MONT 105N – Analyzing Environmental Data Thinking about central measures–“think, pair, share” February 1, 2019

Background

Recall that we have introduced two ways to estimate a “middle” or center of a collection of numerical data: x_{1}, x_{2}, …, x_{N}. The first is the mean or numerical average. We will use the notation

\overline{x} = \frac{x_{1} + x_{2} + ⋯ + x_{N}}{N} = \frac{1}{N} · x_{1} + ⋯ + \frac{1}{N} · x_{N}

for this. The second is the median. The easiest way to describe this is to give the usual method for computing it. To find the median (by hand), we would first sort the numbers into increasing (or non-decreasing) order:

x_{1} ≤ x_{2} ≤ ⋯ ≤ x_{N}

(note that this might involve renumbering the list if it is not sorted to begin with). Then

median = \{ \begin{matrix}x_{M+1} & if N = 2M + 1 is an odd number \\ \frac{x_{M} +x_{M+1}}{2} & if N = 2M is an even number \end{matrix}

Questions

(1) Let N = 8 and

x_{1} = 8.4, x_{2} = 3.5, x_{3} = 5.7, x_{4} = 3.5, x_{5} = 5.7, x_{6} = 6.2, x_{7} = 3.5, x_{8} = 9.3

What are \overline{x} and the median for this data set? (Note that the x_{i} do not need to be all different numbers!) Why are the mean and median different?

(2) What happens to the mean and the median in (1) if x_{1} is changed to 10.2?

(3) The x_{i} in part (1) above can be thought of as a collection of 5 different numbers, with some “repeats.” If you group the repeated numbers together, explain how \overline{x} could also be viewed as a weighted average of the distinct number values, with weights proportional to the number of times each number appears. (Ask me if the idea of a weighted average is not clear!)

(4) Give examples where each of these two methods for estimating a middle might be superior to the other. That is, try to think of situations where the mean of a collection of numbers might be more informative than the median, and other situations where the opposite would be true.