Quartile Deviation for ungrouped data
Quartile Deviation is given by
QD=Q3−Q12
Coefficient of Quartile Deviation is given by
Coefficient of QD=Q3−Q1Q3+Q1
where
- Q1 is the first quartile
- Q3 is the third quartile
The formula for ith quartile is
Qi= Value of (i(n+1)4)th observation, i=1,2,3
where n is the total number of observations.
Example 1
A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital.
5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12.
Find quartile deviation and coefficient of quartile deviation.
Solution
The formula for ith quartile is
Qi= Value of (i(n+1)4)th observation, i=1,2,3
where n is the total number of observations.
Arrange the data in ascending order
5, 6, 8, 9, 9, 10, 10, 10, 10, 12, 12, 13, 13, 14, 15
First Quartile Q1
The first quartle Q1 can be computed as follows:
Q1=Value of (1(n+1)4)th obs.=Value of (1(15+1)4)th obs.=Value of (4)th obs.=9 days.
Thus, 25 % of the patients had length of stay in the hospital less than or equal to 9 days.
Third Quartile Q3
The third quartile Q3 can be computed as follows:
Q3=Value of (3(n+1)4)th obs.=Value of (3(15+1)4)th obs.=Value of (12)th obs.=13 days.
Thus, 75 % of the patients had length of stay in the hospital less than or equal to 13 days.
Quartile Deviation
The quartile deviation is
QD=Q3−Q12=13−92=42=2 days.
Coefficient of Quartile Deviation
The coefficient of quartile deviation is
Coefficient of QD=Q3−Q1Q3+Q1=13−913+9=422=0.1818.
Example 2
Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:
75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.
Find quartile deviation and coefficient of quartile deviation.
Solution
The formula for ith quartile is
Qi= Value of (i(n+1)4)th observation, i=1,2,3
where n is the total number of observations.
Arrange the data in ascending order
72, 73, 73, 73, 75, 75, 76, 76, 78, 78, 79, 80, 82, 83, 84, 85, 86, 87, 97, 99
First Quartile Q1
The first quartle Q1 can be computed as follows:
Q1=Value of (1(n+1)4)th obs.=Value of (1(20+1)4)th obs.= Value of (5.25)th obs.=Value of (5)th obs.+0.25(Value of (6)th obs.−Value of (5)th obs.)=75+0.25(75−75)=75 mg/dl.
Thus, 25 % of the patients had blood sugar level less than or equal to 75 mg/dl.
Third Quartile Q3
The third quartile Q3 can be computed as follows:
Q3=Value of (3(n+1)4)th obs.=Value of (3(20+1)4)th obs.= Value of (15.75)th obs.=Value of (15)th obs.+0.75(Value of (16)th obs.−Value of (15)th obs.)=84+0.75(85−84)=84.75 mg/dl.
Thus, 75 % of the patients had blood sugar level less than or equal to 84.75 mg/dl.
Quartile Deviation
The quartile deviation is
QD=Q3−Q12=84.75−752=9.752=4.875 mg/dl.
Coefficient of Quartile Deviation
The coefficient of quartile deviation is
Coefficient of QD=Q3−Q1Q3+Q1=84.75−7584.75+75=9.75159.75=0.061.