Question:
The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation \(\sqrt{13.44}\), then the standard deviation of the second sample is :
The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation \(\sqrt{13.44}\), then the standard deviation of the second sample is :
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Find the second mean from the combined mean, then use the combined-variance formula \(N\sigma^2=\sum n_i(\sigma_i^2+d_i^2)\).
Updated On: Nov 4, 2026
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The Correct Option is C
Solution and Explanation
Sample 1: \(n_1=100,\ \bar x_1=15,\ \sigma_1=3\). Group: \(N=250,\ \bar x=15.6,\ \sigma^2=13.44\). So \(n_2=150\).
Mean of sample 2: \(100(15)+150\bar x_2=250(15.6)=3900\Rightarrow \bar x_2=\frac{2400}{150}=16\).
Deviations from the combined mean: \(d_1=15-15.6=-0.6\), \(d_2=16-15.6=0.4\).
Combined variance: \(250(13.44)=100(9+0.36)+150(\sigma_2^2+0.16)\).
\(3360=936+150\sigma_2^2+24\Rightarrow 150\sigma_2^2=2400\Rightarrow \sigma_2^2=16\Rightarrow \sigma_2=4\).
Answer: (3) 4
Mean of sample 2: \(100(15)+150\bar x_2=250(15.6)=3900\Rightarrow \bar x_2=\frac{2400}{150}=16\).
Deviations from the combined mean: \(d_1=15-15.6=-0.6\), \(d_2=16-15.6=0.4\).
Combined variance: \(250(13.44)=100(9+0.36)+150(\sigma_2^2+0.16)\).
\(3360=936+150\sigma_2^2+24\Rightarrow 150\sigma_2^2=2400\Rightarrow \sigma_2^2=16\Rightarrow \sigma_2=4\).
Answer: (3) 4
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