If n 1= 10 and n 2= 5 are the sizes, \(\rm\bar{x}_1\) = 7 and \(\rm\bar{x}_2\) = 4 are the means and σ 1= 1 and σ 2= 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 n 1+ n 2is equal to:
3
Step 1 — Deviations from combined mean: \(d_1 = \bar x_1 - \bar x = 7-6 = 1\), \(d_2 = \bar x_2 - \bar x = 4-6 = -2\).
Step 2 — Combined variance formula:
\[\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}\]
Step 3 — Substitute:
\[\sigma^2 = \frac{10(1+1) + 5(1+4)}{15} = \frac{20+25}{15} = \frac{45}{15} = 3\]
The variance of the combined series is 3.
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