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.Real bells do not deviate much from a standard shape.Since we are inthe realm of theory, we can stretch or squash our bell-shaped curve likePlay-Doh.The normal distribution is statistical in nature.Suppose you took theheights of 20 men and plotted them on a graph like Panel A of Figure3.3.Chances are you would get a lumpy curve, looking roughly like abell that had seen battle.There would be gaps your group might notinclude anyone from 6 ft.1 in.to 6 ft.3 in., for example and excesses.There might be a cluster around 5 ft.8 in.The smaller the group, themore irregular the bell-shaped curve.On the other hand, fan out across the city, making thousands of mea-surements, and the curve that you draw at the completion of your laborswill almost certainly be nearly smooth.This is basically what we meanwhen we say that the normal curve is statistical.Enough measurementseven it out.So most things we can measure, from the number of hairs on our headsto the times it rains in July, will show up as a bell-shaped curve on graphpaper.The key word in the previous sentence is "most." For a varietyof reasons, not all distributions are normal.It is tempting to call everything that is not a normal distribution abnor-mal, but nature makes no such distinction.It chooses its distributionsin ways that are still somewhat mysterious.Putting these distributionson graph paper and giving them a name are human artifacts, not thoseof nature. Figure 3.3Graphing Techniques Used in This Book Figure 3.3 (Continued) 54 HOW RICH IS TOO RICH?THE LOGNORMAL DISTRIBUTIONThe spreads of wealth and income do not follow simple bell curves.Itturns out that money matters tend to be best described as a lognormaldistribution, a cousin of the normal or bell-shaped curve.Then why bothertalking about normal distributions in the first place?The normal curve was exemplified by something we all know about,height.So will the lognormal.The first syllable of lognormal is a contrac-tion for logarithm.The dictionary defines logarithm as the index of thepower to which a fixed number (i.e., the base) must be raised to produceanother given number.For example, the logarithm of 100 is 2.This meansthat if we raise 10 to the second power or square it (i.e., 10 x 10) theresult is 100.Logarithms can also be a decimal number like 2.78.In thiscase, we have to turn to a hand calculator to determine 10 raised to thepower 2.78.Most calculators have a key marked "log," so doing the com-putation is much less of a challenge than it once was.What is an example of a lognormal distribution? Consider somethingwe all know about, some of us all too well: weight.Human weight isclose to a lognormal distribution, as opposed to the normal distributionof height.We can recognize a normal distribution by its shape on graphpaper.How can we identify a lognormal distribution? The statisticianssay that a lognormal distribution is skewed, or not symmetrical.It will tendto look like Panel B in Figure 3.3, with a long tail on the right.Suppose we go back to the thousand men whose heights we measured,and this time put them on the scales.After the beam has been tippedfor the last time, suppose we find that more men are around 170 poundsthan any other weight.This is actually close to the national average.InFigure 3.3, Panel B, we can see that the maximum is close to that value.Now we can do as we did with heights; proceed in both directions fromthe maximum.Let us move 30 pounds, to 140 and 200 pounds.We willprobably find more men at 200 than at 140, although we might first ascribethis to ordinary variation in the numbers.This is our first indication thatthe weights are nonsymmetrical.Now let us proceed 70 pounds from the midpoint, to 100 and 240.Thereare very few men of the former tiny size, but we see many of the secondgroup every autumn Sunday afternoon on our TV screens.The numberof 240-pound men is far higher than the 100-pound variety, as we candeduce from the left-hand side of Panel B.In fact, the graph suggestsfew if any 100-pound men.The nonsymmetrical nature of the graph isbecoming evident.Even if we had never seen a game of professional football, we couldprove the skewness of the weight curve.Every so often we read of some-one who, beset by glandular problems, dies at a weight of 400 or morepounds.This is obviously more than 170 pounds from the midpoint of LAW OF INCOME DISTRIBUTION 55170 we assumed above.To produce symmetry in the weight curve, some-one would have to have a negative weight (i.e., more than 170 subtractedfrom 170) to correspond to that unfortunate person.Peter Pan was supposed to have the ability to fly, implying some sortof weightlessness.He remains confined to the realm of imagination.Nobody else has zero or negative weight.As a result, the curve of humanweight is unbalanced or skewed.Not every skewed distribution is lognormal.There are types of distribu-tion other than the two we have discussed, some of which are nonsym-metrical.One, the Pareto distribution describing the very top of theincome pyramid, is a vital part of our argument.RECOGNIZING A LOGNORMAL DISTRIBUTIONHow would we know a lognormal distribution if we were given a seriesof numbers? The simplest way is to take the logarithms of each of thequantities involved [ Pobierz całość w formacie PDF ]