Graphical representation of data

(Using Box-and-Whisker Diagram)

Module Tutor: Rinchen Tshewang,

To Start

-What is median? What are other related positional values?

-Think individually what you know about the box plot.

-Share with your neighbour-Group discussion

BOX-AND-WHISKER PLOT / 5 NUMBER SUMMARY:

They allow people to explore data and to draw informal conclusions when two or more variables are present.

It shows only certain statistics rather than all the data.

Five-number summary is another name for the visual representations of the box-and-whisker plot.

The five-number summary consists of the median, the quartiles, and the smallest and greatest values in the distribution.

Immediate visuals of a box-and-whisker plot are the center, the spread, and the overall range of distribution.

There are two types of box and whisker plot: Traditional box and whisker plot Modified version of the box and whisker plot.

Traditional box and whisker OR The 5 Number Summary

The five number summary is another name for the visual representation of the box and whisker plot.

The five number summary consist of : The median ( 2nd quartile) The 1st quartile The 3rd quartile The maximum value in a data set The minimum value in a data set

Single middle value

Review on the Median

The median is the middle value of a set of data once the data has been ordered.

Example 1. Ugyen hits 11 balls at T/phu driving range. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his/her drives.

85, 125, 130, 65, 100, 70, 75, 50, 140, 95, 70

Median drive = 85 yards

50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 140

Ordered data

Two middle values so take the mean.

Review on The Median

The median is the middle value of a set of data once the data has been ordered.

Example 2. Rinzin hit 12 balls at T/phu driving range. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his/her drives.

85, 125, 130, 65, 100, 70, 75, 50, 140, 135, 95, 70

Median drive = 90 yards

50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 135, 140

Ordered data

Finding the median, quartiles and inter-quartile range.

12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10

4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12

Order the data

Inter- Quartile Range = 9 - 5½ = 3½

Example 3: Find the median and quartiles for the data below.

Lower Quartile = 5½

Q1

Upper Quartile = 9

Q3

Median = 8

Q2

Upper Quartile = 10

Q3

Lower Quartile = 4

Q1

Median = 8

Q2

3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15,

Finding the median, quartiles and inter-quartile range.

6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10 Order the data

Inter- Quartile Range = 10 - 4 = 6

Example 4: Find the median and quartiles for the data below.

4 5 6 7 8 9 10 11 12

MedianLower

QuartileUpper

QuartileLowest Value

Highest Value

BoxWhiskerWhisker

Anatomy of a Box and Whisker Diagram.

Note: plotting the median, lower quartile and upper quartile i.e. the box portion shows the range of middle 50% of the members with the median being the mid-point.

Lower Quartile = 5½

Q1

Upper Quartile = 9

Q3

Median = 8

Q2

4 5 6 7 8 9 10 11 12

4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12

Example 5: Draw a Box plot for the data below

Drawing a Box Plot.

Upper Quartile = 10

Q3

Lower Quartile = 4

Q1

Median = 8

Q2

3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15,

Example 6: Draw a Box plot for the data below

Drawing a Box Plot.

3 4 5 6 7 8 9 10 11 12 13 14 15

Upper Quartile = 180

Qu

Lower Quartile = 158

QL

Median = 171

Q2

Question: Sonam recorded the heights in cm of boys in his class as shown below. Draw a box plot for this data.

Drawing a Box Plot.

137, 148, 155, 158, 165, 166, 166, 171, 171, 173, 175, 180, 184, 186, 186

130 140 150 160 170 180 190cm

2. The boys are taller on average.

Question: The box plots for both boys and girls are shown below. Use the box plots to choose some correct statements comparing heights of boys and girls in the class. Justify your answers.

Drawing a Box Plot.

130 140 150 160 170 180 190

Boys

Girls

cm

1. The girls are taller on average.

3. The girls show less variability in height.

4. The boys show less variability in height.

5. The smallest person is a girl

6. The tallest person is a boy

Problem for the class to do:Suppose you caught 13 fish, during the after-math of the Paro Flood along

the river side and you measured the length of the fish to be: (in cms)

12, 13,5,8,9,20,16,14,14,6,9,12,12Draw a box and whisker plot based on medians.

Assess step by step by step (peer assessment) as display the solution:-

Solution:Step 1: Rewrite the data in order, from smallest length to largest:

5,6,8,9,9,12,12,12,13,14,14,16,20Step 2: Now find the median of all the numbers. Notice that since there

are 13 numbers, the middle one will be the seventh number:i.e. 12

This must be the median (middle number) because there are six numbers on each side.

Step 3: Is to find the lower quartile. This is the middle of the lower six numbers. The exact centre is half-way between 8 and 9 ... which would be 8.5

Step 4: Now find the upper quartile. This is the middle of the upper six numbers. The exact centre is half-way between 14 and 14 ... which must be 14

Now we are ready to start drawing the actual box and whisker diagram

Step 5: Draw an ordinary number line that extends far enough in both directions to include all the numbers in your data: locate the 5 number…

5 10 15 20

Final box and whisker plot:

Short-comings of this plot:

This method of display fails to show if the data does not have a near normal PDF.

For example: Highly skewed and bimodal data are more

difficult to discern using this data display method. The median and the traditional box-and-whisker diagram are often more representative when the data is bimodal.

Homework:

Investigate the Modified Box and whisker plot

Find out the difference between the two types of box and whisker plot

When and where should be use traditional and the modified box and whisker plot?

The End(Tashi Delek &

Have a Great Day)