Answer First
The sample mean is unbiased because its expected value equals the population mean. It is efficient because it has the smallest variance among all unbiased estimators in normal models. It is sufficient because it captures all information about the population mean contained in the sample.
Problem Setup
Let \(X_1, X_2, \ldots, X_n\) be independent observations from a distribution with mean \(\mu\) and variance \(\sigma^2\). The sample mean is:
\[ \bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i \]
Step-by-Step Solution
1. Unbiasedness
The sample mean is unbiased because its expectation equals the population mean:
\[ E[\bar{X}] = \mu \]
This follows from linearity of expectation.
2. Efficiency
The variance of the sample mean is:
\[ Var(\bar{X}) = \frac{\sigma^2}{n} \]
In the normal distribution, the Cramér–Rao lower bound for any unbiased estimator of \(\mu\) is:
\[ \frac{\sigma^2}{n} \]
Because the sample mean achieves this bound, it is the most efficient unbiased estimator of \(\mu\).
3. Sufficiency
For normal data, the joint density factors as:
\[ f(x_1, \ldots, x_n \mid \mu) = g(\bar{X}, \mu)\, h(x_1, \ldots, x_n) \]
By the factorization theorem, this shows that \(\bar{X}\) is a sufficient statistic for \(\mu\). No other statistic contains more information about the mean.
Intuition
The sample mean is powerful because it summarizes the entire sample with one number that preserves the correct center. Averaging reduces noise, making the estimator stable. In normal models, the sample mean captures all information about the mean, making it both sufficient and efficient.
Common Exam Mistakes
- Thinking efficiency means “low variance” (it means minimum possible variance).
- Believing sufficiency holds for all distributions (it does not).
- Confusing unbiasedness with consistency.
- Assuming the sample mean is always the best estimator (not true for heavy‑tailed data).
Why This Matters
The sample mean is the backbone of undergraduate statistics. Confidence intervals, t‑tests, ANOVA, and regression all rely on its unbiasedness, efficiency, and sufficiency. Understanding these properties helps students see why the sample mean appears everywhere in statistical inference.
Final Summary
The sample mean is unbiased because its expectation equals the population mean. It is efficient in normal models because it achieves the Cramér–Rao lower bound. It is sufficient because it captures all information about the mean contained in the sample. These three properties make the sample mean the most important estimator in undergraduate statistics.
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