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An AP Statistics problem reads:

Mr. Lopez teaches one statistics class. The girls have a semester average of 83 with standard deviation 9 and the boys have a semester average of 78 with standard deviation 6. Assuming that the students are all independent of one another, what is the standard deviation of the entire class?

The answer according to the book is 10.8 because the book treats boys and girls as random variables. Thus the answer simply took the square root of the sum of the variances, which gave an answer of 10.8.

However, boys and girls are not really random variables in this case. Theoretically there could be 2 boys and 100 girls in the class. Wouldn't that change the answer?

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Why do you say the boys and girls are not random variables? Yes there could be the numbers you quote, but why does that say they are not random. The book is using the fact that variances add as long as the variables are uncorrelated, which is where the root mean square addition for standard deviations comes from.

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    I didn't think we can assume such a thing given the problem. For example, means also add - assuming that these are random variables. However, if there are 50 girls and 2 boys, then the means would not add. I think it would only be appropriate to view them as random variables if the question asked what would be the average score after choosing one random boy and one random girl. No? – 2017-01-25
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    Yes, the means add. We are given the means, so the mean number of total students is $151$ – 2017-01-25
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    But if there are 2 boys with a mean of 78 and 8 girls with a mean of 83, then we can't just add 78 + 83 = 161. Rather, we would need to do: $\frac{(78*2 + 83*8)}{10}= 82 $ – 2017-01-26
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    I just realized I was reading the semester average and standard deviation as applying to the number of students of each sex attending each day, not as test scores. In that reading it is fine to add the variances. As test scores the book is badly wrong. Even if there are an equal number of boys and girls you need to consider the difference in averages of the two sexes. Imagine you had three boys with mean 10 and standard deviation 0 and three girls with mean 90 and standard deviation 0. The class average would be 50 and the standard deviation would be 40. – 2017-01-26