Questions: Standard Error of Estimators

Question 1

A researcher triples their sample size from 100 to 900. What happens to the standard error of the sample mean?

• A. It triples — SE grows with sample size
• B. It decreases to one-third of its original value
• C. It decreases to one-ninth of its original value
• D. It is cut in half

Answer: (B)

SE = σ/√n. Going from n = 100 to n = 900 multiplies √n by 3 (√900 = 30 vs √100 = 10), so SE is divided by 3 — it becomes one-third of its original value. This is the √n relationship: to halve SE you need to quadruple n; to reduce SE by a factor of k, you need to multiply n by k².

Question 2

A population has σ = 20. You take a random sample of 100 observations. Which statement correctly distinguishes the sample standard deviation from the standard error of the mean?

• A. Both equal σ = 20 for this sample
• B. The standard error is 20/√100 = 2, measuring how variable x̄ would be across repeated samples; the sample SD is approximately 20, measuring how spread out individual observations are
• C. The SE measures spread of individual observations; the sample SD measures precision of x̄
• D. The SE equals the sample SD divided by the sample size

Answer: (B)

These measure fundamentally different things. The sample standard deviation estimates σ and describes how much individual observations vary from their mean — it lives at the data level. The standard error SE = σ/√n = 20/10 = 2 describes how much the sample mean x̄ would vary across different samples of size 100 — it lives at the estimator level. SE shrinks with more data; sample SD does not necessarily.

Question 3

Halving the standard error of the sample mean requires collecting four times as many observations.

• T. True
• F. False

Answer: True

SE = σ/√n. To halve SE: σ/√n' = (1/2)(σ/√n) → √n' = 2√n → n' = 4n. The inverse-square-root relationship means error reduction becomes progressively expensive: going from SE = 1.0 to SE = 0.5 requires 4× more data; going from SE = 0.5 to SE = 0.25 requires another 4×. This has important practical implications for the cost of increasing precision.

Question 4

An estimator with a very small standard error is necessarily unbiased.

• T. True
• F. False

Answer: False

SE measures precision — how tightly the estimator's values cluster around their own mean across samples. It says nothing about whether that mean equals the true parameter. A biased estimator can be extremely precise (small SE) while consistently overshooting or undershooting the truth. Unbiasedness requires E[θ̂] = θ; SE measures the spread of θ̂ around E[θ̂]. A good estimator needs both small bias and small SE — they are independent properties.

Question 5

Why does the standard error of the sample mean decrease as sample size increases, even though the population variance σ² is fixed and individual observations don't become 'less noisy'?

Model answer: The sample mean averages n independent observations. When random errors are independent, they partially cancel: some are above the truth, some below. Var(x̄) = σ²/n because averaging over n values divides the variance by n. As n grows, more errors are available to cancel each other, so the average error shrinks. The individual observations are just as noisy (σ² is unchanged), but the noise in their average diminishes.

The √n denominator in SE = σ/√n is a consequence of the additivity of variance for independent random variables: Var(X₁ + ... + Xn) = nσ², so Var(x̄) = nσ²/n² = σ²/n. The cancellation intuition is correct — larger samples give positive and negative deviations more chances to offset each other. This is why all of inferential statistics — confidence intervals, t-tests, power calculations — depend critically on sample size.