MEASURES OF DISPERSION- INTERQUARTILE RANGE AND QUARTILE DEVIATION

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

1. 
   

   The "Interquartile Range" is from Q1 to Q3:

   
   

   

Example 1

The Interquartile Range is:

Q3 - Q1 = 8 - 4 = 4

EXAMPLE 2

EXAMPLE 3:

Data set in a table

i	x[i]	Quartile
1	102
2	104
3	105	Q1
4	107
5	108
6	109	Q2 (median)
7	110
8	112
9	115	Q3
10	116
11	118

For the data in this table the interquartile range is IQR = 115 − 105 = 10.

Example 4:

The wheat production (in Kg) of 20 acres is given as: 1120, 1240, 1320, 1040, 1080, 1200, 1440, 1360, 1680, 1730, 1785, 1342, 1960, 1880, 1755, 1720, 1600, 1470, 1750, and 1885. Find the quartile devi ation.

Solution:

After arranging the observations in ascending order, we get
1040, 1080, 1120, 1200, 1240, 1320, 1342, 1360, 1440, 1470, 1600, 1680, 1720, 1730, 1750, 1755, 1785, 1880, 1885, 1960.
            Q1=Value of (n+14)th item
                  =Value of (20+14)th item
                  =Value of (5.25)th item
                  =5th item+0.25(6th item−5th item)=1240+0.25(1320−1240)
            Q1=1240+20=1260
            Q3=Value of 3(n + 1)4th item
                  =Value of 3(20 + 1)4th item
                  =Value of (15.75)th item
                  =15th item+0.75(16th item−15th item)=1750+0.75(1755−1750)
            Q3=1750+3.75=1753.75

Q.D=Q3−Q12=1753.75−12602=492.752=246.875

Example 5:

Calculate the quartile deviation from the data given below:

Maximum Load (short-tons)	Number of Cables
9.3−9.7	2
9.8−10.2	5
10.3−10.7	12
10.8−11.2	17
11.3−11.7	14
11.8−12.2	6
12.3−12.7	3
12.8−13.2	1

Solution:

The necessary calculations are given below:

Maximum Load (short-tons)	Number of Cables (f)	Class Boundaries	Cumulative Frequencies
9.3−9.7	2	9.25−9.75	2
9.8−10.2	5	9.75−10.25	2+3=7
10.3−10.7	12	10.25−10.75	7+12=19
10.8−11.2	17	10.75−11.25	19+17=36
11.3−11.7	14	11.25−11.75	36+14=50
11.8−12.2	6	11.75−12.25	50+6=56
12.3−12.7	3	12.25−12.75	56+3=59
12.8−13.2	1	12.75−13.25	59+1=60

            Q1=Value of (n4)th item =Value of (604)th item = 15th item
            Q1 lies in the class 10.25−10.75
            ∴     Q1=l+hf(n4−c)
            Where l=10.25, h=0.5, f=12, n/4=15 and c=7
            ∴     Q1=10.25+0.512(15−7)=10.25+0.33=10.58

            Q3=Value of (3n4)th item =Value of (3×604)th item = 45th item
            Q3 lies in the class 11.25−11.75
            ∴     Q3=l+hf(3n4−c)
            Where l=11.25, h=0.5, f=14, 3n/4=45 and c=36
            ∴     Q1=11.25+0.514(45−36)=11.25+0.32=11.57

Q.D=Q3−Q12=11.57−10.582=0.992=0.495

References:

Math is fun

Wikipedia
www.emathzone.com