Mittwoch, 15. Januar 2014

Distribution of Achievement:

Distribution of Achievement
Arthur Jensen; 1980

As we have seen, there are a number of reasons for believing that mental ability is normally distributed in the population. But this generalization most likely does not extend to manifestations of ability in individual achievements, output, acquired knowledge, developed skills, occupational success, earnings, and the like. There is good reason to believe that achievement, in contrast to more elemental traits and abilities, is not normally distributed in the general population but that it has a markedly skewed distribution, like that in Figure 4.20.
When frequency distributions are plotted for accomplishments that can be counted, and thus are measurable on an absolute scale, the distributions are found to be markedly skewed. Examples are number of patents held by inventors, number of publications of research scientists and university professors, amount of music written by composers, and yearly earnings.
The skewness of the distribution of accomplishments also accords with subjective impressions of the absolute differences between persons lying at various percentile points on the scale of accomplishments in any field. The difference in chess skill between world champions and the average chess player certainly seems greater than the difference be­tween the average player and the chess duffer. An Olympics champion is much farther above the average person in his particular athletic skill than the average person is above those who are just barely capable of displaying the particular skill at all.
A similar skewness is found for scholastic achievement and measures of general knowledge. Most scholastic achievement tests, however, are constructed in such a way as not to reveal the skewed distribution of achievement. In the first place, the usual achieve­ment tests given in any grade in school have too little “ top,” that is, too few hard items, thereby cutting off the long upper tail of the skewed curve. The reason for this is that the usual achievement tests are intended to measure the achievement of a particular grade and do not include information and skills that are a part of the curriculum of higher grades. The eighth-grader who has mastered calculus could never show it on the usual math achievement tests given to eighth-graders. In the second place, most scholastic achieve­ment tests today are in a sense double-duty tests; they are designed to measure not only what the child has learned in a given grade in school, but to test his intelligence as well.

Figure 4.20. Theoretical distribution of achievement as measured on a scale of equal intervals. Notice that, even with this marked positive skewness of the distribution, the median and mean are quite close together but that the mean is pulled in the direction of the skew.

Knowledge acquisition is substantially correlated with intelligence, but not to the ex­tremely high degree suggested by the correlation between the usual group verbal IQ test and the usual achievement test. A large number of the items in achievement tests requires the subject to use his or her acquired knowledge to solve novel problems, to reason, compare, generalize, and figure out answers to questions that may be unlike anything he or she has been taught in school, except that the information required to solve the problems has been taught. The ability to use information in reasoning and problem solving is more a matter of intelligence than of how much information the child has acquired in class, and so a larger proportion of the variance in achievement will represent intelligence variance than variance in how much children have actually learned from their lessons. Thus, when achievement tests are made to resemble intelligence tests in this way, and also are made to restrict the range of informational content sampled by the test, it should not be surprising that the distribution of achievement scores on such tests is much like the distribution of intelligence test scores. Because their informational content is so restricted, the usual grade-level achievement tests can be made difficult enough to spread students out into a normal distribution only by increasing the level of reasoning and problem­ solving ability required by some of the items, making them very much like intelligence test items.
But achievement tests can be constructed to measure knowledge of things taught in school, rather than reasoning ability; the items sample knowledge at all levels over a wide range of fields, so that the test has virtually no ceiling and few if any subjects at any age could obtain a perfect score. An example of such a test is the General Culture Test (Learned & Wood, 1938), which was originally devised to assess all-around scholastic achievement in the high schools and colleges of Pennsylvania. The test contains some 1,200 questions involving information on all the fine arts, all periods of history, social studies, natural sciences, and world literature. The test has plenty of bottom and plenty of top for high school and college students. The total range of scores in a group of 1,503 high school seniors was from 25 to 615. The highest score found in a sample of 5,747 college sophomores was 755; the highest among 3,720 college seniors was 805, which is only about two-thirds of the maximum possible score. The distribution of scores in a large sample of high school seniors was quite skewed (10 percent less skew than in Figure 4.20).
If achievement depends on other normally distributed factors in addition to ability, such as motivation, interest, energy, and persistence, and if all these factors act multi­plicatively, then theoretically we should expect achievement to show a positively skewed distribution. The greater the number of factors (each normally distributed), the more skewed is the distribution of their products. The products of normally distributed variables are distributed in a skewed way such that the distribution of products can be normalized by a logarithmic transformation. A logarithmic transformation of achievement scores in effect makes the component elements of achievement additive rather than multiplicative. Theoretically a multiplicative effect of ability and motivation (or other traits involved in achievement) makes sense. Imagine the limiting case of zero ability; then regardless of the amount of motivation, achievement would equal zero. Also, with zero motivation, regard­less of the amount of ability, achievement would equal zero. Great achievers in any field are always high in a number of relevant traits, the multiplicative interaction of which places their accomplishments far beyond those of the average person—much farther than their standing on any single trait or a mere additive combination of several traits. A superior talent alone does not produce the achievements of a Michelangelo, a Beethoven, or an Einstein. The same can be said of Olympics-level athletic performance, which depends on years of concentrated effort and training as well as certain inborn physical advantages. Thus it is probably more correct to say that a person’s achievements are a product, rather than a summation, of his or her abilities, disposition, and training.


  1. I would have expected a left skew due to diseases and difficulties during birth that would reduce an individual's intelligence, while these factors would never increase it. Perhaps it is a small effect.

  2. As far as I know the intelligence distribution usually does not contain the scores of people with severe organic defects or diseases.