(6) graphical presentation 2

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Applied Statistics and Computing Lab

GRAPHICAL PRESENTATIONS 2(REPRESENTATION OF NUMERICAL DATA)


Indian School of Business

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Learning Goals Why we use graphs

What are the various types of graphs for

presenting numerical data

Which graph to use in which scenario Graphical Distortion

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Why use Graphs: An example A private insurance firm interested in

marketing its insurance products in region A.To target precisely, needs to know age

distribution. Questions-

In which age group does the highest number of

people lie?

Needs to divide population into 4 different agegroups, to sell 4 different products

It has the following data-

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Data on ages of 507 people23,21,23,26,22,27,29,37,55,53,21,19,20,18,32,20,28,19,23,33,40,28,24,36,23,29,34,31,34,42,45,23,46,26,30,25,20,37,24,36,28,29,23,23,25,24,37,42,30,28,29,39,26,

20,21,20,19,20,40,25,45,28,21,22,19,24,24,20,29,27,27,40,43,22,22,21,24,23,23,45,20,25,25,33,21,23,20,34,20,41,25,32,24,65,28,25,38,23,22,20,35,34,67,38,33,26,25,52,21,32,43,24,28,62,45,40,21,23,30,20,28,41,32,26,37,38,27,23,50,25,23,43,33,22,26,37,32,23,37,23,27,23,27,24,21,25,23,23,46,34,25,29,45,44,35,55,25,31,19,45,34,19,20,29,33,37,21,23,51,31,27,27,37,25,37,33,25,29,25,20,25,28,24,31,25,27,23,20,28,40,21,62,44,49,34,25,29,19,20,20,26,19,36,34,24,27,23,20,28,40,21,62,

44,49,34,25,29,19,20,20,26,19,36,34,24,27,22,22,48,21,27,33,34,54,25,35,22,21,41,23,19,29,27,36,21,20,20,24,35,33,25,45,55,49,30,28,25,23,26,21,26,32,32,32,35,19,26,22,23,25,38,30,43,60,32,26,23,24,21,28,25,20,64,39,27,32,23,24,23,29,44,20,24,42,27,43,37,20,47,45,20,28,21,37,27,26,22,21,62,27,27,22,22,52,42,30,19,19,19,24,21,36,32,52,26,56,30,23,21,44,37,51,38,23,44,26,23,20,44,25,18,22,35,24,25,23,22,24,26,26,28,34,24,33,46,51,25,19,35,19,19,20,41,33,44,19,29,35,33,22,33,

44,29,46,19,30,26,20,32,20,27,22,40,42,29,31,22,29,36,37,25,46,25,43,43,24,24,19,46,29,26,32,29,34,26,34,22,25,41,38,21,34,37,56,28,35,29,22,22,24,36,40,40,37,23,34,20,23,40,20,30,32,30,21,39,37,22,39,49,24,20,40,24,39,32,24,22,20,27,21,26,28,26,18,30,22,30,18,52,25,28,42,23,41,32,22,24,25,27,24,27,31,35,21,36,20,23,19,25,31,32,40,41,36,43,34,26,29,23,45,33,29,29,45,48,19,38,26,48,22,32,44,44,19,32,30,

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Inferences

What can you infer from the data? Practically nothing!

How long before you come up with answers?

Probably the first thing you do, is count the observations for each age.

Note down the observations along with the corresponding age

That makes a frequency table for you!

Frequency table- Just like in categorical data, a frequency table for

discrete numerical data lists each possible value (either individually or

grouped into intervals), the associated frequency and sometimes the

corresponding relative frequency.

Note: Age is, in theory, a continuous variable as it can assume any value.

But here the variable is, age, in whole years, which is discrete.

But 44 distinct values in your data!

Hence frequency table with 44 rows and one frequency column

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So, list individually or group? List Individually or group into intervals?-

In the ages data, there are 44 distinct values. If we list individually, wehave data with 44 rows! Cumbersome to interpret

Insurance company interested in selling 4 different products cateringto the needs of 4 different age groups.

Interested in 4 age categories

In general, depending on need and size of data, decide whether togroup or not ( For discrete data).

For continuous data it is necessary to group.

How to make groups?- Find max and min. Choose suitable classwidth= (max-min)/(desired no of classes), round off to the nextinteger, if decimal. If not, then the next integer

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Construction of Frequency table

Do we have the answers in a minute from this table?

The age group 17-29 has the maximum number of people

We also have the exact number of people in each age group

This same data can be represented pictorially in a number of

ways!

