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Chapter 32: Statistics – Exercise 32.3



Statistics – Exercise – 32.3 – Q.1



We have to calculate mean deviation from the median. So, first we calculate the median.




	
		
		
		
	
	
		

			CI
			x
			F
			cf
			d = (x – med)
			fd
		

		

			0 – 10
			5
			5
			5
			20
			100
		

		

			10 – 20
			15
			10
			15
			10
			100
		

		

			20 – 30
			25
			20
			35
			0
			0
		

		

			30 – 40
			35
			5
			91
			10
			50
		

		

			40 – 50
			45
			10
			101
			20
			200
		

		

			 
			 
			50
			 
			 
			450
		

	







 



Statistics – Exercise – 32.3 – Q.2(i)




	
		
		
		
	
	
		

			CI
			x
			f
			xf
			d = (x-mean)
			fd
		

		

			0 – 100
			50
			4
			200
			308
			1232
		

		

			100 – 200
			150
			8
			1200
			208
			1664
		

		

			200 – 300
			250
			9
			2250
			108
			972
		

		

			300 – 400
			350
			10
			3500
			8
			80
		

		

			400 – 500
			450
			7
			3150
			92
			644
		

		

			500 – 600
			550
			5
			2750
			192
			960
		

		

			600 – 700
			650
			4
			2600
			292
			1168
		

		

			700 – 800
			750
			3
			2250
			392
			1176
		

		

			 
			 
			50
			17900
			 
			7896
		

	







 



Statistics – Exercise – 32.3 – Q.2(ii)




	
		
		
		
		
	
	
		

			Classes
			fi
			xi
			di
			fidi
			 
			 
		

		

			95 – 105
			9
			100
			-3
			-27
			28.58
			257.22
		

		

			105 – 115
			13
			110
			-2
			-26
			18.58
			241.54
		

		

			115 – 125
			16
			120
			-1
			-16
			8.58
			137.28
		

		

			125 – 135
			26
			130
			0
			0
			1.42
			36.92
		

		

			135 – 145
			30
			140
			1
			30
			11.42
			342.6
		

		

			145 – 155
			12
			150
			2
			24
			21.42
			257.04
		

		

			 
			N = 106
			 
			 
			Total = -15
			 
			Total = 1272.60
		

	







 



Statistics – Exercise – 32.3 – Q.2(iii)




	
		
		
		
	
	
		

			CI
			x
			f
			xf
			d = (x - mean)
			fd
		

		

			0 – 10
			5
			6
			30
			22
			132
		

		

			10 – 20
			15
			8
			120
			12
			96
		

		

			20 – 30
			25
			14
			350
			2
			28
		

		

			30 – 40
			35
			16
			560
			8
			128
		

		

			40 – 50
			45
			4
			180
			18
			72
		

		

			50 – 60
			55
			2
			110
			28
			56
		

		

			 
			 
			 
			 
			 
			 
		

		

			 
			 
			50
			1350
			 
			512
		

		

			 
			Mean
			27
			 
			 
		

		

			Deviation
			10.24
		

	







 



Statistics – Exercise – 32.3 – Q.3



Find the mean deviation from the mean for the data:




	
		
		
		
		
	
	
		

			Classes
			0 -10
			10 – 20
			20 – 30
			30 – 40
			40 – 50
			50 – 60
		

		

			Frequencies
			6
			8
			14
			16
			4
			2
		

	







 



Statistics – Exercise – 32.3 – Q.4



We have to calculate mean deviation from the median. So, first we calculate the median.




	
		
		
		
	
	
		

			CI
			x
			f
			cf
			D = (x – med)
			fd
		

		

			17 – 19.5
			18.25
			5
			5
			20
			100
		

		

			20 – 25.5
			22.75
			16
			21
			15.5
			248
		

		

			26 – 35.5
			30.75
			12
			33
			7.5
			90
		

		

			36 – 40.5
			38.25
			26
			59
			0
			0
		

		

			41 – 50.5
			45.75
			14
			73
			7.5
			105
		

		

			51 – 55.5
			53.25
			12
			85
			15
			180
		

		

			56 – 60.5
			58.25
			6
			91
			20
			120
		

		

			61 – 70.5
			65.75
			5
			96
			27.5
			137.5
		

		

			 
			 
			96
			 
			 
			980.5
		

	




We have N = 96 ⟹ N/2 = 48



The cumulative frequency just greater than N/2 is 59 and the corresponding value of x is 38.25.



