Statistics: Central Tendency of Variation
The Median isn't the Message
Gould’s The Median Isn’t the Message is the wisest, most humane thing ever written about cancer and statistics. It is the antidote both to those who say that, “the statistics don’t matter,” and those who have the unfortunate habit of pronouncing death sentences on patients who face a difficult prognosis. --- Edward Tufte
lies, damned lies, and statistics. --- Mark Twain (not Disraeli)
... a personal story of statistics, properly interpreted, as profoundly nurturant and life-giving. It declares holy war on the downgrading of intellect by telling a small story about the utility of dry, academic knowledge about science. Heart and head are focal points of one body, one personality.
Statistics recognizes different measures of an “average,” or central tendency. The mean is our usual concept of an overall average – add up the items and divide them by the number of sharers (100 candy bars collected for five kids next Halloween will yield 20 for each in a just world). The median, a different measure of central tendency, is the half-way point. If I line up five kids by height, the median child is shorter than two and taller than the other two (who might have trouble getting their mean share of the candy). A politician in power might say with pride, “The mean income of our citizens is $15,000 per year.” The leader of the opposition might retort, “But half our citizens make less than $10,000 per year.” Both are right, but neither cites a statistic with impassive objectivity. The first invokes a mean, the second a median. (Means are higher than medians in such cases because one millionaire may outweigh hundreds of poor people in setting a mean; but he can balance only one mendicant in calculating a median). --- Stephen Jay Gould
Platonic heritage, with its emphasis in clear distinctions and separated immutable entities, leads us to view statistical measures of central tendency wrongly, indeed opposite to the appropriate interpretation in our actual world of variation, shadings, and continua. In short, we view means and medians as the hard “realities,” and the variation that permits their calculation as a set of transient and imperfect measurements of this hidden essence. If the median is the reality and variation around the median just a device for its calculation, the “I will probably be dead in eight months” may pass as a reasonable interpretation.
variation itself is nature’s only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions.
