Let me give you an example: Take five people who are five feet tall; add one person who is six feet tall. They AVERAGE five feet two inches. FIVE of the SIX are below average.

Sometimes the word "average" gets used to mean "sort of in the middle" but not when you add that it is what a statistician would say.

Median is a frequently used measure of "average", and exactly half the population is below the median, by definition. Arithmetic mean may or may not coincide with the median, but it is not the only measure of "average", and anyway it probably does at the least come close to the median.

There are tree different measures often called the "average":

There is the mean , which is the measure which you described.

There is the median , which by definition means precisely at which half are above, and half are below that value.

There is also the mode , which is the commonest value (in your example, that would be 5 feet).

Sometimes, all three measures are the same, but they can be, and often are, different -- which is fine, as long as one remembers the distinction, and under what circumstances each is the most pertinent or useful measure.

In this case, the author clearly meant the median average, and his "interpretation" was in fact, by definition, correct.

This is a common mistake (I see it all the time among the undergraduates I teach) that has proven difficult to counter as more people use the terms average, mean, median, and mode incorrectly. Mathematically, the average of a set of numbers is the sum of the numbers divided by the number in the set. That's it--it is identical to the arithmetic mean of the numbers, which is usually just abbreviated to mean unless confusion with the geometric mean is possible.

The median and modes are measures of central tendancy, but are not averages--since nothing is being averaged in these numbers.

If Patterson had merely been speaking colloquially, he would have been fine, but once he invoked statisticians, he should have used the precise definition. Someday the "average can mean anything I want it to mean" crowd might get their usage into the mathematics textbooks, but until then, Patterson was wrong.

I disagree that 'average' necessarily means 'arithmetic mean'. I'm not a statistician, but I've had three courses in statistics and a couple courses that heavily depend on statistics (ugh, turbulence modeling...) and in my experience, even those who very much know the precise meanings of what they're talking about, use average to indicate central tendency or expected value, and when they want to refer to one particular way of measuring this, they use the appropriate term. I personally frequently use arithmetic mean and root mean square, but I have no problem with median, geometric mean, or one of several other measures of center also being referred to as "average".

If the method of determining central tendency is not mentioned explicitly, then I would assume that the method used would be the one most commonly used in the type of analysis in question. In the case of demographic studies, indeed median is the most commonly used statistic, such as with income, life expectancy, local housing values, etc. etc.

I'd expect to see "average" equated with arithmetic mean in a grade school text, but in a college text I'd expect the subject to be handled in a more succinct and complete manner. I have two college level statistics texts on my bookshelf, and neither uses the word "average", in the interest of clarity I suspect.

Expected value refers to the underlying probability distribution, and has no meaning for a set of numbers. You typically estimate the expected value for your statistical model by using the average of the data, but the two refer to separate things.

"More succinct" and "complete" are usually at cross-purposes, unfortunately. I agree, though, your texts probably avoid the use of average because the use of the word has become corrupted and they don't want to confuse introductory students. However, words do have precise meanings in mathematics, and it does no one any good to pretend that such definitions are as mutable as the rest of the English language, because they are not.