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Question

Below is given distribution of profit (in Rs.) per day of a shop in a certain town.
Calculate median profit of shops.
Profit (in Rs.)500 - 10001000 - 15001500 - 20002000 - 25002500 - 30003000 - 35003500 - 4000
No. of shops
818272120188


A
Rs. 1867
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B
Rs. 1967
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C
Rs.2167
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D
Rs.2567
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Solution

The correct option is B Rs.$$ 2167$$
Consider the following table, to calculate median:
 $$c_i$$ $$f_i$$ $$cf$$
 500-1000 8 8
 1000-1500 18 8+18=26
 1500-2000 27 26+27=53
 2000-2500 21 53+21=74
 2500-3000 20 74+20=94
 3000-3500 18 94+18=112
 3500-4000 8 112+8=120
Median $$=l +\dfrac {(\dfrac{N}{2}-cf)}{ f_m}\times c$$
Here, 
Class Interval $$c=500$$
$$N=\Sigma f_i=120$$   
$$\dfrac {N}{2}=60  $$
$$\Rightarrow$$ Median class $$=2000-2500$$
$$\Rightarrow$$Frequency of median class$$ f_m=21$$
Lower boundary of median class $$l=2000$$
previous cumulative frequency of median class $$cf=53$$

$$\therefore$$ Median $$=2000 +\dfrac {(\dfrac{120}{2}-53)}{ 21}\times 500$$

$$\therefore$$ Median $$=2000 +\dfrac {(7)}{ 21}\times 500$$

$$\therefore$$ Median $$=2166.67 \Rightarrow 2167$$ 
Hence. option $$C$$ is correct.

Mathematics

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