The box-and-whisker plot (also known as box plot) is an exploratory graphic, created by John W. Tukey, used to show the distribution of a data set. Statistics assumes that data points are clustered around some central value. The "box" in the box-and-whisker plot contains, and thereby highlights, the middle half of these data points.It is more like a histogram and is usually drawn alongside a number line.

Box plot is used to represent differences among populations as we don't assume the distribution to which it belongs which helps us to identify the degree of dispersion (spread) and skewness in the data and also outliers for the given data.They can be drawn vertically as well as horizontally.
The advantage of using boxplot is it takes less space and it is helpful to compare the data.
Box and Whisker Plot

Box and whisker plot is very similar to five number summary, where in five number summary we have the, minimum and maximum value, the median and the quartiles(first and third). However, the visuals of box and whisker plot are the center, the spread, and the overall range of distribution.
Given below are the steps to be followed while constructing a box and whisker plot.
  • For a given data first we have to find the median and the quartiles (lower and upper).
  • Median gives the middle number, if it is an even number then the median is the mean of the two middle numbers.
  • Lower and upper quartiles are the medians of the lower half and upper half respectively.
  • Next we find Inter quartile range(IQR).
  • The inter quartile range gives us the difference between the upper quartile and the lower quartile, which is used to summarize the extent of the spread for the given data.It is the length of the box in the box and whisker plot.

Solved Examples

Question 1: Draw a box-and-whisker plot for the following data set:
90, 94, 53, 68, 79, 84, 87, 72, 70, 69, 65, 89, 85, 83, 72
Arrange the given data starting from the lowest value to the highest value.
Solution:
 
53 65 68 69 70 72 72 79 83 84 85 87 89 90 94

Then we find the median, which is the value exactly in the middle of the data.
   79 is the median

Now we consider the values which is left to the median, 53 65 68 69 70 72 72
From the above we see that 69 is the value which is exactly in the middle from the given set of observations.
Therefore, 69 is the lower quartile (Q1).

On the similar lines to the right of the median we have, 83 84 85 87 89 90 94
Here we see that 87 is in the middle which is the upper quartile.
Therefore, 87 is the upper quartile (Q3).

Now we find the Inter quartile range, IQR = (Q3) - (Q1)
                                        = 87 - 69
                                        = 18
Therefore,18 is the inter quartile range.
From the given data we see that 53 and 94 are the extreme values( lowest and highest).
Given below is the visual display of box and whisker plot.

Box and Whisker Plot Example

We can clearly see that 50% of students have scored between 69 and 87 points while 75% of the students scored lower than 87 points, and 50% scored above 79.
 

Question 2: Draw a box and whisker plot  for 4 17 7 14 18 12 3 16 10 4 4 11
Put the data in order
Solution:
 
3 4 4 4 7 10 11 12 14 16 17 18
In the given data there are 12 points,so the median will be at position $\frac{10+11}{2}$ =10.5
Therefore, Q2 = 10.5

From the above we can see that there are six data points on either side of the median, so Q1 will be 4 and Q3 will be 15
Therefore,lower quartile = 4 and upper quartile = 15

Now IQR = (15-4) = 11.

11 is the  inter quartile range for the given data. Below is the visual plot for the above:

Visual Boxplot