Reading Box and Whisker Plots

Box plot (also called ‘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.

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Basic Terms

Mentioned below are the basic terms relating to Graphs, Box and Whisker plots:

1. Whispers

Whiskers are spread over a wider range apart from the quartile groups.

From the minimum to the maximum data represents the whisker area.

2. 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.

3. 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.

4. 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.

5. Inter-quartile Range

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

Steps in Reading Box and Whisker Plots

Solved Examples

Example 1:

Find the median, quartile 1, and 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

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  value.

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

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

So, median, Q₂ = == 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 =  = = 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 = =  = 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).

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

Check Point

1. The following numbers are the data of people who performed the different hours of voluntary service done in a week by teenage kids in a particular area. Find the median and interquartile range.

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

2. 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