What is the importance of/an application for interquartile range and quartile deviation?
The interquartile range and the quartile deviation refer to the same thing. They both mean the difference between the third quartile (Q3) and the first quartile (Q1). Both are also called midspread or middle fifty.
Some of its applications include determining the spread of data. It is used in the construction of a box plot. It is a good indicator of spread because it is robust with breakpoint of 25%. A breakpoint percentage indicates the number of incorrect observations, before a parameter starts giving a wrong description of the data set. A 25% breakpoint is robust, as it needs a quarter of the data to be incorrect, before it reflects an incorrect spread.
The IQR is also used to determine outliers to the data set. This is in conjuction with the box plot (or the box-and-whisker plot). Outliers are defined as values that are below Q1-1.5*IQR or above Q3+1.5*IQR. There are other methods that could be used to determine whether outliers can be eliminated from the data set.