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 2000 | 10 |
| Less than 3000 | 27 |
| Less than 4000 | 46 |
| Less than 5000 | 75 |
| Less than 6000 | 115 |
| Less than 7000 | 140 |
| Less than 8000 | 163 |
| Less than 9000 | 182 |
| Less than 10000 | 200 |
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 Families | Cumulative Frequency |
|---|---|---|
| Less than 2000 | 10 | 10 |
| Less than 3000 | 27-10=17 | 27 |
| Less than 4000 | 46-27=19 | 46 |
| Less than 5000 | 75-46=29 | 75 |
| Less than 6000 | 115-75=40 | 115 |
| Less than 7000 | 140-115=25 | 140 |
| Less than 8000 | 163-140=23 | 163 |
| Less than 9000 | 182-163=19 | 182 |
| Less than 10000 | 200-182=18 | 200 |
| Total | 200 |
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 Families | Cumulative Frequency |
|---|---|---|
| Less than 2000 | 10 | 10 |
| Less than 3000 | 27-10=17 | 27 |
| Less than 4000 | 46-27=19 | 46 |
| Less than 5000 | 75-46=29 | 75 |
| Less than 6000 | 115-75=40 | 115 |
| Less than 7000 | 140-115=25 | 140 |
| Less than 8000 | 163-140=23 | 163 |
| Less than 9000 | 182-163=19 | 182 |
| Less than 10000 | 200-182=18 | 200 |
| Total | 200 |
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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