Question

If $ C $ is the union of $ A $ and $ B $ , how do you calculate the standard deviation of population $ C $ if you know the standard deviations of $ A $ and $ B $ ? What if $ A $ and $ B $ are of different sizes?

Answer 1

Hint: As we know that the standard deviation measures the spread of the statistical data. The deviation of data from the beverage position or its mean value is measured by its distribution. To find the value of the standard deviation i.e. $ S.D $ we take the square root of the variance. It is represented by $ \sigma . $

Complete step by step solution:
Here in the question we have $ C $ is the union of $ A $ and $ B $ , it can be written as $ C = A \cup B $ .
According to the formulas we have
 $ \operatorname{var} (A + B) = \operatorname{var} (A) + \operatorname{var} (B) + 2\operatorname{cov} (A,B) $ , but since we cannot apply it here.
We can write
 $ \sigma (A + B) = \sqrt {\operatorname{var} (A + B)} $ , as we know that Standard deviation is the square root of the variance. It can be written as
 $ \sqrt {\operatorname{var} } = S.D $ .
We can break the parts inside the square root i.e.
 $ \sigma (A + B) = \sqrt {\operatorname{var} (A) + \operatorname{var} (B)} $
Also according to the question we have $ C = A + B $ , therefore by putting the values and applying the relation of variance and standard deviation as above we can write,
 $ \sigma (C) = \sigma (A) + \sigma (B) $ .
Hence the required answer is $ \sigma (C) = \sigma (A) + \sigma (B) $ .
So, the correct answer is “ $ \sigma (C) = \sigma (A) + \sigma (B) $ ”.

Note: Before solving this kind of questions we should have the proper knowledge of standard deviations, variance and their formulas. It is a numerical value about the variation of the data. Variance tells us the degree of spread of the data. We can write the standard deviation for population is $ \sigma = \sqrt {{\sigma ^2}} $ .