If you have a population with variance σ^2 = 16 and you take a sample of size n = 4, what is the variance of the sample mean?

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Multiple Choice

If you have a population with variance σ^2 = 16 and you take a sample of size n = 4, what is the variance of the sample mean?

Explanation:
A primary idea is that the variability in the average of several independent observations shrinks as you average more data. For independent observations from a population with variance σ^2, the variance of the sample mean equals σ^2 divided by the sample size n. Here, σ^2 = 16 and n = 4, so Var(sample mean) = 16 / 4 = 4. That means the standard deviation of the sample mean, often called the standard error, is sqrt(4) = 2. This shows why averaging reduces variability: the dispersion around the true mean gets smaller as you average more observations.

A primary idea is that the variability in the average of several independent observations shrinks as you average more data. For independent observations from a population with variance σ^2, the variance of the sample mean equals σ^2 divided by the sample size n. Here, σ^2 = 16 and n = 4, so Var(sample mean) = 16 / 4 = 4.

That means the standard deviation of the sample mean, often called the standard error, is sqrt(4) = 2. This shows why averaging reduces variability: the dispersion around the true mean gets smaller as you average more observations.

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