# Constructing a Boxplot

Here is the data; it is necessary to sort the data.

08820 10800 12000 12500 13000 14000 15000 16000 16500 16600 16700 16900 16900 17000 17000 17600 17880 18000 18000 18000 18000 18000 18000 18000 18000 18000 18000 18500 18680 19100 20000 20000 20000 20000 20000 20300 20900 22000 23000 23000 23000 23000 23400 24000 25000 25000 26000 26000 27000 30000 30000 32500 37000 48000

Begin by finding values for the median, first quartile and third quartile.

 M=18250 Q1=17000 Q3=23000

Use the quartiles to find the IQR.

IQR = Q3 - Q1 = 23000-17000 = 6000

Your measurement axis should extend from slightly below the minimum to slightly above the maximum. Place tick marks at fairly round values (I've used 10000, 20000, ...).

Draw a rectangle with ends at the quartiles. Place a bar through the rectangle at the median.

Compute 1.5 × IQR. On each side of the box extend a "whisker" to the most extreme observations within 1.5 × IQR of the appropriate quartile.

1.5 × IQR = 1.5 × 6000 = 9000

Let's do the left side of the box first. Q1 = 17000. Going 9000 below this gets us to 17000 - 9000 = 8000. The most extreme value within 9000 of 17000 is the most extreme value above 8000. Of course, all the values are above 8000. So, the most extreme value above 8000 is the smallest one--8820. We draw the left-side whisker to 8820.

Now, the right side. Q3=23000. Going 9000 above Q3 gets us to 23000 + 9000 = 32000. Scroll back up to the data set and notice that 3 values (colored red) are above 32000; the most extreme value within 9000 of 23000---less than 32000---is 30000 (colored yellow blue in the data set above). The right-side whisker is drawn to 30000.

Complete the boxplot by plotting each point outside these extremes with a special symbol. These special symbols indicate outliers. Here that's the three red values from the data set: 32500, 37000 and 48000. Note that there are no outliers detected to the left.

There's the finished product!