Question

The mean of the following frequency distribution is 35. Find the values of x and y, if the sum of frequencies is 25 :

Hint:

To simplify the Mean calculation for large numbers, you can use the Assumed Mean Method. In this case, choosing \(a=35\) would make the arithmetic much faster.

Answer 1

Step 1: Understanding the Concept:
We use the sum of frequencies and the mean formula. We have two unknowns, so we need two independent equations.
Step 2: Key Formula or Approach:
1. \(\sum f_i = N\)
2. Mean \(\bar{x} = \frac{\sum f_ix_i}{N}\)
Step 3: Detailed Explanation:
1. Equation 1 (Sum of Frequencies): \(1 + x + 5 + 7 + y + 3 + 1 = 25 \implies x + y + 17 = 25 \implies x + y = 8 \dots (1)\). 2. Equation 2 (Mean): Midpoints (\(x_i\)): 5, 15, 25, 35, 45, 55, 65. \(\sum f_ix_i = (1 \cdot 5) + (x \cdot 15) + (5 \cdot 25) + (7 \cdot 35) + (y \cdot 45) + (3 \cdot 55) + (1 \cdot 65)\) \(\sum f_ix_i = 5 + 15x + 125 + 245 + 45y + 165 + 65 = 15x + 45y + 605\). Mean \(= 35 \implies \frac{15x + 45y + 605}{25} = 35\). \(15x + 45y + 605 = 875 \implies 15x + 45y = 270\). Divide by 15: \(x + 3y = 18 \dots (2)\). 3. Solving Equations: Subtract (1) from (2): \((x + 3y) - (x + y) = 18 - 8 \implies 2y = 10 \implies y = 5\). Then \(x + 5 = 8 \implies x = 3\).
Step 4: Final Answer:
The values are \(x = 3\) and \(y = 5\).