Q.Two sets each of 20 observations, have the same standard deviation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the set obtained by combining the given two sets.

Concept: Effect of Scaling Variance — When combining two groups, the pooled variance depends on both the individual variances and the squared difference of the means.

Step 1: For each set, $n_1 = n_2 = 20$, $\sigma_1 = \sigma_2 = 5$, so each variance is $25$.

Step 2: The sum of squares for each set is:

• Set 1: $n_1(\sigma_1^2 + \bar{x}_1^2) = 20(25 + 17^2) = 20(25 + 289) = 20 \times 314 = 6280$
• Set 2: $n_2(\sigma_2^2 + \bar{x}_2^2) = 20(25 + 22^2) = 20(25 + 484) = 20 \times 509 = 10180$

Step 3: Combined sum of squares $= 6280 + 10180 = 16460$. Combined mean $\bar{x} = \frac{20 \times 17 + 20 \times 22}{40} = \frac{340 + 440}{40} = 19.5$. …

Given $n_1=n_2=20$, $\sigma_1=\sigma_2=5$ (so $\sigma_1^2=\sigma_2^2=25$), means $\bar{x}_1=17$, $\bar{x}_2=22$.

Combined mean.

Combined variance (population form):

Here $d_1=-2.5$, $d_2=2.5$, so $d_1^2=d_2^2=6.25$: …