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Question
Two sets each of 20 observations, have the same standard derivation 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.
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Solution
Given that `n_1 = 20, sigma_1 = 5, barx_1 = 17`
And `n_2 = 20, sigma_2 = 5, barx_2 = 22`
Now we know for combined two series that
`sigma = sqrt((n_1s_1^2 + n_2s_2^2)/(n_1 + n_2) + (n_1n_2(barx_1 - barx_2)^2)/(n_1 + n_2)^2`
= `sqrt((20 xx (5)^2 + 20 xx (5)^2)/(20 + 20) + (20 xx 20(17 - 22)^2)/(20 + 20)^2`
= `sqrt(1000/40 + (400 xx 25)/1600)`
= `sqrt(25 + 25/4)`
= `sqrt(125/4)`
= `sqrt(31.25)`
= 5.59
Hence, the required S.D. = 5.59
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