What's the variance of the average of n points of X.

All X's are independent and identically distributed.

That is, what's V(xbar)?

Question

What's the variance of the average of n points of X.

All X's are independent and identically distributed.

That is, what's V(xbar)?

BlackTom AI · Accepted Answer

First, restate the problem in my own words to ensure clarity: we have n independent and identically distributed random variables X1, X2, ..., Xn all distributed like X, and we consider the sample mean Xbar = (X1 + X2 + ... + Xn)/n. We want Var(Xbar).

Because the Xi are independent, the variance of their sum is the sum of their variances: Var(X1 + X2 + ... + Xn) = Var(X1) + Var(X2) + ... + Var(Xn) = n Var(X).

Now, Xbar is the sum divided by n, so Var(Xbar) = Var((1/n)(X1 + X2 + ... + Xn)) = (1/n^2) Var(X1 + X2 + ... + Xn).

Substituting the previous result gives Var(Xbar) = (1/n^2) · n Var(X) = Var(X) / n.

Key point: the derivation relies on two standard facts—(1) variance of a sum of independent variables adds, (2) scaling a random variable by a constant scales its variance by the square of that constant. Applying these to the sample mean yields Var(Xbar) = Var(X)/n.

In words, as you average more IID observations, the variability of the average shrinks at a rate of 1/n, assuming the underlying variance Var(X) exists and is finite.

Note on the provided answer field: the single option given—"1/n * V(X)"—is consistent with the derived result, since V(X) denotes Var(X). If alternatives were present, they would need to distinguish between Var(X)/n and expressions like Var(X)/n^2, which would be incorrect, or between Var(X) and Var(Xbar), which would be misinterpretations of the scaling behavior.

If you want to connect to practical intuition: averaging independent observations reduces noise, and the amount of reduction depends inversely on the number of observations, here captured by the factor 1/n in the variance expression.

ERMCPS5350_006_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 7. Quiz for Class 7

What's the variance of the average of n points of X. All X's are independent and identically distributed. That is, what's V(xbar)?

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