Class Interval Frequency

17-29 298

30-42 142

43-55 56

56-68 10

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Types of Graph

Graphs for presenting Numerical data:

Bar chart (for discrete variable)

Histogram

Frequency Polygon

Ogive

Line Diagram

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Bar Chart (Numerical Data) Graph of the frequency distribution

Similar to bar chart for categorical data Each frequency or relative frequency is represented by

a rectangle centered over the corresponding value (orrange of values for grouped data)

Area of the rectangle is proportional to thecorresponding frequency or relative frequency

We could name the groups group 1, group 2, group 3and group 4 and plot the corresponding frequencies,exactly like in case of categorical data (Exercise)

Conceptually hence there is no difference between thetwo

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Histogram( for continuous

numerical data)-

Graph of the frequency distribution

of continuous data Suppose given the ages of 507

people in continuous form- (Nowage not reported in whole years,can take any value on real line)

We draw histogram instead of barchart

Similar to bar chart for numericaldata except that there are no gaps

between the bars Length of each rectangle represents

frequency of each equal class-interval , so that area representedby histogram= total frequency

If class-intervals are not equal, thenlength represents relativefrequency,(= class frequency/classinterval) then total area enclosed byhistogram=1

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Inferences: Maximum concentration is in the age group 20-25

Gives an idea about shape of the distribution-

for eg, we can say that the distribution of ages is not symmetric, it is highly right skewed

(See module on Skewness and kurtosis)

Extent of spread or variation (see module on dispersion)

What is bin width?- Bin width refers to the length of each class interval.

How to choose bin width?- Well, R chose a bin width for you!

The default bin width in R is given by Sturges Rule

Some other Thumb Rules- Doanes Formula, Rice Rule, Scott Rule, Freedman Diaconis Rule,

All you need to do is specify the option in breaks= in R (see histogram in R-code slide)

For more details on these rules- http://en.wikipedia.org/wiki/Histogram11


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Frequency Polygon (for

representing continuousdata)

A frequency polygon is formed

by plotting the frequencies ofeach class against their

midpoints and joining the points

by straight lines

To get a closed polygon, we take

two additional classes, one at

each end, that have zero

frequencies. ( The midpoints

corresponding to these classes

thus have zero frequencies)

Basically, if superimposed on a

histogram, it joins the midpointsof each rectangular bar by

straight line segments

We draw the frequency polygon

for the ages data over thehistogram itself 13


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Inferences

But is there any additional information you can derive from a frequency polygon, over

and above which the histogram gives?

Not really! In fact histogram gives more information since while it lists the entire

class intervals, a frequency polygon only shows the midpoint. To appreciate fully, lookat a frequency polygon without the corresponding histogram-

In the construction we have made a simplification by drawing the class frequency

corresponding to the mid point of the class interval thereby losing more information14


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Why use Frequency Polygon?

For comparing between two sets of data the corresponding

frequency polygons can be drawn on the same graph

Drawing two histograms on the same diagram for comparison

purposes is confusing

The insurance company is looking at the profitability of

investing in two regions- region A and region B. Region with a higher proportion of 50 plus population

demands more insurance.

The ages.both.regions.csv data gives the ages of a randomsample of 507 people in both region A and in region B

Draw two histograms on the same diagram and try to

compare-

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Why use Frequency polygonQ. What can you infer? Practically nothing, right!

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Why Frequency Polygon (Contd)Draw two frequency polygons on the same diagram and compare.

What can you infer? Can you infer better?

Which region should have the higher insurance demand?17


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Ogives- Cumulative Frequency Curves

Now suppose the insurance company wants answers to more

particular questions- In region A, how many are 50 years or more?

In region A, how many people are 20 years or less?

In region A, how many people are 60 years or more?

(Similar questions for region B)

It wants to design separate products for the age groups 20 or

less, 20-50, 50 and above and a few additional schemes for 60

plus people

Clearly, it needs to know the cumulative frequency for each

age group!

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Ogives for Region A A cumulative frequency curve

or ogive is obtained by plottingcumulative, rather thanindividual class frequencies.There maybe two types ofogives- A curve showing the number

of observations equal to orgreater than the lower classlimit of each correspondingclass- referred to as more

than type ogive A curve showing the number

of observations equal to orless than the upper class limitof each corresponding class-

referred to as less than typeogive

Each successive point is joinedby line segments to give theogive

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Ogives- For Region A

The black plot gives the less than type Ogive

The purple plot gives the more than type Ogive

From diagram insurance company readily has the answers- From the less than type Ogive,

observe that there are 114 people aged less than 22, around 100 people aged less than 20 491 people aged less than 52, roughly 500 people less than 50

From the more than type ogive, we infer there are very few people above 60, something

around 5

Q. Draw the ogives for Region B and try to answer the above questions. Compare with the

age distribution for region A20


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Line Diagram

Year Households

with computer

1985 8.2

1990 15

1994 22.8

1995 24.1

1998 36.6

1999 42.1

2000 51

Source: Falling Through The Net: Toward Digital Inclusion

( U.S Department of Commerce,October 2000)

The ages data is an example of cross section data.

Use any of the above diagrams depending on nature of cross section data.

But what if given a time series data- a series of observations given corresponding to

each time point?

For eg, consider the following data

How to represent this graphically?

Need to represent each value

corresponding to each given year

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Line Chart Plot years on the horizontalaxis and mark the values

corresponding to each year

on the vertical axis

Join the points by linesegments. We have our line

graph ready!

Think: Can we construct a

histogram , ogive or bar chart

with this data? Why or why

not?

