Quartiles and
Box and Whisker Plots
Topic Index | Algebra Index | Regents Exam Prep Center

 

When describing a set of data, without listing all of the values, we have seen that we can use measures of location such as the mean and median.  It is also possible to get a sense of the data's distribution by examining a five statistical summary (or five number summay),

the (1) minimum, (2) maximum, (3) median (or second quartile), (4) the first quartile, and (5) the third quartile.  Such information will show the extent to which the data is located near the median or near the extremes.

 

See how to use your TI-83+/TI-84+ graphing calculator  with five number studies.
Click calculator.

Quartiles

We know that the median of a set of data separates the data into two equal parts.  Data can be further separated into quartiles. Quartiles separate the original set of data into four equal parts. Each of these parts contains one-fourth of the data. 
Quartiles are
percentiles that divide the data into fourths.

The first quartile is the middle (the median) of the lower half of the data.  One-fourth of the data lies below the first quartile and three-fourths lies above.
(the 25th percentile)
The second quartile is another name for the median of the entire set of data.
Median of data set = second quartile of data set.
(the 50th percentile)
The third quartile is the middle (the median) of the upper half of the data.  Three-fourths of the data lies below the third quartile and one-fourth lies above.
(the 75th percentile)

A quartile is a number, it is not a range of values.  A value can be described as "above" or "below" the first quartile, but a value is never "in" the first quartile.

Consider:  Check out this five statistical summary for a set of tests scores.

minimum   first quartile     second quartile (median)      third quartile    maximum
65 70 80  90 100

While we do not know every test score, we do know that half of the scores is below 80 and half is above 80.  We also know that half of the scores is between 70 and 90. 
(The difference between the third and first quartiles is called the interquartile range.)


A five statistical summary can be represented graphically as a

box-and-whisker plot.

The first and third quartiles are at the ends of the box, the median is indicated with a vertical line in the interior of the box, and the maximum and minimum are at the ends of the whiskers.


Box-and-whisker plots are helpful in interpreting the distribution of data.

 

How to construct a box-and-whisker plot:

Construct a box-and-whisker plot for the following data:
The data:
 
Math test scores 80, 75, 90, 95, 65, 65, 80, 85, 70, 100

Write the data in numerical order. Find the first quartile, the median, the third quartile, the minimum (smallest value) and the maximum (largest value).  These are referred to as a five statistical summary.

median (2nd quartile) = 80
first quartile = 70
third quartile = 90
minimum = 65
maximum = 100

Place a circle beneath each of these values in relation to their location on an equally spaced number line.

Draw a box with ends through the points for the first and third quartiles.  Then draw a vertical line through the box at the median point.  Now, draw the whiskers (or lines) from each end of the box to these minimum and maximum values.

 

Special Case:

You may see a box-and-whisker plot, like the one below, which contains an asterisk.

Sometimes there is ONE piece of data that falls well outside the range of the other values.  This single piece of data is called an outlier.  If the outlier is included in the whisker, readers may think that there are grades dispersed throughout the whole range from the first quartile to the outlier, which is not true.  To avoid this misconception, an * is used to mark this "out of the ordinary" value. 


See how to use your TI-83+/TI-84+ graphing calculator  with box and whisker plots.
Click calculator.