The mean and variance of a group of 100 observations are 8 and 9, respectively. Out of 100 observations, the mean and standard deviation of 60 observations are 10 and 2, respectively. Find the standard deviation of remaining 40 observations?
(A) 4.5
(B) 3.5
(C) 2.5
(D) 1.5

Question

Accepted Answer

> [!TLDR]
> The remaining 40 observations have variance 1.5 (option D).

### Step 1 — Find the missing group mean
Total sum = 100 × 8 = 800; first group sum = 60 × 10 = 600, so remaining sum = 200 and its mean = 200/40 = 5.

### Step 2 — Apply the combined-variance formula
With grand mean 8: d₁ = 10 − 8 = 2, d₂ = 5 − 8 = −3.
$$n\sigma^2 = n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)$$
$$100(9) = 60(4 + 4) + 40(\sigma_2^2 + 9)$$

### Step 3 — Solve for σ₂²
$$900 = 480 + 40\sigma_2^2 + 360 \Rightarrow 40\sigma_2^2 = 60 \Rightarrow \sigma_2^2 = 1.5$$ …

Combined Standard Deviation of Groups

How it works

Method

Quick example

Step 1 — Find the missing group mean

Step 2 — Apply the combined-variance formula

Step 3 — Solve for σ₂²

Step 1 — Find the missing group mean

Step 2 — Apply the combined-variance formula

Step 3 — Solve for σ₂²

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