How To Build Box and Whisker Plot?

Box and whisker plots are suitable for contrasting distributions since the core, spread, and total array are immediately identifiable. A box and whisker plot summarises a set of data calculated on an interval scale. This technique is also used for the analysis of explanatory data. IIT JAM exam and CUET exam aspirants must be well aware of the concepts. The box and whisker plot is a graph exhibiting data from the five-numbers set, including central tendency metrics. The distribution is not as precise as the results of a histogram or a stem and leaf plot. The following is a box and whisker diagram:

Understanding how to build a box and whisker plot is extremely important for aspirants of the NEET exam and the JEE Main entrance exam since the statistics chapter holds good weightage. Let us understand the steps through which you can build a box and whisker plot:

1. Arrange the given set of data in ascending order, i.e., smallest to largest.
2. Calculate the median of the set of data.
3. Find the quartiles of the data set, i.e., the median of the upper and lower halves.
4. Calculate the extremes of the data, i.e., the greatest and least values.
5. Now, plot the values of median, extremes, and quartiles below a number line. Moreover, you can use a ruler to mark the points at even intervals.
6. Draw the whiskers and the plots.  

The following points highlight the key uses of box and whisker plots:

• Summarising Data: Box plots provide a concise summary of key statistical measures. These display median, quartiles and potential outliers to provide a quick overview of central tendency and data spread.
• Identifying the Outliers: In box plot, the outliers are represented as individual points beyond whiskers which makes it easier to identify anomalies and unusual observations. IISER entrance exam often ask questions around the uses of box and whisker plot. 
• Distribution Comparison: Box plots are useful to compare the distributions of multiple datasets. By placing box plots side by side, it is possible to compare the medians, spreads and ranges of different groups and categories.
• Skewness assessment: Shape of box and position of the median line within the box indicate the skewness of data. If the median is closer to the bottom of box, data might be positively skewed. In case the median is closer to the top, the data might be negatively skewed.
• Spread Visualization and Variability: Length of box (IQR, interquartile range) and length of whiskers provide information about spread and data variability.
• Process Improvement and Quality Control: In quality control, box plots are used for monitoring process performance and identifying variations/deviations from desired specifications.]
• Exploratory Data Analysis (EDA): Box plots allow quick visualisation to understand the underlying data structure, identify patterns and generate hypotheses for further investigation.

Let us understand what is histogram and how it is different from box plot through the following table of comparison between two methods:

Let us go through some Math questions based on the box and whisker plot:

1. Evaluate the first & third quartiles of the set of data {3, 7, 8, 5, 12, 14, 21, 13, 18).

Solution:
Arrange the data in ascending order: 3, 5, 7, 8, 12, 13, 14, 18, 21
The numbers of terms are 9 i.e., odd in number.
 Therefore, Median = 12
 The Q₁ = median of lower half
     = 5+7/2 = 12/2 = 6
The Q₃ = median of upper half 
              = 14+18/2 = 32/2 = 16
 So, Q₁ and Q₃ = 6 and 16

 2. Find the interquartile range and  range of {7, 3, 8, 12, 21, 5, 14, 18, 15, 13, 14}

Solution:
Arrange the data in ascending order: 3, 5, 7, 8, 12, 14, 14, 15, 18, and 21
Range = Maximum – Minimum = 21-3 = 18.
Q₁= 7 and Q₃ = = 15.
Therefore, Interquartile Range = Q₃ = – Q₁ = 15-7 = 8
So, the range =18 and the interquartile range = 8.

3. Find Q₁, Q₂, and Q₃ = for the given data and also draw a box-and-whisker plot. {2,6,7, 11, 8,8,12,13,14,15,22,23}
Solution:

Total number of observations = 12
The Middle terms are 11 and 12.
Therefore, Median Q₂ = 11+12/2 = 23/2 = 11.5
Q₁ =  Lower halves median i.e. (2, 6, 7, 8, 8, 11) = 7.5
Q₃ = Upper halves median i.e. (12, 13, 14, 15, 22, 23) = 14.5
Extremes = Minimum and Maximum values of the data set.
                   i.e., 2 and 23

Let us go through important questions on Box and whisker plot which are important from both school and entrance exam point of view: