# What is the interquartile range of the set of data: 67, 58, 79, 85, 80, 72, 75, 76, 59, 55, 62, 67, 80?

Nov 29, 2017

$I Q R = 19$
(Or 17, see note at end of explanation)

#### Explanation:

The interquartile range (IQR) is the difference between the 3rd Quartile value (Q3) and the 1st Quartile value (Q1) of a set of values.

To find this, we need to first sort the data in ascending order:

55, 58, 59, 62, 67, 67, 72, 75, 76, 79, 80, 80, 85

Now we determine the median of the list. The median is generally known as the number is the "center" of the ascending ordered list of values. For lists with an odd number of entries, this is easy to do as there is a single value for which an equal number of entries are less than or equal and greater than or equal. In our sorted list, we can see that the value 72 has exactly 6 values less than it and 6 values greater than it:

$\textcolor{b l u e}{55 , 58 , 59 , 62 , 67 , 67 ,} \textcolor{red}{72 ,} \textcolor{g r e e n}{75 , 76 , 79 , 80 , 80 , 85}$

Once we have the median (also sometimes referred to as the 2nd Quartile [Q2]), we can determine the Q1 and Q3 by finding the medians of the lists of values below and above the median, respectively.

For Q1, our list (colored in blue above) is 55, 58, 59, 62, 67, and 67. There is an even number of entries in this list, and therefore a common convention to use for finding the median in an even list is to take the two "center most" entries in the list and find their mean [arithmetic average]. Thus:

$Q 1 = \frac{59 + 62}{2} = \frac{121}{2} = 60.5$

For Q2, our list (colored in green above) is 75, 76, 79, 80, 80, and 85. Again, we will find the mean of the two center most entries:

$Q 3 = \frac{79 + 80}{2} = 79.5$

Finally, the IQR is found by subtracting $Q 3 - Q 1$:

$I Q R = Q 3 - Q 1 = 79.5 - 60.5 = 19$

Special note:

Like many things in statistics, there are often many accepted conventions for how to calculate something. In this case, it is common for some mathematicians, when calculating Q1 and Q3 for an even number of entries (such as we did above), to actually include the median as a value in the grouping in order to avoid taking the mean of the sublists. Thus, in that case, the Q1 list would actually be 55, 58, 59, 62, 67, 67, and 72, leading to a Q1 of 62 (rather than 60.5). The Q3 would likewise be calculated to be 79 instead of 79.5, with a final IQR of 17.