Mean and median. One of the most basic tasks in statistics is to summarize a set of observ...

Mean and median. One of the most basic tasks in statistics is to summarize a set of observations (x1; x2,..., xn}⊆ ℝ by a single number. Two popular choices for this summary statistic are:

• The median, which we’ll call μ1

• The mean, which we’ll call μ2

(a) Show that the median is the value of μ that minimizes the function

You can assume for simplicity that n is odd. (Hint: Show that for any μ ≠ μ1, the function decreases if you move μ either slightly to the left or slightly to the right.)

(b) Show that the mean is the value of μ that minimizes the function

One way to do this is by calculus. Another method is to prove that for any μ ∈ ℝ,

Notice how the function for μ2 penalizes points that are far from μ much more heavily than the function for μ1. Thus μ2 tries much harder to be close to all the observations. This might sound like a good thing at some level, but it is statistically undesirable because just a few outliers can severely throw off the estimate of μ2. It is therefore sometimes said that μ 1 is a more robust estimator than μ 2. Worse than either of them, however, is μ ∞, the value of μ that minimizes the function

(c) Show that μ∞ can be computed in O (n) time (assuming the numbers xiare small enough that basic arithmetic operations on them take unit time).