(9) Basic Box-Plot

7/30/2019 (9) Basic Box-Plot

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Applied Statistics and Computing Lab

BASIC BOX-PLOT


Indian School of Business


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Learning goals

What are the components of a basic box-plot?

How is a basic box-plot constructed?

How to interpret it?

What are its salient features?

What are its limitations?

How is it related to Histogram? What is the effect of translation on a box-plot?

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Where does a box-plot fit in?

Exploratory Data-analytic tool for continuous

data

Visual display of certain important summary

statistics

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Why Box-plot?

Useful in Studying

Location

Spread

Distribution

Symmetry

Tail behaviour Skewness

Useful in comparison of different batches of Data, or a batch of data with

factors

Useful to study information of observations at the tails

Easy to compute and draw, yet informative

User-friendly

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Constructing a Basic Box-plot Data set 1. Suppose we have data on a batch (variable)

90, 41, 22, 135, 15, 72, 50, 26, 105

Step 1: Arrange the data in the increasing order:

15, 22, 26, 41, 50, 72, 90, 105, 135

Step 2: Get the Five-point Summary, consisting of (i) the Minimum, (ii) First

quartile ( ), (iii) Median, (iv) Third quartile ( ) and (v) the Maximum

For the above data, the Five-point Summary is:

Minimum= 15

First Quartile= 26

Median= 50

Third Quartile= 90

Maximum= 1355


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Constructing a Basic Box-plot (contd.) Step 3: Draw a boxof length equal to ( ). For now, we can choose the

width as per convenience. The lower and upper hinges of the box represent the

first and third quartiles. (In this case, the width is the vertical distance and hingesare the right and the left extremes)

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Constructing a Basic Box-plot (contd.) Step 4: From the middle of the lower hinge draw a line (parallel to the

lines corresponding to the length of the box) up to the minimum. Similarly

draw a line from the middle of the upper hinge (parallel to the lines

corresponding to the length of the box) up to the maximum. These lines

are called the whiskers.

Step 5: Draw a line at the median parallel to the hinges, dividing the box

into two parts.

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Horizontal and vertical Box-plots

The box-plot we saw is a horizontal box-plot

(here the scale is on the horizontal axis)

One can also have a vertical plot (where the

scale is on the vertical axis) There is no specific advantage of one over the

other, in general

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What can we get from a basic Box-

plot?

We can obviously get the position of the

location measure, spread and where themiddle 50% of observations are concentrated,

in a visual display

After all, a picture is worth a thousand words

What else can we get ?

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What can we get from a basic Box-

plot? (contd.)

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Visuals from Aczel A., Sounderpandian J. Complete business statistics


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Interpretation of the basic box-plot

Thus the distribution of the Dataset 1 is right skewed.

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Basic Box-plot: Features and limitations

Features

Basic box-plot displays the location (median), intervals required for thefirst, second, third and fourth quarters of the data

It visually shows where the middle 50% of data is located

It tells us whether the data is symmetric or left skewed or right skewed

Limitations We cannot get modal information

We cannot identify unusual observations

It is hard to identify the tail behaviour

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Histogram and box-plot

In histogram, the width of the interval is fixed and the height

of the vertical bar is proportional to the (relative) frequency inthat interval

In a box-plot the relative frequency is fixed at 25% and the

intervals correspond to the first, second, third and fourth 25%

of the relative frequencies. More precisely, The left and right whiskers correspond to the first and fourth 25%

The part of the box from the first quartile to the median corresponds

to the second 25%, and

The part of the box from the median to the third quartile corresponds

to the third 25%

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Scores dataset

Comprises 50 students scores in their second

semester exam of the second course inQuantitative methods

We also have their GPA of first semester exam

and scores in 3 minors of the subject

= 60 =

20 = 1,2,3

= ( 10)

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Histogram and box-plot (contd.)

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Box-plot of scoresHistogram of scores


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Histogram and box-plot (contd.)

Box-plot readily gives the information on the 5-point

summary and the information about skewness. However, it is

not possible to get information about the mode(s)

Histogram readily give information about mode(s). But it takes

some effort to extract information about the 5-point summary

Thus the two plots complement each other!

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Effect of translation Consider the box-plot of a variable X. Suppose we translate

from X to Y = aX + b where a is positive, the features of the

box-plot do not change

If we translate from X to Y = aX + b where a is negative, then

the features of the box-plot would be the same as those of

boxplot of (X)

We shall demonstrate this by getting the box-plots of ,

(3 +5) and (-3 +5) for the Scores data set ( is the score

in the 2nd

minor)

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Effect of translation

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Box-plot of (3 +5)Box-plot of Box-plot of (-3 +5)


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R-codes The R-code for box-plot is boxplot(variable

name)

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Thank you

Documents

(9) Basic Box-Plot