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Question

Find the mean, median and mode of the following data:
Classes:$$0-50$$$$50-100$$$$100-150$$$$150-200$$$$200-250$$$$250-300$$$$300-350$$
Frequency:$$2$$$$3$$$$5$$$$6$$$$5$$$$3$$$$1$$


Solution

 Class intervalMid value
  $$x_i$$ 
Frequency 
  $$f_i$$
$$f_ix_i$$ $$cf$$ 
 $$0-50$$$$35$$ $$2$$ $$50$$ $$2$$ 
 $$50-100$$$$75$$ $$3$$ $$225$$ $$5$$ 
 $$100-150$$$$125$$ $$5$$ $$625$$ $$10$$ 
 $$150-200$$$$175$$ $$6$$ $$1050$$ $$16$$ 
 $$200-250$$$$225$$ $$5$$ $$1127$$ $$21$$ 
 $$250-300$$$$275$$ $$3$$ $$825$$ $$24$$ 
 $$300-350$$$$325$$ $$1$$ $$325$$ $$25$$ 
   $$\sum f_i=25$$ $$\sum f_ix_i=4225$$ 
$$\Rightarrow$$  $$Mean=\dfrac{\sum f_ix_i}{\sum f_i}=\dfrac{4225}{25}=169$$

$$\Rightarrow$$  We have $$N=25$$$. Then, $$\dfrac{N}{2}=12.5$$
$$\Rightarrow$$ So, median class is $$150-200$$.


$$l =$$ lower limit of the modal class

$$h =$$ size of the class intervals

$$f =$$ frequency of the modal class

$$f_1 =$$ frequency of the class preceding the modal class

$$f_2 = $$ frequency of the class succeed in the modal class.


$$\therefore$$  $$l=150,\,h=200-150=50,\,f=6,\,cf=10$$

$$\Rightarrow$$  $$Median=l+\dfrac{\dfrac{N}{2}-cf}{f}\times h$$

$$\Rightarrow$$  $$Median=150+\dfrac{12.5-10}{6}\times 50$$

$$\Rightarrow$$  $$Median=150+\dfrac{125}{6}$$

$$\therefore$$   $$Median=150+20.83=170.83$$.

 $$\Rightarrow$$  Here maximum frequency is $$6$$, then the corresponding class $$150-200$$ is the modal class.

$$\Rightarrow$$  $$l=150,\,h=50,\,f=6,\,f_1=5\,f_2=5$$

$$\Rightarrow$$  $$Mode=l+\dfrac{f-f_1}{2f-f_1-f_2}\times h$$

$$\Rightarrow$$  $$Mode=150+\dfrac{6-5}{2\times 6-5-5}\times 50$$

$$\Rightarrow$$  $$Mode=150+\dfrac{50}{2}$$

$$\therefore$$   $$Mode=150+25=175$$

Mathematics
RS Agarwal
Standard X

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