Case Study Question 08

Mathematics

Chapter 14: Statistics

Class 10

Read the following passage carefully and answer the following questions:

A survey was conducted by an NGO to know the monthly expenditure of families living in slums in Delhi. A total of 200 families were interviewed and it was found that their minimum monthly expenditure was Rs.1000. The result is tabulated as given below:

Monthly Expenditure (in Rs.)Number of Families
Less than 200010
Less than 300027
Less than 400046
Less than 500075
Less than 6000115
Less than 7000140
Less than 8000163
Less than 9000182
Less than 10000200

Question.1.
Find the number of families whose monthly expenditure is more than or equal to Rs. 8000. 

We can see from the given frequency distribution that the number of families having monthly expenditure less than Rs. 8000 is 163 out of a total of 200 families. Therefore, number of families whose monthly expenditure is more than or equal to Rs. 8000 = 200 – 163 = 37 

Question.2.
Find the number of families whose monthly expenditure is in the range Rs. (6000 – 7000).

Let us prepare a frequency distribution table from the given cumulative frequency table as below:

Monthly Expenditure (in Rs.)Number of FamiliesCumulative Frequency
Less than 20001010
Less than 300027-10=1727
Less than 400046-27=1946
Less than 500075-46=2975
Less than 6000115-75=40115
Less than 7000140-115=25140
Less than 8000163-140=23163
Less than 9000182-163=19182
Less than 10000200-182=18200
Total200 

Therefore, number of families whose monthly expenditure is in the range Rs. (6000-7000) is 25

Question.3.
Find the lower limit of median class.

Median class is the class whose cumulative frequency is just greater than half of sum of all frequencies. Here `\frac{N}{2}=100`. As the cululative frequency of the class 5000 – 6000 is 115, which is just greater than 100, therefore the median class is 5000 – 6000 and thus the lower limit of the median class is 5000.

Question.4.
Find the median monthly expenditure of the families as per the frequency distribution table.

The median class is 5000 – 6000. The formula for calculating the median is:
Median = `l+\left(\frac{\frac{n}{2}-cf}{f}\right)\times h`
Where, `l`= lower limit of the median class,
`h` = size of clas interval,
`n` = number of observations
`cf` = cumulative frequency of class preceding the median class,
`f` = frequency of the median class

Using the table,

Monthly Expenditure (in Rs.)Number of FamiliesCumulative Frequency
Less than 20001010
Less than 300027-10=1727
Less than 400046-27=1946
Less than 500075-46=2975
Less than 6000115-75=40115
Less than 7000140-115=25140
Less than 8000163-140=23163
Less than 9000182-163=19182
Less than 10000200-182=18200
Total200 

we have `l`=5000, `h`=1000, `cf`=75, `f`=40 and `n` = 200.
Putting these values in the formula for calculating median, we get
Median = `5000+\left(\frac{\frac{200}{2}-75}{40}\right)\times 1000`
⇒ Median = `5000+\left(\frac{25}{40}\right)\times 1000`
⇒ Median = `5000+625`
⇒ Median = `5625`

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