250 apples of a box were weighed and the distribution . . .

Question : 250 apples of a box were weighed and the distribution of masses of the apples is given in the following table : 

Mass (in grams)	80-100	100-120	120-140	140-160	160-180
Number of apples	20	60	70	x	60

i) Find the value of x and the mean mass of the apples. [3 Marks]
ii) Find the modal mass of the apples. [2 Marks]

CBSE 2023

Solution : 

i) Total apples = 250
20+60+70+x+60=250
210+x=250
x=250-210
x=40

Mass (C.I.)	No. of Apples (fi)	Class Marks (xi)	di=xi-a	ui=di/a	fiui
80-100	20	90	-40	-2	-40
100-120	60	110	-20	-1	-60
120-140	70	130 = a	0	0	0
140-160	40	150	20	1	40
160-180	60	170	40	2	120
	Σfi=250				Σfiui=60

Using Step Deviation Method
Mean (x̄) 
= a+(Σfiui/Σfi)×h
= 130+(60/250)×20
=130+(1200/250)
=130+4.8
=134.8 grams

Hence, the mean mass of the apple is 134.8 grams

ii) Modal Mass

Mass (C.I.)	No. of Apples (fi)
80-100	20
100-120	60 - f0
120-140	70 - f1
140-160	40 - f2
160-180	60
	Σfi=250

Here
Maximum Frequency = 70
Modal Class = 120-140
Lower Limit of the model class (l) = 120
Class size (h) = 140-120
=20
f₀=60
f₁=70
f₂=40

Mode = l+[(f₁-f₀)/2f₁-f₀-f₂)]×h
= 120+[(70-60)/(140-60-40)]×20
= 120+[10/(140-100)]×20
= 120+[10/40]×20
= 120+10/2
= 120+5
= 125 grams

Hence, the modal mass of the apple is 125 grams.