(11) notched and variable width box-plots

7/30/2019 (11) Notched and Variable Width Box-Plots

1/16

Applied Statistics and Computing Lab

NOTCHED AND VARIABLE WIDTH BOX-PLOT


Indian School of Business


2/16


Learning goals What is a notched box-plot?

How does one construct such a plot? What is a variable width box-plot?

How does one construct such a plot?

How are they useful? Can one combine these two features of a box-plot?

How does one construct a box-plot for data with factors?

What is its use?

2


3/16


Dataset For this study of Notched and Variable Width box-plots, we

consider a slightly modified version of the scores dataset Suppose the score record is blank for some students, during

some or all the exams

The student could have been absent for the exam

There could be a data entry error

In either case, we do not have 50 scores for each of the exams

3

Variable name First minor Second minor Third minor First semester

GPA

Second

semester

scores

# of

observations

available

48 45 48 47 47


4/16


Notched Box-plots As per Oxford Advanced Learners Dictionary, one of the meanings

of notch is a V-shaped cut in an edge or a surface.

This is used to test whether two or more population medians are

equal at 5% level

In a notched box-plot, a notch appears on either side of the

median. The interval corresponding to the notch is the confidenceinterval for the population median

If the notches of the box-plots of variables in the same frame do

not overlap, then we conclude that the population medians are

different (using a test at 5% level of significance)

4


5/16


Notched box-plots (contd.)

5


6/16


Width of the box If there is only one batch (variable), the width can be arbitrary.

If there are several batches, each having the same number ofobservations, then again the width can be the same for all the

variables.

If there are several batches with varying numbers of observations, it

is desirable that the Box-plots produced in the same frame exhibitthis information.

This can be done using varwidth option in R

When this option is used, the width of each box is proportional

to the square root of the number of observations

6


7/16


Variable width box-plot

7


8/16


What if we combine the features of notches

and variable width, to make a variable width

notched box-plot?

8


9/16


Variable width Notched Box-plot

9


10/16


Comments on the Box-plot

Earlier we remarked that the medians of the three minors

appear to be close. From the preceding plot, it is clear thatthe notch of First.minor does not overlap with those of the

other two

Thus the earlier belief is refuted

The upper end of the notch of the Box-plot of Second.minor

barely coincides with the lower end of the notch of

Third.minor

Thus it cannot be said that the medians of the minors atpopulation level are the same

10


11/16


Box-plot for data with factors

Sometimes we have data on a batch with factors

Research has shown that in the fast-paced world of electronics, thekey factor that separates the winners from the losers is actually

how slowa firm is in making decisions: The most successful firms

take longer to arrive at strategic decisions on product development,

adopting new technologies, or developing new products The following values are the number of months taken to arrive at a

decision, for firms ranked high, medium and low in terms of

Performance:

11

High 3.5 4.8 3 6.5 7.5 8 2 6 5.5 6.5 7 9 5 10 6

Medium 3 5.5 6 4 4 4.5 6 2 9 4.5 5 2.5 7

Low 1 2.5 2 1.5 1.5 6 3.8 4.5 0.5 2 3.5 1 2


12/16


Box-plot for data with factors (contd.)

Notice that in such cases, typically one does analysis of

variance to test the equality of means. Here, the batch is thedata on the number of months taken to arrive at a decision

and the factor is the performance: high, medium and low

In such cases one can use a variable width notched Box-plot

to examine the equality of medians. This can be usedindependently or in conjunction with the analysis of variance

in arriving at meaningful conclusions on the location behavior

of different factors

12


13/16


Box-plot for data with factors (contd.)

13


14/16


Comments on the Box-plot

From the plot it is clear that the medians in the population are

most unlikely to be equal

For the Box-plot for high performance, the notch is within thefirst and third quartiles. However, for the plots corresponding

to low and medium performances, the lower end of the notch

is below the first quartile. Thus the population median could

fall below the observed first quartile in these two cases

It is also worth noting that the sampling variability of the

median (as observed by the length of the notch) is about the

same for the three factors (performance groups).

14


15/16


R-codesPlot R-code

Notched box-plot boxplot(data name, notch=TRUE)

Variable width box-plot install.packages(aplpack)

library(aplpack)

boxplot(data name, varwidth=TRUE)

Variable width notched boxplot boxplot(data name, varwidth=TRUE,

notch=TRUE)

Box-plot for data with factors Boxplot(numeric variable~factor

variable, varwidth=TRUE, notch=TRUE)

15


16/16


Thank you

(11) notched and variable width box-plots

Documents