5
Aug 16

## The pearls of AP Statistics 12 Are you sure you are massaging the right muscle in the brain?

They say: The median is the middle value of the observations when the observations are ordered from the smallest to the largest (or from the largest to the smallest) (Agresti and Franklin, p.47) Half the observations are smaller than it, and half are larger.

Generally, if the shape is: a) perfectly symmetric, the mean equals the median, b) skewed to the right, the mean is larger than the median, c) skewed to the left, the mean is smaller than the median (same source, p.51)

I say: those who don't think will swallow this without demur. I have two problems. Firstly, the way the definition of the median is given makes me think that the median takes one of the observed values, which it does not, in general. Of course, you can provide a caveat. But why not just say: The median is such a point that half of the observations lie to the left of it and another half to the right. Secondly, the part about relationship between the mean and median is just terrible because it appeals to memorization. I went to great lengths to explain what is internal vision. Here are questions that are aimed at developing it: 1. Suppose you have a sample $x_1,..., x_n$. If all values move to the right by a constant c, what happens to the mean, median, mode, sample variance, range and IQR?
2. Suppose you have a sample $x_1, ..., x_n, x_{n+1}, ..., x_{2n}$ with an even sample size $2n$. If the first $n$ values move to the left by a constant c and the last $n$ values move to the right by the same constant c, what happens to the mean, median, mode, sample variance, range and IQR?

To feel how a formula works, try to change its elements.