Box and Whisker Plot : Explained

A box plot (also known as box and whisker plot) is a type of chart often used in descriptive data analysis to visually show the distribution of numerical data and skewness by displaying the data quartiles (or percentiles) averages.

Box plots show the five-number summary of a set of data: the minimum score, first (lower) quartile, median, third (upper) quartile, and maximum score.

First Quartile (Q1/25th Percentile): It is the middle number between the smallest and median of the dataset.

Median (Q2/50th Percentile): Median is the middle value of the dataset.

Third quartile (Q3/75th Percentile): The middle value between the median and the dataset’s highest value.

Interquartile range (IQR): 25th to the 75th Percentile.

Whiskers: The whiskers go from each quartile to the minimum or maximum. The upper and lower whiskers represent values outside the middle 50% (i.e. the lower 25% of values and the upper 25% of values).

Outliers: Outlier is an observation that is numerically separated from the rest of the data.

Minimum: The lowest value, excluding outliers. “minimum”: Q1 -1.5*IQR

Maximum:  The highest value, excluding outliers. “maximum”: Q3 + 1.5*IQR

Page Contents

An Example of Box and Whisker Plot

Draw a box and whisker plot for the data set {3, 7, 8, 5, 12, 14, 21, 13, 18,50}.

Step 1: Order the data in ascending order.

3,7,8,5,12,14,21,13,18,50

Step 2: Find the median.

The median is the mean of the middle two numbers:

3,7,8,5,12,14,21,13,18,22

$\frac{12+14}{2}= 13$

The median is 13

Step 3: Find the quartiles.

The first quartile is the median of the data points to the left of the median.

3,7,8,5,12

Q1=8

The third quartile is the median of the data points to the right of the median.

14,21,13,18,22

Q3=13

Step 4: Complete the five-number summary by finding the min and the max.

The min is the smallest data point, which is 3. The max is the largest data point, which is 50.

Step 6: Scale and label an axis that fits the datasets.

Interpretation of  Box and Whisker Plot

Normal Distribution or Symmetric Distribution: If a box plot has equal proportions around the median and the whiskers are the same on both sides of the box then the distribution is normal.

Positively Skewed: When the median is closer to the lower or bottom quartile (Q1) then the distribution is positively skewed.

Negatively Skewed: When the median is closer to the upper quartile (Q3) and the whisker is shorter on the upper end of the box, then the distribution is negatively skewed.

Interquartile range (IQR): It is the box plot showing the middle 50% of scores and can be calculated by subtracting the lower quartile from the upper quartile (e.g. Q3−Q1).

Outlier: If a data point is higher than the 1.5*IQR above the upper quartile (Q3), the value will be considered an outlier.

Q3 + 1.5 * IQR

Similarly, if a value is lower than the 1.5*IQR below the lower quartile (Q1), the value will be considered an outlier.

Q1 – 1.5 * IQR

How to create a Box and Whisker plot in SAS?

You can create a BOX Plot in SAS using the SG PLOT procedure. First, let us look at a very simple example. I have used the same data set as above in this article.

We use the VBOX or HBOX Statement in PROC SGPLOT and specify the analysis variable.

data inp;
input var @@;
datalines;
3 7 8 5 12 14 21 13 18 50
;
run;

To create a vertical BOX Plot use the VBOX statement as below.

proc sgplot data=inp;
vbox var;
run;

To create a horizontal BOX Plot use the HBOX statement as below.

proc sgplot data=inp;
hbox var;
run;

Using Category = option

In PROC SGPLOT,  you can specify a categorical variable in the category= option.

proc sgplot data=sashelp.cars;
title "Price by Car Type";
hbox msrp / category=type;
run;

You can see the result from PROC SGPLOT  below. This is a horizontal box plot. The square in the box indicates the group mean. The vertical line inside the box is the median (50’th percentile).

The two vertical lines that constitute the top and bottom of the box also know as the whiskers are the 25’th (Q1) and 75’th (Q3) percentiles respectively.

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Subhro Kar is an Analyst with over five years of experience. As a programmer specializing in SAS (Statistical Analysis System), Subhro also offers tutorials and guides on how to approach the coding language. His website, 9to5sas, offers students and new programmers useful easy-to-grasp resources to help them understand the fundamentals of SAS. Through this website, he shares his passion for programming while giving back to up-and-coming programmers in the field. Subhro’s mission is to offer quality tips, tricks, and lessons that give SAS beginners the skills they need to succeed.

6 thoughts on “Box and Whisker Plot : Explained”

1. Are there different conventions on whisker length (i.e. IQRx1.5, IQRx3, up to 98th percentile). Does outlier identification differ under these different situations?

• IQRX1.5 is one of the methods to identify outliers in a dataset. There can be only 4 Quartiles in a dataset. Q1-Q4, Even you find data 98th percentile of a dataset that will fall under Q4. The upper quartile(Q3) is the number dividing the third and fourth quartile.
So, any data point beyond Q3+1.5⋅IQR is an outlier on a higher side.

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