Question

Find mean and mode of the following frequency distribution :

Hint:

For mean calculation, if numbers are large, use the 'Assumed Mean Method' to reduce the size of the values and minimize the chance of calculation errors.

Answer 1

Step 1: Calculating the Mean:
Class mid-points (xᵢ):
10, 20, 30, 40, 50, 60

Frequencies (fᵢ):
11, 20, 25, 22, 12, 10

Total frequency:
Σfᵢ = 11 + 20 + 25 + 22 + 12 + 10
= 100

Now compute Σfᵢxᵢ:
= (11×10) + (20×20) + (25×30) + (22×40) + (12×50) + (10×60)
= 110 + 400 + 750 + 880 + 600 + 600
= 3340

Mean = Σfᵢxᵢ / Σfᵢ
= 3340 / 100
= 33.4

Step 2: Calculating the Mode:
Highest frequency = 25
So modal class = 25 – 35

For modal class:
l = 25
f₁ = 25
f₀ = 20
f₂ = 22
h = 10

Mode formula:
Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

= 25 + [(25 − 20) / (50 − 20 − 22)] × 10
= 25 + (5 / 8) × 10
= 25 + 6.25
= 31.25

Final Answer:
Mean = 33.4
Mode = 31.25