Hence, median = 38.25






 



Statistics – Exercise – 32.3 – Q.5



M.D from Median




	
		
		
		
		
	
	
		

			Marks
			Students
			xi
			Cum. Freq
			
			fidi
		

		

			0 – 10
			5
			5
			5
			55/3
			275/3
		

		

			10 – 20
			8
			15
			13
			25/3
			200/3
		

		

			20 – 30
			15
			25
			28
			3-May
			75/3
		

		

			30 – 40
			16
			35
			44
			35/3
			560/3
		

		

			40 – 50
			6
			45
			50
			65/3
			390/3
		

		

			 
			N = 50
			 
			 
			 
			Total = 500
		

	







M.D from mean




	
		
		
		
		
		
	
	
		

			Marks
			Students
			xi
			 
			fidi
			|xi - 27|
			fi|xi - 27|
		

		

			0 – 10
			5
			5
			-3
			-15
			22
			110
		

		

			10 – 20
			8
			15
			-2
			-16
			12
			96
		

		

			20 – 30
			15
			25
			-1
			-15
			2
			30
		

		

			30 – 40
			16
			35
			0
			0
			8
			128
		

		

			40 – 50
			6
			45
			1
			6
			18
			108
		

		

			 
			N = 50
			 
			 
			Total = 40
			 
			Total = 472
		

	







 



Statistics – Exercise – 32.3 – Q.6



Converting the given data into continuous frequency distribution by sub tracing 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.




	
		
		
		
		
		
	
	
		

			Age
			x:
			f:
			Cumulative frequency
			|d:|=|x:-38|
			f:|d:|
		

		

			15.5 – 20.5
			18
			5
			5
			20
			100
		

		

			20.5 – 25.5
			23
			6
			11
			15
			90
		

		

			25.5 – 30.5
			28
			12
			23
			10
			120
		

		

			30.5 – 35.5
			33
			14
			37
			5
			70
		

		

			35.5 – 40.5
			38
			26
			63
			0
			0
		

		

			40.5 – 45.5
			43
			12
			75
			5
			60
		

		

			45.5 – 50.5
			48
			16
			91
			10
			160
		

		

			50.5 – 55.5
			53
			9
			100
			15
			135
		

		

			 
			 
			N = Σf: = 100
			 
			 
			Σf:|d:| = 735
		

	




Clearly, N = 100 ⟹ N/2 = 50.



Cumulative frequency is just greater than N/2 is 63 and the corresponding class is 35.5 - 40.5.



I = 35.5, f = 26, h = 5, F = 37






 



Statistics – Exercise – 32.3 – Q.7




	
		
		
		
		
	
	
		

			Classes
			fi
			xi
			fixi
			|xi - 9.2|
			fi|xi - 9.2|
		

		

			0 – 4
			4
			2
			8
			7.2
			28.8
		

		

			4 – 8
			6
			6
			35
			3.2
			19.2
		

		

			8 – 12
			8
			10
			80
			0.8
			6.4
		

		

			12 – 16
			5
			14
			70
			4.8
			24
		

		

			16 – 20
			2
			18
			36
			8.8
			17.6
		

		

			 
			N = 25
			 
			Total = 230
			 
			Total = 96.0
		

	







 



Statistics – Exercise – 32.3 – Q.8




	
		
		
		
		
	
	
		

			Classes
			fi
			xi
			fixi
			|xi - 14.1|
			fi|xi - 14.1|
		

		

			0 – 6
			4
			3
			12
			11.1
			44.4
		

		

			6 – 12
			5
			9
			45
			5.1
			25.5
		

		

			12 – 18
			3
			15
			45
			0.9
			2.7
		

		

			18 – 24
			6
			21
			126
			6.9
			41.4
		

		

			24 – 30
			2
			27
			54
			12.9
			25.8
		

		

			 
			N = 20
			 
			Total = 282
			 
			Total = 139.8


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