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?