# BOX PLOTS

A box plot is a display that shows how the values in a data set are distributed, or spread out. To make a box-plot, first find five values for the data set :

• the least value
• the lower quartile — the median of the lower half of the data
• the median
• the upper quartile — the median of the upper half of the data
• the greatest value

The picture given below clearly illustrates a box-plot.

In the above figure, the box spans interquartile range (central 50%).

To draw box-plot graph for a data set, we have to know lower quartile, upper quartile and median.

## Steps to Construct Box and Whisker Plot

To construct box-plot for the given data set, we have to do the following steps.

Step 1 :

Write the observations of the given data set in ascending order.

Step 2 :

Find lower quartile, upper quartile and median using the formulas given below.

Lower Quartile = [⁽ⁿ ⁺ ¹⁾⁄₄]th value

Upper Quartile = [³⁽ⁿ ⁺ ¹⁾⁄₄]th value

Median = [⁽ⁿ ⁺ ¹⁾⁄₂]th value

Here, n = number of observations in the given data set.

Step 3 :

Using lower quartile, upper quartile and median, we have to construct box-plot as given in the above picture.

Example 1 :

Make a box-plot for the data given below.

4.3,  5.1,  3.9,  4.5,  4.4,  4.9,  5.0,  4.7,  4.1,  4.6,  4.4,  4.3,  4.8,  4.4,  4.2,  4.5,  4.4

Solution :

Making box plot :

Step 1 :

Let us write the observations in the data in ascending order.

3.9,  4.1,  4.2,  4.3,  4.3,  4.4,  4.4,  4.4,  4.4,  4.5,  4.5,  4.6,  4.7,  4.8,  4.9,  5.0,  5.1

Step 2 :

Number of observations (n) = 17.

Let us find lower quartile, upper quartile and median.

Lower Quartile :

To find the lower quartile, find the value of ⁽ⁿ ⁺ ¹⁾⁄₄.

= ⁽ⁿ ⁺ ¹⁾⁄₄

= ⁽¹⁷ ⁺ ¹⁾⁄₄

= ¹⁸⁄₄

= 4.5

Lower quartile comes in between 4th and 5th observations.

So, lower quartile is equal to average of 4th and 5th observations.

Lower quartile :

= ⁽⁴.³ ⁺ ⁴.³⁾⁄₂

= .⁶⁄₂

= 4.3

Upper Quartile :

To find the upper quartile, find the value of ³⁽ⁿ ⁺ ¹⁾⁄₄.

= ³⁽ⁿ ⁺ ¹⁾⁄₄

= ³⁽¹⁷ ⁺ ¹⁾⁄₄

= ⁵⁴⁄₄

= 13.5

Upper quartile comes in between 13th and 14th observations.

So, upper quartile is equal to average of 13th and 14th observations.

Upper quartile :

= ⁽⁴.⁷ ⁺ ⁴.⁸⁾⁄₂

= .⁵⁄₂

= 4.75

Median :

To find the upper quartile, find the value of ⁽ⁿ ⁺ ¹⁾⁄₂.

= ⁽ⁿ ⁺ ¹⁾⁄₂

= ⁽¹⁷ ⁺ ¹⁾⁄₂

= ¹⁸⁄₂

= 9

Median is exactly the 9th observation.

So, median = 4.4.

Step 3 :

Using lower quartile, upper quartile and median, we can make box-plot graph as given below.

Example 2 :

The heights of several students are shown.

Make a box-plot for the data.

Solution :

Making box-plot :

Step 1 :

Order the data and find the needed values.

Step 2 :

Draw the box-plot.

Draw a number line that includes all the data values.

On the number line, draw dots above the least value, the lower quartile, the median, the upper quartile, and the greatest value.

## Reflect

In the example, what percent of the data values are included in the box portion ? What percent are included in each of the “whiskers” on the ends of the box ?

50%, 25%, 25%

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