Math Problem Statement
Arthimetic mean of salaries
Compute the arithmetic mean of the monthly salaries of employees from the given distribution.
Solution
To compute the arithmetic mean of the salaries from the given distribution, we need to follow these steps:
Step 1: Identify the midpoints (class marks) of each salary class
The midpoint for each class interval is calculated by taking the average of the lower and upper boundaries of each class.
Here are the class intervals and their midpoints:
- : Midpoint =
- : Midpoint =
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- : Midpoint =
- : Midpoint =
- : Midpoint =
- : Midpoint =
- : Midpoint =
Step 2: Multiply the midpoints by the number of employees
Now we multiply each midpoint by the corresponding number of employees in that salary range:
\hline \text{Salary Class} & \text{Midpoint (x)} & \text{No. of Employees (f)} & f \times x \\ \hline 0 - 500 & 250 & 6 & 250 \times 6 = 1500 \\ 500 - 1000 & 750 & 8 & 750 \times 8 = 6000 \\ 1000 - 1500 & 1250 & 12 & 1250 \times 12 = 15000 \\ 1500 - 2000 & 1750 & 25 & 1750 \times 25 = 43750 \\ 2000 - 2500 & 2250 & 25 & 2250 \times 25 = 56250 \\ 2500 - 3000 & 2750 & 20 & 2750 \times 20 = 55000 \\ 3000 - 3500 & 3250 & 12 & 3250 \times 12 = 39000 \\ 3500 - 4000 & 3750 & 8 & 3750 \times 8 = 30000 \\ 4000 - 4500 & 4250 & 2 & 4250 \times 2 = 8500 \\ \hline \end{array}$$ ### Step 3: Sum up the products and the number of employees - Total sum of $$f \times x$$ = $$1500 + 6000 + 15000 + 43750 + 56250 + 55000 + 39000 + 30000 + 8500 = 253000$$ - Total number of employees = $$6 + 8 + 12 + 25 + 25 + 20 + 12 + 8 + 2 = 118$$ ### Step 4: Calculate the arithmetic mean The arithmetic mean is given by: $$\text{Mean} = \frac{\sum(f \times x)}{\sum f}$$ Substituting the values: $$\text{Mean} = \frac{253000}{118} = 2144.07$$ ### Conclusion: The **arithmetic mean** of the salaries is approximately **₹2144.07**. Would you like any more details or have any questions? --- Here are 5 related questions to expand on this topic: 1. How would the mean change if one of the salary ranges had more employees? 2. What is the significance of the midpoint in calculating the mean? 3. How can we use this information to find the median salary? 4. What would happen to the mean if there were a higher range of salaries? 5. How does the arithmetic mean differ from the median in salary distributions? **Tip:** When working with grouped data like this, always remember to verify the midpoints and class intervals before performing calculations to avoid errors.
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Math Problem Analysis
Mathematical Concepts
Statistics
Arithmetic Mean
Grouped Data
Formulas
Midpoint = (Lower limit + Upper limit) / 2
Mean = Σ(f × x) / Σf
Theorems
Arithmetic Mean Theorem
Suitable Grade Level
Grades 8-10