The mean and variance of the marks of Division 1 which has 60 students is 45 and 4 respectively and the mean and variance of the marks of Division 2 which has 40 students is 55 and 9 respectively. What will be the variance of the marks of the two divisions combined?

The mean and variance of the marks of Division 1 which has 60 students is 45 and 4 respectively and the mean and variance of the marks of Division 2 which has 40 students is 55 and 9 respectively. What will be the variance of the marks of the two divisions combined? Correct Answer 30

Given

n₁ = 60

n₂ = 40

x̅₁ = 45

x̅₂ = 55

σ²₁ = 4

σ²₂ = 9

Formula used

Combined mean = X̅ = (n₁ × x̅₁ + n₂ × x̅ ₂)/(n₁ + n₂)

Variance of combined mean = n₁(σ²₁ + d²₁) + n₂(σ²₂ + d²₂)/(n₁ + n₂)

n₁ and n₂ are group 1 and group 2 observation

x̅₁ and x̅₂ are mean of group 1 and group 2

σ²₁ and σ²₂ are variance of group 1 and group 2

d₁ = x̅₁ – X̅

d₂ = x̅₂ – X̅

Calculation

The combined mean is X̅

⇒ (60 × 45 + 40 × 55)/(60 + 40)

⇒ (2700 + 2200)/100

∴ Combined mean is 49

⇒ d₁ = 45 – 49

⇒ d₁ = -4

⇒ d²₁ = 16

⇒ d₂ = 55 – 49 = 6

⇒ d²₂  = 36

The combined variance of both group

⇒ 60(4 + 16) + 40(9 + 36)/(60 + 40)

⇒ (60 × 20 + 40 × 45)/(100)

⇒ (1200 + 1800)/100

⇒ 3000/100

∴ The combined variance is 30