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 are given a grouped frequency distribution with two unknown frequencies x and y.
Total frequency (N) is 25 and mean is 35.
Since there are two unknowns, we form two independent equations:
1) Using total frequency
2) Using mean formula

Step 2: Forming First Equation (Sum of Frequencies):
Total frequency = 25

1 + x + 5 + 7 + y + 3 + 1 = 25
x + y + 17 = 25
x + y = 8 → (1)

Step 3: Forming Second Equation (Using Mean Formula):
Mean formula:
Mean = (Σfᵢxᵢ) / N

Midpoints (xᵢ) are:
5, 15, 25, 35, 45, 55, 65

Now calculate Σfᵢxᵢ:
= (1×5) + (x×15) + (5×25) + (7×35) + (y×45) + (3×55) + (1×65)
= 5 + 15x + 125 + 245 + 45y + 165 + 65
= 15x + 45y + 605

Given mean = 35
(15x + 45y + 605) / 25 = 35

Multiply both sides by 25:
15x + 45y + 605 = 875
15x + 45y = 270

Divide entire equation by 15:
x + 3y = 18 → (2)

Step 4: Solving the Two Equations:
Equation (1): x + y = 8
Equation (2): x + 3y = 18

Subtract (1) from (2):
(x + 3y) − (x + y) = 18 − 8
2y = 10
y = 5

Substitute y = 5 into (1):
x + 5 = 8
x = 3

Final Answer:
x = 3
y = 5