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Business Statistics

Lecture 2: Descriptive Statistical

Graphs and Plots

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• Graphical descriptive statistics

• Histograms (and bar charts)

• Boxplots

• Scatterplots

• Time series plots

• Mosaic plots

• Continue our introduction to JMP

Goals for this Lecture

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• A histogram is a graph of the observed frequencies for a statistic

• Use for either continuous or categorical variables

• Categorical histogram called a bar chart

• With continuous data, histograms show

• Shape

• Location or central tendency

• Spread or amount of variation

Histogram

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0

2

4

6

8

10

12

170 180 190 200 210 220 230 240 250 260

10 observations

between 215 and 225

Reading a Histogram

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Histograms help answer:

• What is the overall shape of the data?

• Are there any unusual observations?

• Where is the “center” or “average” of the

data located?

• What is the spread of the data? Is the data

spread out or close to the center?

Histogram

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0

2

4

6

8

10

12

170 180 190 200 210 220 230 240 250 260

Histograms show overall shape of data

Histograms Show the Shape

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0

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8

10

12

170 180 190 200 210 220 230 240 250 260

Shows the central tendency or average:

Where the histogram balances

Histogram and the Mean

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0

2

4

6

8

10

12

170 180 190 200 210 220 230 240 250 260

0

2

4

6

8

10

12

170 180 190 200 210 220 230 240 250 260

Shows

differences

in variability

Histogram and the SD

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• The distribution of a variable describes:

• What values the variable can assume, and

• The frequency that those values occur

• The histogram is an empirical distribution

• It’s the distribution we observed in the sample

• Often, we want to know (something about)

the population distribution

• Could be real or an abstraction

• The “Normal” distribution occurs frequently

• More about normal distribution in next class

Histograms & Distributions

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Distribution of Wharton GMAT Scores

400

500

600

700

800

400 500 600 700 800

You can display the

histogram horizontally or

vertically in JMP

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Distribution of Executive Compensation

0

100000000

200000000

0

10000000

20000000

30000000

40000000

50000000

Eisner

of Disney

With Eisner

Removed

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An Aside: Data Transformation

• Sometimes transforming data is useful:

• It helps us see patterns in the data and/or

simplifies the interpretation

• Logs particularly useful for financial data

• “Log base 10” or “log” or “log10” easy

• If Y=10X, then logY=X

• Examples: Y logY

1 0

10 1

100 2

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Data Transformation, Part 2

• Natural logs a little more confusing

• “Log base e” or “ln” or “loge”

• If Y=eX, then lnY=X (where e=2.71)

• Examples: Y lnY

1 0

2.71 1

7.39 2

20.09 3

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Data Transformation, Part 3

• The picture:

-5

-4

-3

-2

-1

0

1

2

3

4

0.0

1

1.0

5

2.0

9

3.1

3

4.1

7

5.2

1

6.2

5

7.2

9

8.3

3

9.3

7

10.4

11.4

12.5

13.5

14.6

15.6

Y

log(Y)

ln(Y)

15

5

6

7

8

Executive Compensation Redux

0

100000000

200000000 Eisner

of Disney

Sc

ale

ea

sy t

o in

terp

ret $100M

$10M

$1M

$100K

Histograms vs. Bar Charts

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Bar chart of CEO’s

undergraduate degrees

(Forbes94.jmp)

Histogram of the year of

CEO’s undergraduate

degrees (Forbes94.jmp)

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Boxplots

• A boxplot shows distribution in one

dimension

• Use with continuous variables only

• Most useful when comparing distributions

of a continuous variable between

categorical groups

• Will not show multiple modes

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Example Boxplot

5.0 6.0 7.0 8.0

median

quartiles

whiskers outliersoutliers

least half

• Boxplot:

• Center spread, skewness, outliers

• Requires only one dimension

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Boxplot for GMAT Scores

• Boxplot is to a

histogram what a

contour map is to a

mountain

• Imagine you are

looking down on the

histogram

400

500

600

700

800

20

5.0

6.0

7.0

8.0

Aerospacedefense

Business

Capital goods

Chemicals

ComputersComm

Construction

Consumer

Energy

Entertainment

Financial

Food Forest

Health

Insurance

Metals

Retailing

Transport

TravelUtility

WideIndustry

Side by Side Boxplots

• Do CEOs in some industries make more

than others?

• One continuous and one categorical variable

Scatterplots

• A scatterplot shows the relationship

between two variables

• Use with continuous variables only

• Scatterplots can help determine whether

there is

• A positive relationship between two variables

• As variable #1 increases, variable #2 increases

• A negative relationship between two variables

• As variable #1 increases, variable #2 decreases

• A linear relationship between two variables21

5.0

6.0

7.0

8.0

30 40 50 60 70 80

Age

Shows joint distributionof two variables

5.0

6.0

7.0

8.0

30 40 50 60 70 80

More information than two marginal distributions

Do older CEO’s make more than younger CEO’s?

log10com

p

Scatterplot Example

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Time Series

• A time series plots one variable over

time

• Use with continuous variables only

• Time series plots can help determine

whether there is

• A trend in time

• E.g., stock prices are going up or down

• Whether the data cycles in time

• E.g., sales are always up during Christmas

season

About Time Series Data

• Often one observation tells something

about the next observation

• It’s what makes time series (longitudinal)

data interesting

• Later we’ll say that the data are not

“independent”

• How to tell if time series?

