Quartiles are the values that divide the given data arranged in ascending order into subdivisions of twenty five percent, fifty percent and seventy five percent. First quartile is the twenty fifth percentile also known as the lower quartile. Second quartile is the 50th percentile also known as the median and the third quartile is the seventy fifth percentile also known as the upper quartile. The lower quartile or the first quartile is the middle value or the median of the first half of the data values arranged in the numerical order. It is denoted as Q1. Q1 =( ¼). (n+1)th value of the data set, here n is the total number of data values. The second quartile denoted by Q2 is the median of the data set arranged in the numerical order. Q2 = median = (1/2)(n+1)th value of the data set. The third quartile or the upper quartile is denoted by Q3. Q3 is the (3/4)(n+1)th value which is the middle value or median of the upper half of the data set.
We know that range of a given data set is the value got by calculating the difference between the highest and lowest values in the data set. So, range = highest score – lowest score. Now Quartile Range is a bit similar to range, it is the difference between the upper quartile (Q3) and lower quartile (Q1). Quartile Range is given as (Q3 – Q1) for a particular data set which is also called the Inter Quartile Range denotes as IQR. So, IQR = (Q3 – Q1). Let us learn about Quartile Deviation which is the absolute measure of dispersion. It is also called the semi Inter Quartile Range and is half of the Inter Quartile Range. It is written as Q.D in short, Q.D. = (1/2) (Q3 – Q1)
For example let us find the lower quartile, median, upper quartile, inter quartile range and quartile deviation of the data set 15, 18, 14, 20, 26, 16, 18.
First arrange the data values in the numerical order:
14, 15, 16, 18, 18, 20, 26
The number of values, n = 7
Lower Quartile = (1/4)(n+1) = (1/4)(7+1) = 8/4 = 2nd value which is 15(Q1) in the data set
Second Quartile = Median = (1/2)(n+1)= (1/2)(7+1)=8/2= 4th value which is 18(Q2)in the data set
Third Quartile= (3/4)(7+1)=3. 8/4 = 3.2 = 6th value which is 20(Q3) in the data set
Inter Quartile Range = IQR = (Q3 – Q1)= (20 – 15) = 5
Semi Inter Quartile Range = Quartile Deviation= Q.D. = (1/2)(IQR)= (1/2) (Q3 – Q1) = 5/2 = 2.5