Line diagram is meant forrepresenting chronological

data. It exhibits the

relationship of the variable

with time. 22


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Line Chart: Inferences

Shows an increasing trend over the years- that is, from 1985 to 2000, the percentageof households with computers consistently rising

From under 10% in 1985 it has crossed to over 50% in 2000, signifying an over 400%

increase from 1985 to 2000

Useful for analysing time trend- that is, the long-term movement of time series data

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Graphical Distortion of Data

As much as graphs can be used to summarize and represent various aspects of

data succinctly it can also be used to distort data

First might be inadequate representation of data. Consider the following line graph

showing the population above poverty line of a hypothetical country A-

Seeing this graph, we conclude that poverty has been falling in this country as the

number of people above poverty line is rising.

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0

200

400

600

800

1000

1200

1990 1995 2000 2005 2010

People above poverty line

People above poverty line


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Graphical Distortion: Continued

But now, this graph used inadequate information- this is the table from which the

graph has been produced

Draw a line chart showing the relative share of people above poverty line

We see that the relative share of people above poverty line is actually decliningand thus the relative share of people below poverty line is actually rising

Our earlier conclusion, based on representation of inadequate data, led to a

fallacious conclusion

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.70.8

1990 1995 2000 2005 2010

Relative share of people above

poverty line

Relative share of

people above poverty

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Graphical Distortion of data Contd..

The above is just an example. There might be

numerous ways in which data can be misrepresented

For eg, one common misuse might be distortion withscale

With the explosion of data visualization techniques

and sophisticated displays like 3-D charts datadistortion can be easier to achieve

For more information read-

http://lilt.ilstu.edu/gmklass/pos138/datadisplay/bad

chart.htm

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R Codes

Histogramdata=read.csv('ages.continuous.csv',header=TRUE,sep=',')

View(data)

age=data$age

max(age)

colors=c("red", "bisque", "darkslategray", "violet", "orange",

"blue", "pink", "cyan","brown","cornsilk")

# hist for histogram,right=TRUE means right-closed, left-open intervals

hist(age,right=TRUE,col=colors)

# To specify bin widths on your own

bins=seq(17,67,by=5)

hist(age,right=TRUE,breaks=bins,col=colors)

#Example of Histogram with too small binwidth


# Example of Histogram with too large binwidth


hist(age,right=TRUE,breaks=bins,col=colors)

# Drawing a frequency polygon over a histogram


hist(age,right=TRUE,breaks=bins,col=colors,xlim=c(10,75)) # draw the histogram

lines(c(12,seq(22,62,by=10),72),c(0,as.vector(table(cut(age,seq(17,67,by=10)))),0),lwd=2) #draw the frequency

polygon27


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R CodesFrequency Polygon

RegionA.age=data$RegionA.age

RegionB.age=data$RegionB.age

max(RegionA.age)

min(RegionA.age)

max(RegionB.age)min(RegionB.age)

bins.A=seq(17,67,by=10)

bins.B=seq(15,75,by=10)

#To draw two frequency polygons on the same graph

plot(c(12,seq(22,62,by=10),72),c(0,as.vector(table(cut(RegionA.age,seq(17,67,by=10)))),0),type="b",main="Frequency

distribution of age",xlab="age ",ylab="frequency", xlim=c(10,80),ylim=c(0,270))

lines(c(10,seq(20,70,by=10),80),c(0,as.vector(table(cut(RegionB.age,seq(15,75,by=10)))),0),lwd=2,col="violet")

Line Chart

data=read.csv('Households with computer.csv',header=TRUE,sep=',')

household.comp=data$Households.with.computer.percentage

Year=data$Yearx=c(0,0,0,0,0)

y=c(0,0,0,0,0)

plot(x,y,xlab="Year",ylab="Percentage of Households with Computer",type="b",xlim=c(1985,2000),ylim=c(5,65))

lines(Year,household.comp,type="b",col="blue")

title("Line Chart")

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R CodesOgives

min(data)

max(data)

NumberOfClasses = 10

ClassInterval = (67 - 17)/10

ClassInterval

ClassEnds = seq(17,67,5)

classes=cut(data[,1], breaks=ClassEnds)

FrequencyDistribution = table(classes)

CumulativeFrequencies = c(cumsum(FrequencyDistribution))

cbind(CumulativeFrequencies)

#Less than type Ogive

plot(ClassEnds,c(0,as.vector(CumulativeFrequencies)),type="b",xlim=c(10,70),ylim=c(0,700),main="Ogives",xlab="ClassIntervals",y

lab="Cumulative Frequency of Age")

#More than type Ogive

cbind(FrequencyDistribution)Frequency=as.vector(FrequencyDistribution)

cbind(as.vector(FrequencyDistribution))

More.than.cum.freq=cumsum(rev(Frequency))

Upper.limit=rev(ClassEnds)

lines(Upper.limit,c(0,More.than.cum.freq),type="b",col="violet")

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Thank you

Applied Statistics and Computing Lab30

(6) graphical presentation 2

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