Question

Suppose we have 3 independent observations x₁, x₂, x₃ from a distribution with mean μ and standard deviation σ. What is the variance of the mean of these 3 values: (x₁ + x₂ + x₃)/3?

a) σ²/3
b) σ²/9
c) σ²/2
d) σ²/6

Answer 1

The variance of the mean of three independent observations is σ²/3. The correct answer is option a).a

Step-by-step explanation:

To find the variance of the mean of three independent observations, we need to find the variance of the sum of these observations and then divide it by the square of the sample size (3 in this case). The variance of the sum of these observations is equal to the sum of the variances of each observation, assuming they are independent.

Since the observations are from the same distribution with a standard deviation of σ, the variance of each observation is σ². Therefore, the variance of the sum of these observations is 3σ².

Finally, to calculate the variance of the mean, we divide the variance of the sum by the square of the sample size: (3σ²)/3 = σ²/3.