X1, ,Xn are independent with variance 2. What is Var( X) for n observations?

Write the mean as a scaled sum. X = 1n i Xi, so Var( X)= 1n2 Var ( i Xi ). Use independence. Independence gives Var ( i Xi )=n 2.

Variance of the Sample Mean

Statistics

Sampling Distributions

Easy

X_{1}, \dots, X_{n}

are independent with variance

σ^{2}

. What is

Var (X ˉ)

for

n

observations?

Hint 1

Write the sample mean as a scaled sum of the observations.

Hint 2

Use independence so the variance of the sum is the sum of variances.

Show worked solution

1. Write the mean as a scaled sum
$X ˉ = \frac{1}{n} \sum_{i} X_{i}$ , so $Var (X ˉ) = \frac{1}{n ^{2}} Var (\sum_{i} X_{i})$ .
2. Use independence
Independence gives $Var (\sum_{i} X_{i}) = n σ^{2}$ .
$Var (X ˉ) = \frac{n σ ^{2}}{n ^{2}} = \frac{σ ^{2}}{n}$

Answer: sigma^2/n