Statistical Inference: A Short Course

3.9 Quantiles

In general, we can view quantiles as positional measures that divide the observations on a variable X into a number of equal portions (given that the data values are in an increasing sequence). We have already encountered the median of a data set; it is a positional value that splits the data into two equal parts. Others are:

Quartiles: Split the data into four equal parts

Deciles: Split the data into 10 equal parts

Percentiles: Split the data into 100 equal parts

There are three computed quartiles that will divide the observations on a variable X into four equal parts: Q₁, Q₂, and Q₃. In this regard, 25% of all observations on X lie below Q₁; 50% of all observations on X lie below Q₂ ( = median); and 75% of all observations on X lie below Q₃. Remember that quartiles are “positional values.” Hence the following:

(3.12a)

(3.12b)

(3.12c)

(provided, of course, that our data have been arranged in an increasing sequence). Given Equation (3.12), we can easily calculate the interquartile range (IQR) as IQR = Q₃ − Q₁—it is the range between the first and third quartiles and serves to locate the middle 50% of the observations on a variable X. Then the quartile deviation ...