Quartile Deviation

Quartile Deviation

Quartile Deviation

For the quartile deviation, let us have an idea about the quartiles. We know that the median divides the given observations arranged in ascending or descending order into two equal parts. Similarly, the quartiles divide the given observations into four equal parts after arranging them in ascending or descending order. So there are three quartiles denoted by Q1, Q2 and Q3 known as the first, the second and the third quartile respectively.

If N be the number of observations of discrete data arranged in the ascending or descending order, then the quartiles are given by the following formula:

Q1 = value of ("N+1" /"4" )^"th" item 	Q2 = value of ("N+1" /"2" )^"th" item 	Q3 = value of ("3" 〖"(N+1)" 〗^"th" )/"4"  item

In case of continuous frequency distribution, the quartiles are calculated by the following formula:

Q1 = L + "i" /"f"  ("N" /"4"  " – cf" )  	Q2 = L + "i" /"f"  ("N" /"2"  " – cf" ) Q3 = L + "i" /"f"  ("3N" /"4"  " – cf" ) Where, L = lower limit of the quartile class    f = frequency of the quartile class           cf = cumulative frequency of preceding class    i = height of class-interval

Inter Quartile Range: The difference between the third and the first quartile is called inter quartile range.

i.e.    Inter Quartile Range = Q3 – Q1

 

Quartile Deviation: The average of deviation of the first quartile and the third quartile taken from median (M) is called the quartile deviation or semi-inter quartile range.

i.e. Quartile Deviation = (("Q" _"3" - "M" )" + (M " -〖" Q" 〗_"1"  ")" )/"2"                  = ("Q" _"3" - "Q" _"1" )/"2"  ∴ Quartile Deviation, Q.D. = ("Q" _"3" - "Q" _"1" )/"2"

Coefficient of Quartile Deviation: As the unit of quartile deviation is same as that of the given observations, the two distributions with different units cannot be compared by quartile deviation. To facilitate the comparison of the quartiles of two or more than two series, a relative measure, the coefficient of quartile deviation is used, which is calculated as,

Coefficient of Quartile Deviation = (("Q" _"3 " -" " "Q" _"1" )/"2" )/(("Q" _"3"  " + " "Q" _"1" )/"2" )   = ("Q" _"3"   - "Q" _"1" )/("Q" _"3"   + "Q" _"1"  ) ∴	Coefficient Q.D. = ("Q" _"3"   - "Q" _"1" )/("Q" _"3"   + "Q" _"1"  )


Workout Examples

Example 1: Calculate the quartile deviation and its coefficient of the following data: 24, 32, 46, 48, 39, 42, 28, 25, 26, 24, 38 Solution: Here, Arranging the data in ascending order,  24, 24, 25, 26, 28, 32, 38, 39, 42, 46, 48 No. of data (N) = 11 ∴	First quartile (Q1) = ("N+1" /"4" )^"th" item 			    = ("11+1" /"4" )^"th" item 			    = 3rd item 			    = 25 ∴	Third quartile (Q3) = ("3" 〖"(N+1)" 〗^"th" )/"4"  item 			    = ("3" 〖"(11+1)" 〗^"th" )/"4"  item 			    = 9th item 			    = 42 ∴	Quartile deviation (Q.D.) = ("Q" _"3"   - "Q" _"1" )/"2"  				   = ("42 " - "25" )/"2"  				   = "17" /"2"  				   = 8.5 ∴	Coefficient of Q.D. = ("Q" _"3"   - "Q" _"1" )/("Q" _"3"   + "Q" _"1"  ) 		                = ("42 " - "25" )/"42 + 25"  			       = "17" /"67"  			       = 0.25 Hence, quartile deviation = 8.5 and its coefficient = 0.25.


Example 2: Calculate the quartile deviation and its coefficient from the following data:	        Ages	12	13	14	15	16	17	18 No. of Students	12	21	15	20	17	10	5 Solution: Here, Ages (in years)	No. of Students (f)	Cumulative frequency (cf) 12 13 14 15 16 17 18	12 21 15 20 17 10 5	12 33 48 68 85 95 100 	N = 100	 Here, N = 100 ∴	First quartile (Q1) = ("N+1" /"4" )^"th" item 		               = ("100+1" /"4" )^"th" item 		               = 25.25th item 		               = 13 ∴	Third quartile (Q3) = ("3" 〖"(N+1)" 〗^"th" )/"4"  item 			    = ("3" 〖"(100+1)" 〗^"th" )/"4"  item 			    = 75.75th item 			    = 16 ∴	Quartile deviation (Q.D.) = ("Q" _"3"   - "Q" _"1" )/"2"  				  = ("16 " -" 13" )/"2"  				  = "3" /"2"  				   = 1.5 ∴	Coefficient of Q.D. = ("Q" _"3"   - "Q" _"1" )/("Q" _"3"   + "Q" _"1"  ) 			    = ("16 " - "13" )/"16 + 13"  			    = "3" /"29"  			    = 0.103 Hence, quartile deviation = 1.5 and its coefficient = 0.103.


Example 3: Calculate the quartile deviation and its coefficient from the following data:	        Marks	0-20	20-40	40-60	60-80	80-100 No. of Students	12	20	25	18	5 Solution: Here, Marks	No. of Students  (f)	Cumulative frequency  (cf) 0-20 20-40 40-60 60-80 80-100	12 20 25 18 5	12 32 57 75 80 	N = 80	 Here, N = 80 Q1 class = ("N" /"4" )^"th" class = ("80" /"4" )^"th" class           = 20th class   = 20 – 40 ∴ L= 20, f = 20, cf = 12, i= 20 ∴ 	Q1 = L + "i" /"f"  ("N" /"4" -"cf" )     = 20 + "20" /"20"  (20 – 12)             = 20 + 8     = 28 Q3 class = ("3N" /"4" )^"th" class = ("3×80" /"4" )^"th" class   = 60th class   = 60 – 80 ∴ L= 60, f = 18, cf = 57, i= 20 ∴ 	Q3 = L + "i" /"f"  ("3N" /"4" -"cf" )                 = 60 + "20" /"18"  (60 – 56)                 = 60 + 3.33                 = 63.33 ∴	Quartile deviation (Q.D.) = ("Q" _"3"   - "Q" _"1" )/"2"  				  = ("63.33 " -" 28" )/"2"  				  = "35.33" /"2"  				  = 17.67 ∴	Coefficient of Q.D. = ("Q" _"3"   - "Q" _"1" )/("Q" _"3"   + "Q" _"1"  ) 		        = ("63.33 " -" 28" )/"63.33 + 28"  			= "35.33" /"91.33"  			= 0.39 Hence, quartile deviation = 17.67 and its coefficient = 0.39.

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