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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 $f$.
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency $f$.
Solution :
Given: The frequency distribution of daily pocket allowance of children, where the mean ($\bar{x}$) is Rs 18. The data is as follows:
| Daily Pocket Allowance (Rs) | 11-13 | 13-15 | 15-17 | 17-19 | 19-21 | 21-23 | 23-25 |
|---|---|---|---|---|---|---|---|
| Number of Children ($f_i$) | 7 | 6 | 9 | 13 | $f$ | 5 | 4 |
To Find: The value of the missing frequency $f$.
Step 1: Constructing the Frequency Distribution Table
To calculate the mean using the Direct Method, we first find the class mark ($x_i$) for each interval, where $x_i = \frac{\text{Upper Limit} + \text{Lower Limit}}{2}$.
| Class Interval | Frequency ($f_i$) | Class Mark ($x_i$) | $f_i \cdot x_i$ |
|---|---|---|---|
| 11-13 | 7 | 12 | 84 |
| 13-15 | 6 | 14 | 84 |
| 15-17 | 9 | 16 | 144 |
| 17-19 | 13 | 18 | 234 |
| 19-21 | $f$ | 20 | $20f$ |
| 21-23 | 5 | 22 | 110 |
| 23-25 | 4 | 24 | 96 |
| Total | $\sum f_i = 44 + f$ | - | $\sum f_i x_i = 752 + 20f$ |
Step 2: Applying the Mean Formula
The formula for the mean ($\bar{x}$) using the direct method is given by:
$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
Step 3: Substituting the Values and Solving for $f$
Given $\bar{x} = 18$, we substitute the values from our table:
$18 = \frac{752 + 20f}{44 + f}$
Multiply both sides by $(44 + f)$ to clear the denominator:
$18(44 + f) = 752 + 20f$
Distribute the 18 on the left side:
$792 + 18f = 752 + 20f$
Rearrange the terms to isolate $f$ on one side:
$792 - 752 = 20f - 18f$
$40 = 2f$
Divide by 2:
$f = \frac{40}{2}$
$f = 20$
Final Answer: The missing frequency $f$ is 20.
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.
- 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?
- 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.
- 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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