Given two unbiased estimators of the same parameter, how do you choose between them?

## Efficiency is the tiebreaker Unbiasedness fixes where an estimator is centred and says nothing about how far it typically strays. When two estimators are both centred on the same parameter, the comparison collapses to variance, and the smaller-variance one is called **more efficient**. The **relative efficiency** of estimator B with respect to estimator A is ``` RE = Var(A) / Var(B) ``` at a given sample size — greater than 1 means B is tighter. Because for unbiased estimators mean squared error equals variance, ranking by variance is the same as ranking by mean squared error here. An equivalent and more intuitive reading is in units of data: if B has relative efficiency 0.64 against A, then B needs roughly 1 / 0.64 = 1.57 times as many observations to match A's precision. Efficiency is a sample-size exchange rate. ## The canonical comparison: mean versus median For a **symmetric** population, both the sample mean and the sample median are centred on the same point, so they are genuinely competing estimators of the same parameter and the comparison is fair. - **Normal population.** The sample mean has variance `sigma^2 / n` exactly. The sample median has variance approximately `(pi / 2) * sigma^2 / n`, about `1.57 * sigma^2 / n`, in large samples. So the median's efficiency relative to the mean is about `2 / pi`, roughly 0.64. Under normality the mean wins decisively, and no estimator of the centre does better. - **Heavy-tailed population.** The picture reverses. Occasional very large observations contribute enormously to the mean's variance while shifting the median by at most one rank position. For a double-exponential population the median is the more efficient of the two. For a Cauchy population the situation is extreme: the sample mean of n Cauchy observations has the same distribution as a single observation, so it is not even consistent, while the sample median converges normally. The headline is therefore: **efficiency is conditional on the distribution**. "Is the mean or the median better?" has no answer until you say what the data looks like. ## The robustness dimension Beyond variance under a clean assumed distribution, there is the question of what happens when the assumption is slightly wrong — a small proportion of contaminated or mis-recorded values. The mean is maximally sensitive: one arbitrarily large value drags it arbitrarily far. The median tolerates almost half the sample being corrupted before it can be dragged to an arbitrary value. That is why practitioners often accept the mean's roughly 36% efficiency advantage being given up: they are buying insurance against contamination they cannot rule out. A middle path exists. A trimmed mean — discarding a fixed fraction from each end and averaging the rest — sits between the two, keeping most of the mean's efficiency under normality while bounding the damage from a few extreme values. ## Careful: are they estimating the same thing? The entire comparison assumed symmetry. For a **skewed** population the mean and the median are different quantities: the mean of a right-skewed income distribution exceeds its median, and neither is "wrong". Comparing their variances as if they were rival estimators of one parameter is then a category error. The first question in any mean-versus-median discussion is not "which is more efficient" but "which quantity does the decision actually need" — total budget requires a mean, a typical customer's experience is usually better served by a median. ## The practical checklist When an interviewer hands you two unbiased estimators, work through: 1. **Do they target the same quantity?** If the population is skewed, the mean and median do not. 2. **Which has smaller variance under the distribution you believe?** That is efficiency, and it can be quoted as a sample-size exchange rate. 3. **How badly does the winner degrade if that belief is wrong?** Efficiency computed under an assumption you cannot defend is a fragile argument. 4. **What does the downstream use need?** Computational cost, streamability, interpretability and auditability all enter. A median over a very large stream is harder to maintain than a running mean. 5. **Would relaxing unbiasedness beat both?** Sometimes the right answer is neither of the two offered estimators. ## Saying it in an interview Name efficiency as the tiebreaker and define relative efficiency as a variance ratio. Give the normal-data numbers for the mean and median — variance `sigma^2 / n` against about `1.57 * sigma^2 / n`, an efficiency of roughly 0.64 — then flip to heavy tails and say the ranking reverses. Finish with the caveat that under skew the two are not estimating the same parameter at all. That last point is what separates a memorised answer from an understood one.