> a student has to get about this far to realize whether or not there is some potential for a math-based career
To my mind, there is a pretty clear boundary dividing "basic math" from "upper level math." Solid basic math skills give power to burn for most people in their lives and jobs. And the same solid basic math skills form an absolutely essential foundation for all higher math. So my "thing" has always been that the schools gun for mastery of basic math for all students, no matter how many years it takes. I'd be happy to discuss my views on what constitutes "basic math", perhaps in another thread.
> Regarding mean and median, however, your own "outlier" example (or exaggerated even further) shows why the mean may be more misleading than the median. In a very real sense, the 1 or 2 really do give better information than the 11.
For a start, 2 is not even a contender; the median and mode were both 1. If everyone agrees that "$1" does the best job of describing sales in an upscale art gallery that typically makes 3 or 4 sales per day in the $400 range, then I guess I sit down and shut up. (To maintain my sanity, I'd have to cling to the hope of a "silent majority"...)
It's not hard to think up examples with clumpy data. Suppose that one day a beggar collects a bunch of nickels and dimes, plus a $10 bill and a $20 bill. What was the "representative" contribution?
Given nothing else, YOU CANNOT DO BETTER than cranking the average. You CANNOT know that the ten and twenty were once in a lifetime occurrences. (If they were, it was ludicrous of the problem poser to include them.) There is NO reason not to suppose that the beggar has a few donors who are more generous, perhaps because they know and like him.
In this paean to the average, am I saying the average is always good and useful? No, only that it is the ONLY useful one of our three measures of central tendency. It's easy to think up worthless averages, such as the average combined shoe size and age in decades of everyone in the Metropolitan Museum of Art last Saturday.
> I disagree with your algebra examples; a student has > to get about > this far to realize whether or not there is some > potential for a > math-based career (especially after our generally > horrible K-6 > arithmetic competence, both computational and > straightforward, > "cookbook" word problems). > > Regarding the mean, median, and mode mantra, my > greatest objection is > those are all the precollegiate curriculum does over > and over again > (except for AP Statistics that is mostly used as an > excuse for > pretending students are anywhere close to AP Calculus > in math > competence). The mode is truly useless. On small > data sets, it is > just meaningless. On large data sets, the idea is > obvious at a > glance and even more informative if the data happens > to be > bimodal. Regarding mean and median, however, your > own "outlier" > example (or exaggerated even further) shows why the > mean may be more > misleading than the median. In a very real sense, > the 1 or 2 really > do give better information than the 11. Consider our > own national > economy, for example. A relatively small group of > very rich people > pay most of the nation's income tax but the average > (mean) misleads > people into thinking that these very rich people are > not paying their > "fair share", whatever that might be. The median > gives a much more > accurate picture of what is underway. Already a much > higher percent > of the outrageously high salaried income (sometimes > over 60% > combining federal, state, and local income tax) > whereas none of us > here hit half that total percent. > > Wayne > > At 11:16 AM 1/11/2014, Donald Sauter wrote: > >I've just put a web page with my thoughts on math's > three-headed > >monster, the Mean, Median and Mode. > > > > http://www.donaldsauter.com/mean-median-mode.htm > > > >I explain why the last two need chopping off. > > > >Donald Sauter