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प्रश्न
The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure.
| Expenditure (in Rs) | Number of families |
| 1000 − 1500 | 24 |
| 1500 − 2000 | 40 |
| 2000 − 2500 | 33 |
| 2500 − 3000 | 28 |
| 3000 − 3500 | 30 |
| 3500 − 4000 | 22 |
| 4000 − 4500 | 16 |
| 4500 − 5000 | 7 |
उत्तर
It can be observed from the given data that the maximum class frequency is 40, belonging to 1500 − 2000 intervals.
Therefore, modal class = 1500 − 2000
Lower limit (l) of modal class = 1500
Frequency (f1) of modal class = 40
Frequency (f0) of class preceding modal class = 24
Frequency (f2) of class succeeding modal class = 33
Class size (h) = 500
Mode = `l+((f_1-f_0)/(2f_1-f_0-f_2))xxh`
=`1500+((40-24)/(2(40)-24-33))xx500`
=`1500+((16)/(80-57))xx500`
=`1500+8000/23`
Therefore, modal monthly expenditure was Rs 1847.83.
To find the class mark, the following relation is used.
`"class mark" =("Upper class limit + Lower class limit")/2`
Class size (h) of the given data = 500
Taking 2750 as assumed mean (a), di, ui, and fiuiare calculated as follows
aking 2750 as assumed mean (a), di, ui, and fiuiare calculated as follows.
| Expenditure (in Rs) |
families fi | xi | di = xi − 2750 |
ui = di/500 |
fiui |
| 1000 − 1500 | 24 | 1250 | −1500 | −3 | −72 |
| 1500 − 2000 | 40 | 1750 | −1000 | −2 | −80 |
| 2000 − 2500 | 33 | 2250 | −500 | −1 | −33 |
| 2500 − 3000 | 28 | 2750 | 0 | 0 | 0 |
| 3000 − 3500 | 30 | 3250 | 500 | 1 | 30 |
| 3500 − 4000 | 22 | 3750 | 1000 | 2 | 44 |
| 4000 − 4500 | 16 | 4250 | 1500 | 3 | 48 |
| 4500 − 5000 | 7 | 4750 | 2000 | 4 | 28 |
| Total | 200 | −35 |
From the table, we obtain
`sumf_i = 200`
`sumf_iu_i = -35`
Mean `barx = a+ ((sumf_iu_i)/(sumf_i))xxh`
`barx=2750+((-354)/200)xx500`
= 2750 - 87.5
= 2662.5
Therefore, mean monthly expenditure was Rs 2662.50.
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