Question

E3. Five-Number Summary and Boxplot 14 10 11 26 30 22 23 10 12 18 ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? Maximum Problem 25 20 12 Information. You are given the 28 23 16 data in the table on the right ?? ?? ?? Requirement. Construct the boxplot ?? ?? ?? Solution Step 1. Sort data Q Step 2. Insert min. and max values Step 3. Calculate values for quartiles Step 4. Insert the quartiles Step 5. Draw the box Step 6. Connect the maximum and minimum values Minimum ?? ?? ?? ?? ?? H

Answer 1

To construct box-plot

Given data is as follow

25 20 12 10 26 22 14 10

28 23 16 11 30 23 18 12

Step 1 . To sort data

Sorted data as as follow :-
10 10 11 12 12 14 16 18 20 22 23 23 25 26 28 30

Step 2 . To insert min and max values

From sorted data we can see

Minimum Value min = 10

Maximum Value max = 30

Step 3 . To calculate values for quartile

Now here Total observation n = 16   ( Even )

Sorted data is as follow

10 10 11 12 12 14 16 18 20 22 23 23 25 26 28 30

Medain Q2 = [ (n/2)th obs + (n/2 +1)th obs ] / 2

                 = [ (16/2)th obs + (16/2 +1)th obs ] / 2

                 = [ (8)th obs + (9)th obs ] / 2           { from sorted data }

               = [ 18 + 20 ] / 2

Medain Q2 = 19         

1st Quartile Q1 = [ (n/4)th obs + (n/4 +1)th obs ] / 2

                  = [ (16/4)th obs + (16/4 +1)th obs ] / 2

                  = [ (4)th obs + (5)th obs ] / 2           { from sorted data }

                  = [ 12 + 12 ] / 2

1st Quartile Q1 = 12

3rd Quartile Q3 = [ (3*n/4)th obs + (3*n/4 +1)th obs ] / 2

                   = [ (3*16/4)th obs + (3*16/4 +1)th obs ] / 2

                 = [ (12)th obs + (13)th obs ] / 2           { from sorted data }

               = [ 23 + 25] / 2

3rd Quartile Q3 = 24

Hence we have

Q1 = 12    ;     Q2 = 19   ;    Q3 = 24

Thus five number summary is as follow

Min    Q1    Q2    Q3    Max

10      12   19 24     30

Step 4 . To insert quartiles

Step 5 . To draw boxplot

Step 6 . To connect maximum and minimum values

So final boxplot is as follow

(Note that R-software is used to obtain rough graph given below)

Q1 12 Q2 19 Q3 24 Min 10 Max 30 10 15 20 25 30