Question

If n₁ = 10 and n₂ = 5 are the sizes, \(\rm\bar{x}_1\) = 7 and \(\rm\bar{x}_2\) = 4 are the means and σ₁ = 1 and σ₂ = 1 are the standard deviations of two series of data. If combined mean \(\rm\bar{x}\) = 6, then the variance of the combined series with size n₁ + n₂ is equal to:
1. 3
2. 1
3. 2
4. 9

Answer 1

Correct Answer - Option 1 : 3

Given

Two series of data are given

N₁  = 10 and N₂ = 5

X̅₁ = 7 and X̅₂ = 4

σ₁ = 1 and σ₂ = 1

Combined mean = X̅ = 6

x̅₁ and x̅₂ are mean of two series

Formula

Variance of (X₁ + X₂) = [N₁(σ²₁ + D²₁) + N₂(σ²₁ + D²₂)]/(N₁ + N₂)

D₁ = X̅₁ – X̅

D₂ = X̅₂ – X̅

Calculation

⇒ D₁ = 7 – 6 = 1

⇒ D₂ = 4 – 6 = -2

⇒ Var(X₁ + X₂) = [10(1² + 1²) + 5(1² + (-2)²]/(10 + 5)

⇒ (20 + 25)/15

⇒ 45/15

∴ The combined variance is 3

Variance is square of standard deviation = σ²

Coefficient of variation = (σ/x̅) × 100