What is the standard deviation of the average of 100 independent random variables each with a standard deviation of 6?

Question

BlackTom AI · Accepted Answer

Consider the problem setup: we have 100 independent random variables, each with standard deviation 6. When we form the average of these 100 variables, the variability of the average is determined by the standard error of the mean. For independent variables, the variance of the sum is the sum of variances, so Var(sum) = 100 * 6^2 = 3600. The average is (1/100) times the sum, so Var(average) = Var(sum) / 100^2 = 3600 / 10000 = 0.36. The standard deviation of the average is the square root of this variance, which is sqrt(0.36) = 0.6.

A helpful way to remember the general rule: the standard deviation of the sample mean of n independent identically distributed variables with individual standard deviation sigma is sigma / sqrt(n). Here, sigma = 6 and n = 100, so 6 / sqrt(100) = 6 / 10 = 0.6.

Common pitfalls to avoid: confusing the standard deviation of the sum with that of the average (the sum’s sd would be 6 * sqrt(100) = 60), or assuming dependence between variables changes the result (independence is key for Var(sum) = sum Var). Also note that the result does not depend on the actual means of the variables, only on their standard deviations and independence.

What is the standard deviation of the average of 100 independent random variables each with a standard deviation of 6?简答题

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