Read Online Full House by Stephen Jay Gould - Free Book Online Page B
Authors: Stephen Jay Gould
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idealized "correct" values, with random variation on either side—another consequence of lingering Platonism. But nature does not match our expectations very often.)
Actual distributions are often asymmetrical, or skewed. In a skewed distribution, as illustrated by my personal story, variation stretches out farther on one side than the other—called either "right" or "left" skewed depending on the direction of elongation (Figure 6). The reasons for skewing are often fascinating and full of insight about the nature of systems— for skewing measures departure from randomness. Since this book treats the nature of variation, and the reasons for changes in spread through time, skewing becomes an important principle in all my examples.
FIGURE 5 An idealized bell curve or normal frequency distribution, showing that all measures of central tendency (mean, median, and mode) coincide.
MEASURES OF CENTRAL TENDENCY AND THEIR MEANING. I have discussed the three standard measures of central tendency, or "average" value—the mean (or conventional average calculated by adding all values and dividing by the number of cases), the median (or halfway point), and the mode (or most common value). In symmetrical distributions, all three measures coincide — for the center is, simultaneously, the most common value, the halfway point (with equal numbers of cases on either side), and the mean. This coincidence, I suspect, has led most of us to ignore the vital differences among these measures, for we view "normal curves" as, well, normal—and regard skewed distributions (if we grasp the principle at all) as peculiar and probably rare. But measures of central tendency differ in skewed distributions—and a major source of employment for economic and political "spin doctors" lies in knowing which measure to choose as the best propaganda for the honchos who hired your gun.
FIGURE 6 Left- and right-skewed distributions.
I have already shown how the higher mean and lower mode of a right-skewed distribution in incomes can be so exploited (see page 37). In general, when a distribution is prominently skewed, mean values will be pulled most strongly in the direction of skew, medians less, and modes not at all. Thus, in right-skewed distributions, means generally have higher values than medians, and medians higher than modes. Figure 7 should make these relationships clear. If we start with a symmetrical distribution (with equal mean, median, and mode), and then pull the variation to form a right-skewed distribution, the mean will change most in the direction of skew—for one new millionaire on the right tail can balance hundreds of indigent people on the left tail. The median changes less, for a single pauper will now compensate the millionaire when we are only counting noses on either side of a central tendency. (The median might not move at all if only the wealth, and not the number, of people increases on the right side of the distribution. But if the number of wealthy people at the right tail increases as well, then the median will also shift to the right— but not so far as the mean.) The mode, meanwhile, may well stay put and not vary at all, as mean and median grow in an increasingly right-skewed distribution. Twenty thousand per year may remain the most common income, even while the number of wealthy people constantly increases.
"WALLS," OR LIMITS TO THE SPREAD OF VARIATION. As a major reason for skew, variation is often limited in the extent of potential spread in one direction (but much freer to expand in the other). The reasons for such limits may be trivial or logical—as in my cancer story where a person can’t die of mesothelioma before he gets the disease, and zero time between onset and death therefore becomes an irreducible minimum. The reasons may also be subtle and more interesting—as in the examples of batting averages and life’s history to be presented in Parts Three and Four of this book.
FIGURE 7 In a right-skewed
Actual distributions are often asymmetrical, or skewed. In a skewed distribution, as illustrated by my personal story, variation stretches out farther on one side than the other—called either "right" or "left" skewed depending on the direction of elongation (Figure 6). The reasons for skewing are often fascinating and full of insight about the nature of systems— for skewing measures departure from randomness. Since this book treats the nature of variation, and the reasons for changes in spread through time, skewing becomes an important principle in all my examples.
FIGURE 5 An idealized bell curve or normal frequency distribution, showing that all measures of central tendency (mean, median, and mode) coincide.
MEASURES OF CENTRAL TENDENCY AND THEIR MEANING. I have discussed the three standard measures of central tendency, or "average" value—the mean (or conventional average calculated by adding all values and dividing by the number of cases), the median (or halfway point), and the mode (or most common value). In symmetrical distributions, all three measures coincide — for the center is, simultaneously, the most common value, the halfway point (with equal numbers of cases on either side), and the mean. This coincidence, I suspect, has led most of us to ignore the vital differences among these measures, for we view "normal curves" as, well, normal—and regard skewed distributions (if we grasp the principle at all) as peculiar and probably rare. But measures of central tendency differ in skewed distributions—and a major source of employment for economic and political "spin doctors" lies in knowing which measure to choose as the best propaganda for the honchos who hired your gun.
FIGURE 6 Left- and right-skewed distributions.
I have already shown how the higher mean and lower mode of a right-skewed distribution in incomes can be so exploited (see page 37). In general, when a distribution is prominently skewed, mean values will be pulled most strongly in the direction of skew, medians less, and modes not at all. Thus, in right-skewed distributions, means generally have higher values than medians, and medians higher than modes. Figure 7 should make these relationships clear. If we start with a symmetrical distribution (with equal mean, median, and mode), and then pull the variation to form a right-skewed distribution, the mean will change most in the direction of skew—for one new millionaire on the right tail can balance hundreds of indigent people on the left tail. The median changes less, for a single pauper will now compensate the millionaire when we are only counting noses on either side of a central tendency. (The median might not move at all if only the wealth, and not the number, of people increases on the right side of the distribution. But if the number of wealthy people at the right tail increases as well, then the median will also shift to the right— but not so far as the mean.) The mode, meanwhile, may well stay put and not vary at all, as mean and median grow in an increasingly right-skewed distribution. Twenty thousand per year may remain the most common income, even while the number of wealthy people constantly increases.
"WALLS," OR LIMITS TO THE SPREAD OF VARIATION. As a major reason for skew, variation is often limited in the extent of potential spread in one direction (but much freer to expand in the other). The reasons for such limits may be trivial or logical—as in my cancer story where a person can’t die of mesothelioma before he gets the disease, and zero time between onset and death therefore becomes an irreducible minimum. The reasons may also be subtle and more interesting—as in the examples of batting averages and life’s history to be presented in Parts Three and Four of this book.
FIGURE 7 In a right-skewed
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