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.

Midpoint=Lower limit+Upper limit2\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}

Here are the class intervals and their midpoints:

  • 05000 - 500: Midpoint = 0+5002=250\frac{0 + 500}{2} = 250
  • 5001000500 - 1000: Midpoint = 500+10002=750\frac{500 + 1000}{2} = 750
  • 100015001000 - 1500: Midpoint = 1000+15002=1250\frac{1000 + 1500}{2} = 1250
  • 150020001500 - 2000: Midpoint = 1500+20002=1750\frac{1500 + 2000}{2} = 1750
  • 200025002000 - 2500: Midpoint = 2000+25002=2250\frac{2000 + 2500}{2} = 2250
  • 250030002500 - 3000: Midpoint = 2500+30002=2750\frac{2500 + 3000}{2} = 2750
  • 300035003000 - 3500: Midpoint = 3000+35002=3250\frac{3000 + 3500}{2} = 3250
  • 350040003500 - 4000: Midpoint = 3500+40002=3750\frac{3500 + 4000}{2} = 3750
  • 400045004000 - 4500: Midpoint = 4000+45002=4250\frac{4000 + 4500}{2} = 4250

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:

Mean contribution=Midpoint×No. of employees\text{Mean contribution} = \text{Midpoint} \times \text{No. of employees}

\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