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Q6:
The table below shows the daily expenditure on food of 25 households in a locality.
Find the mean daily expenditure on food by a suitable method.
The table below shows the daily expenditure on food of 25 households in a locality.
Find the mean daily expenditure on food by a suitable method.
Solution :
Given: The frequency distribution of daily expenditure on food for 25 households.
| Daily Expenditure (in ₹) | 100-150 | 150-200 | 200-250 | 250-300 | 300-350 |
|---|---|---|---|---|---|
| Number of Households ($f_i$) | 4 | 5 | 12 | 2 | 2 |
To Find: The mean daily expenditure on food using a suitable method.
Methodology: Since the class intervals and frequencies are provided, we will use the Step-Deviation Method to calculate the mean, as it simplifies the arithmetic calculations significantly.
Step 1: Construct the Frequency Distribution Table
Let the assumed mean ($a$) be the midpoint of the central class interval. Here, the class intervals are $100-150, 150-200, 200-250, 250-300, 300-350$.
The class mark ($x_i$) is calculated as: $x_i = \frac{\text{Upper Limit} + \text{Lower Limit}}{2}$.
The class size ($h$) is $150 - 100 = 50$.
Let assumed mean $a = 225$.
The deviation $u_i = \frac{x_i - a}{h}$.
| Class Interval | Frequency ($f_i$) | Class Mark ($x_i$) | $u_i = \frac{x_i - 225}{50}$ | $f_i u_i$ |
|---|---|---|---|---|
| 100-150 | 4 | 125 | -2 | -8 |
| 150-200 | 5 | 175 | -1 | -5 |
| 200-250 | 12 | 225 | 0 | 0 |
| 250-300 | 2 | 275 | 1 | 2 |
| 300-350 | 2 | 325 | 2 | 4 |
| Total | $\sum f_i = 25$ | - | - | $\sum f_i u_i = -7$ |
Step 2: Apply the Step-Deviation Formula
The formula for the mean ($\bar{x}$) using the step-deviation method is:
$\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$
Step 3: Substitute the Values
Given: $a = 225$, $\sum f_i u_i = -7$, $\sum f_i = 25$, $h = 50$.
$\bar{x} = 225 + \left( \frac{-7}{25} \right) \times 50$
[Simplifying the fraction: $\frac{50}{25} = 2$]
$\bar{x} = 225 + (-7 \times 2)$
$\bar{x} = 225 - 14$
$\bar{x} = 211$
Final Answer: The mean daily expenditure on food is ₹ 211.
More Questions from Class 10 Mathematics Statistics EXERCISE 13.1
- Q1: A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. Which method did you use for finding the mean, and why?
- Q2: Consider the following distribution of daily wages of 50 workers of a factory. Find the mean daily wages of the workers of the factory by using an appropriate method.
- Q3: The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency $f$.
- Q4: Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women, choosing a suitable method.
- Q5: In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
- Q7: To find out the concentration of $SO_2$ in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below: Find the mean concentration of $SO_2$ in the air.
- Q8: A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
- Q9: The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
CBSE Solutions for Class 10 Mathematics Statistics
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