Boxplot of High School GPA

The five values (2.37, 2.72, 3.11, 3.52, 4.00)
that divide the data into quarters and form the fences and
whiskers of the boxplot
are collectively called the *five-number-summary* of the data. They
are often denoted as MIN, *Q*_{1}, MED, *Q*_{3}, and MAX respectively.

1. MIN is called theminimum, and is the smallest of the ordered observations.

2.Q_{1}is the upper boundary of the first quarter, and is called thefirst quartile.

3. MED is the upper boundary of the second quarter, and is called thesecond quartile. However, it also divides the data into lower and upperhalves, and is more often called themedian.

4.Q_{3}is the upper boundary of the third quarter, and is called thethird quartile.

5. MAX is the largest of the ordered observations and is called themaximum.

Boxplots are quite useful for comparing two distributions side-by-side.
Below, we present a boxplot of second year GPA
alongside the boxplot of high school GPA. The boxplots are presented
vertically this time, but the interpretation remains the same.
Note that there is a slight difference in location
as measured by the medians, but there is a radical difference in
spread between the two distributions. The most noteworthy feature
of 2nd year GPA is the long left tail, which is evidence that some
students are not doing very well in college.
(Note: Some computing packages use a special symbol to denote outlying
values, or *outliers*.
The boxplot for second year GPA has an
extremely low outlier, denoted by a circle. The left (or bottom) whisker
ends at the second smallest observation).

Different statistical computing packages often have different ways of computing
the quartiles. In this class, we compute the quartiles as follows. First, arrange
the observations from smallest (1st ordered observation) to
largest (*n*th ordered observation). Then

If .25(Q_{1}is the .25(n+1)st ordered observation.

MED is the .50(n+1)st ordered observation.

Q_{3}is the .75(n+1)st ordered observation.

2.37 2.43 2.46 2.55 2.57 2.58 2.59 2.60 2.60 2.60 2.61 2.63 2.67 2.71 2.73 2.75 2.78 2.78 2.78 2.79 2.81 2.81 2.82 2.90 2.91 2.93 2.94 3.08 3.14 3.16 3.19 3.20 3.21 3.29 3.32 3.33 3.35 3.36 3.36 3.36 3.44 3.50 3.54 3.54 3.57 3.58 3.60 3.62 3.72 3.73 3.76 3.76 3.77 3.83 3.86 4.00

Since .25(56+1)=14.25, then *Q*_{1} is computed as the average of the
14th and 15th ordered observations (2.71+2.73)/2=2.72. Similarly,
MED=(3.08+3.14)/2 = 3.11, and *Q*_{3}= (3.50+3.54)/2=3.52.