Distributions & GraphsDistributions & Graphs
Variable TypesVariable Types
Discrete (nominal)Discrete (nominal) Sex, race, football numbersSex, race, football numbers
Continuous (interval, ratio)Continuous (interval, ratio) Temperature, Test score, Reaction timeTemperature, Test score, Reaction time
Frequency DistributionsFrequency Distributions
Graphic representation of dataGraphic representation of data Easier to understand than raw numbersEasier to understand than raw numbers Helps communicate to othersHelps communicate to others
Basic kinds of frequency distributionsBasic kinds of frequency distributions Ungrouped – simple tallyUngrouped – simple tally Grouped – used to simplifyGrouped – used to simplify
UsesUses Relative and cumulative frequencyRelative and cumulative frequency ShapeShape
Common of GraphsCommon of Graphs
Bar Chart (discrete)Bar Chart (discrete)
Histogram (continuous)Histogram (continuous) Scatterplot (2 variables)Scatterplot (2 variables)
Bars show counts
f m
Sex
0
2
4
6
Fre
qu
ency
59.00 60.00 61.00 62.00 63.00 64.00
Height Inches
1
2
3
4
Fre
qu
ency
50 100 150 200
Horsepower
10
20
30
40
Mile
s p
er G
allo
n
Distribution ShapesDistribution Shapes
NormalNormal CenterCenter SpreadSpread ShouldersShoulders SkewSkew
There will be numerical ways to describe all these, but for now, just consider the shape visually.
NormalNormal
Central TendencyCentral Tendency
Variability (spread)Variability (spread)
Central tendency and Variability
SkewSkew
The tail!
Kurtosis - shouldersKurtosis - shoulders
Odd ShapesOdd Shapes
Modern Stat GraphsModern Stat Graphs
Box plotBox plot Stem-leafStem-leaf
BoxplotBoxplot
17N =
D1
9
8
7
6
5
4
3
2
1
Median
25 %tile
75 %tile
Middle50 Percent
Largest Case not an Outlier
Smallest Case not an Outlier
Whiskerortail
Whiskerortail
Boxplot for a normal distribution
Same distribution as a (sort of) histogram
Boxplot with outliersBoxplot with outliers
21N =
20
10
0
-10
19
21
20
22
Outlier
Extreme Outlier
Outlier
Extreme Outlier
Boxplot with Skewed Boxplot with Skewed DistributionDistribution
227N =volcano heights
30000
20000
10000
0
10000
222227223226225224
Exam and Course Distributions Psych Stats Fall 2007
Course Grades Sp 2008Course Grades Sp 2008
Stem-leaf diagramStem-leaf diagram
Frequency Stem & Leaf 1.00 2 . 0 2.00 3 . 00 3.00 4 . 000 4.00 5 . 0000 3.00 6 . 000 2.00 7 . 00 1.00 8 . 0 Stem width: 1.00 Each leaf: 1 case(s)
(Approximately) Normal distribution
17N =
D1
9
8
7
6
5
4
3
2
1
Median
25 %tile
75 %tile
Middle50 Percent
Largest Case not an Outlier
Smallest Case not an Outlier
Whiskerortail
Whiskerortail
Stem-leaf of volcano heightsStem-leaf of volcano heights Frequency Stem & Leaf 8.00 0 . 25666789
8.00 1 . 01367799 23.00 2 . 00011222444556667788999 21.00 3 . 011224445555566677899 21.00 4 . 011123333344678899999 24.00 5 . 001122234455666666677799 18.00 6 . 001144556666777889 26.00 7 . 00000011112233455556678889 12.00 8 . 122223335679 14.00 9 . 00012334455679 13.00 10 . 0112233445689 10.00 11 . 0112334669 9.00 12 . 111234456 5.00 13 . 03478 2.00 14 . 00 3.00 15 . 667 2.00 16 . 25 2.00 17 . 29 6.00 Extremes (>=18500) Stem width: 1000.00 Each leaf: 1 case(s)
Final Grade Pct Sp 08Final Grade Pct Sp 08
TotPct08 Stem-and-Leaf Plot
Frequency Stem & Leaf
12.00 Extremes (=<.48) 3.00 5 . 669 12.00 6 . 011333344444 18.00 6 . 555566677777889999 28.00 7 . 0000111122222223333333444444 34.00 7 . 5555555566666667777777888888889999 29.00 8 . 00000111111112222223333344444 13.00 8 . 5555566667788 16.00 9 . 0000011111123344 3.00 9 . 566
Stem width: .10 Each leaf: 1 case(s)
Note that SPSS has the boxplot with larger numbers at the top, but the stem-leaf shows larger numbers at the BOTTOM, so one is backwards from the other.
DefinitionDefinition
Political party is an example of what Political party is an example of what kind of variable?kind of variable? 1 continuous 1 continuous 2 discrete2 discrete 3 intensity3 intensity 4 objective4 objective
DefinitionDefinition
A distribution with a long tail to the A distribution with a long tail to the right (high) end is called ________right (high) end is called ________
1 leptokurtic1 leptokurtic 2 negatively skewed2 negatively skewed 3 platykurtic3 platykurtic 4 positively skewed4 positively skewed
GraphsGraphs
Which boxplot shows a skewed Which boxplot shows a skewed distribution?distribution?
24N =
VAR00001
12
10
8
6
4
2
406N =
Vehicle Weight (lbs.
6000
5000
4000
3000
2000
1000
019N =
VAR00002
10
8
6
4
2
0
19N =
VAR00001
8
7
6
5
4
3
2
1
0
1 2 3 4
Discussion QuestionDiscussion Question
When might you prefer a graph to a When might you prefer a graph to a table of numbers for presenting a table of numbers for presenting a result?result?
Name a variable you think would be Name a variable you think would be interesting for college students to interesting for college students to see as a graph. What kind of data see as a graph. What kind of data would you put in the graph? Why would you put in the graph? Why would it be of interest?would it be of interest?