• Special knowledge (common sense?)

• Look for trends

• Look for cycles24

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Trends in Data

• Data trends in one direction or another

• Tends to go up or down over time

• Could be a linear trend

• Example:

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30 35 40

time

26-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35 40

Cycles in Data

• Data shows a repeating pattern

• Retail sales often show weekly and annual

cycles (e.g., sales go up on the weekends)

• Length of a cycle is called a period

• Must see several periods to determine a

cycle

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35 40

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Can Have Both Cycles & Trends

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40

Trend Cycle around the trend

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How Does GM’s Stock Change?

30

35

40

45

50

55

Pri

ce

92 93 94

Time

Trends?

Cycles?

How predictable is it from one period to the next?

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How Predictable in the Short-term?

• Use relative change to get some insight:Price Today Price Yesterday

RelChangePrice Yesterday

-0.07

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.07

Re

lCh

ang

e

92 93 94

Time

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A Note on Relative Changes

• Natural log of price ratio:

• For small changes

• See BBS page 33 for an example

LogRelative RelChange

Price TodayLogRelative ln

Price Yesterday

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Mosaic Plots

AerospacedefenseBusinessCapital goodsChemicalsComputersCommConstructionConsumerEnergyEntertainmentFinancialFoodForestHealthInsuranceMetalsRetailingTransportTravelUtility

50 100 150

Count Axis

AerospacedefenseBusinessCapital goodsChemicalsComputersCommConstructionConsumerEnergyEntertainment

Financial

FoodForestHealth

InsuranceMetalsRetailingTransportTravelUtility

Bar Chart Mosaic Plot

(1 dimension)

Mosaic Plots for Two Variables

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Graphical Summaries for

Continuous Variables• Histogram:

• Empirical distribution of contiuous variable

• Center, spread, skewness, bimodality, outliers

• Plots in two dimensions

• Boxplot:• Center spread, skewness, outliers

• Plots in one dimension

• Scatterplot• Plot of one variable against another

• Gives some idea about relationships between the two

• Time series plot• Plot of one variable against time

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Graphical Summaries for

Categorical Variables

• Bar chart:

• Shows frequency of each type of observation

• Two dimensions

• Mosaic chart:

• Stacked bar chart showing proportions

• Can do side by side bars

Notes on Business Stats Reading

• In chapter 2, don’t worry about:

• Details for calculating histograms by hand

• We’ll let the software do the work for us

• Just skim the Grouped Data section

• Histograms with unequal bar widths – ugh!

• Skip stem and leaf plots

• Never used in the real world

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JMP Practice (1)

• Start JMP and load the data by double clicking on Forbes94.jmp dataset

• Reproduce histogram of year of CEO undergraduate degrees (“UGDate”)• Analyze Distribution, highlight UGDate, select

“Y, Columns” & “OK”

• Note the quantiles and moments

• Mean = average = x-bar

• Explore display and histogram options

• How would you create histograms of CEO age (“Age”) and UGDate simultaneously?

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JMP Practice (2)

• Reproduce bar chart of CEO undergraduate degrees (“UGDegree”)

• Analyze Distribution, highlight UGDegree, select “Y, Columns” & “OK”

• It’s a categorical variable (how do you

know?)

• How is the display different?

• What is the mean and standard

deviation for this variable?

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JMP Practice (3)

• Create a scatterplot of CEO age and salary

• Pull down menu: Analyze Fit Y by X

• Highlight Salary, select “Y, Columns”

• Highlight Age, select “X, Factor”

• What does this plot show?

• Are there “outliers”? Can you identify them?

• Convention is X “explains” Y

• How you could simultaneously plot multiple Xs against one Y?

JMP Practice (4)

• Create log transformation of CEO total

compensation (“TotalComp”)

• Create a new variable log10Comp:

• Columns New Column

– Input column name

– Under “Column Properties” choose “Formula”

• Formula dialog box:

– Click on TotalComp

– Click Transcendental Log10

– Once formula appears, click “Apply” and “OK”

• Now reproduce scatterplot from slide 2239

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JMP Practice (5)

• Reproduce side-by-side boxplots of

CEO compensation (log10Comp) by

industry (WideIndustry)

• Pull down menu: Analyze “Fit Y by X”

• Highlight log10Comp, select “Y, Columns”

• Highlight WideIndustry, select “X, Factor”

• Select OK

• Pull down menu under red triangle, select

“Display Options” and “Quantiles”

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What we have learned so far…

• Types of data and why data vary

• Descriptive Statistics

• Numerical summaries of data

• Graphical summaries in one and two

dimensions

• Histograms, boxplots, and scatterplots

• Bar plots and mosaic plots

• JMP software