the normal curve
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SEG example
The instructor and/or section leader The instructor and/or section leader The instructor and/or section leader The instructor and/or section leader The instructor and/or section leader 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 0.69 -1.5 6.8
Encourages questions class participation 6.1 0.66 -0.88 3.7
Stimulates interest in the subject 5.9 0.76 -1.1 4.7
Is available outside of class for questions 5.9 0.68 -1.3 6.3
Overall rating of teaching 5.9 0.67 -1.2 5.5
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Graph some SEG variables
The instructor and/or section leader The instructor and/or section leader The instructor and/or section leader The instructor and/or section leader The instructor and/or section leader 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
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Binary data
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Commands in STAT for getting univariate statistics

• summarize
• summarize, detail
• graph, bin() normal
• graph, box
• tabulate NB compare to table

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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 1
2 21M.480 5.69231 13 17.906 5.28571 14 2.51 5.882
35 17
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Graph Q9

• . graph q9

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Divide into 7 bins and have them span 1, 1..2,
2..3, 6..7

• . graph q9,bin(7) xscale(0,7)

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Add ticks at each integer score

• . graph q9,bin(7) xscale(0,7) xlabel(0,1,2,3,4,5,6
  ,7)

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Add a finer grain to the bars

• . graph q9,bin(14) xscale(0,7) xlabel(0,1,2,3,4,5,
  6,7)

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Even finer grain

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

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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

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Do the previous graph with only larger classes (n
gt 20)

• . graph q9 if ngt20,bin(28) xscale(0,7)
  xlabel(0,1,2,3,4,5,6,7)

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Draw the previous graph with a box plot

• . graph q9 if ngt20,box ylabel

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Draw the box plots for small (0..20), medium
(21..50), and large (50) classes

• . gen size 0 if n lt20
• (237 missing values generated)
• . replace size1 if n gt 20 n lt100
• (196 real changes made)
• . replace size 2 if n gt 100
• (41 real changes made)
• . sort size
• . graph q9 ,box ylabel by(size)
• . graph q9 ,box ylabel by(size)

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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

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Simple graph
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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)
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Graph of recoded data
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Density plot of data