Variance of a sum of 100 independent items

Question

You sum 100 independent, identically distributed items, each with variance 3. What is the variance of the sum?

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For a sum of n independent pieces each with variance \sigma^2, the variances simply add: \operatorname{Var}(	ext{sum}) = n\sigma^2 = 100 	imes 3 = 300. The standard deviation of the sum is \sqrt{300} \approx 17.32, which equals \sigma\sqrt{n} = \sqrt{3}	imes 10. Note that the variance grows in direct proportion to n, while the standard deviation grows only like \sqrt{n}. Takeaway. Independent variances add, so the sum of 100 items each with variance 3 has variance 300 and sd about 17.32.

Variance of a sum of 100 independent items

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