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Reading Box-and-Whisker Plots

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 Box plot (also called as box and whisker plot) is a basic method to represent a given data on a graph.

We draw a rectangle on it to show the second and third quartiles, with a vertical line within to show the median value. The lower and upper quartiles are shown as horizontal lines on both sides of the rectangle. This can be drawn both horizontally and vertically.

reading box and whisker plots

Basic terms relating to graphs, box and whisker plots:

  1. Whiskers

Whiskers are spread over a wider range apart from the quartile groups. From the minimum to the maximum data represents the whisker area.

  1. Median
    The middle quartile is the mid-point of the data and is shown by the line that divides the box into two parts. This is called the median. Median is also called Quartile 2 or Q2.

Half of the data is greater than or equal to this value and half is lesser than this value.

  1. Lower Quartile or First Quartile
    It is the median of the lower half of the data set. 25% of values in the data set fall below the lower quartile (Q1) as shown in the above diagram.
  2. Upper Quartile or Third Quartile
    It is the median of the upper half of the data set. 75% of values in the data set fall below the upper quartile (Q3) as shown in the above diagram.
  3. Inter-quartile Range

The difference between the lower and upper quartiles is called as the inter-quartile range. IQR = Q3 – Q1

reading box and whisker plots

Example 1:

Find the median, quartile 1, quartile 3 for the given set of data.

9, 7, 8, 15, 14, 13, 11, 16

Solution:

Arrange the given data in ascending order: 7, 8, 9, 11, 13, 14, 15, 16

reading box and whisker plots

 For such data without range we can just split the data into respective quartiles and get the answer.

Example 2:

What is the inter quartile range of the given data set?
3, 5, 6, 9, 11, 15, 16, 18, 21, 23, 25, 27

Draw the box and whisker plot for the above data.

Solution:

Step 1: Arrange the data from least to greatest.

3, 5, 6, 9, 11, 15, 16, 18, 21, 23, 25, 27

Median of the above data is the middlemost data \left ( \frac{n +1}{2} \right )^{th} value.

We can see that, no. of items, n = 12.

Hence, median is the sum of 6th and 7th value in the given data set, divided by 2.

So, median, Q2 =\frac{15+16}{2} =\frac{31}{2}= 15.5

Lower quartile, Q1 = it’s the median value of the lower half of the data, 3, 5, 6, 9, 11, 15

Median of the above data =  \frac{6+9}{2}= \frac{15}{2}= 7.5

Upper quartile, Q3 = it’s the median value of the data set, 16, 18, 21, 23, 25, 27

Median of 16, 18, 21, 23, 25, 27 = \frac{21+23}{2}= \frac{44}{2} = 22

So, interquartile range (IQR) = Q3 – Q1 = 22 – 7.5 = 14.5

The “box” goes from Q1 to Q3, with a line drawn inside the box to show the location of the median, Q2.

Then whiskers are drawn to the endpoints (the lowest and the highest value in the given data set).

reading box and whisker plots

 

 Note: Box and whisker plot can be drawn horizontal, as drawn above and vertical also.

Check Point

  1. The following numbers is the data of people who performed the different hours of voluntary service done in a week by teen age kids in a particular area. Find the median, and inter quartile range.

4, 6, 3, 2, 8, 5, 9, 11, 22, 13, 15, 5, 8

 

  1. From the following data

15, 20, 25, 30, 35, 40, 45, 50, 55, 60

Find the

a.Lower quartile

b.Upper quartile

c.Inter quartile range

d.Represent it graphically

Answer key
  1. Median is 8. Inter quartile range is 7.5 (IQR = Q3 – Q1 = 12 – 4.5 = 7.5)

2.

a.LQ or Q

1 = 22.5

b.UQ or Q3 = 52.5

c.IQR = 30

d.

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