Average

Example

Most students, however innumerate, are able to calculate their grade point averages because it is typically the arithmetic mean of all their work over a course that provides their final grade. Table 1 shows a hypothetical set of results for students for three pieces of coursework. The table gives the mean and median for each student and also for the distribution of marks across each piece of work. It shows the different relationships the mean and median have with the different distributions, as well as the different ways that students can achieve a mark above or below the average. Dudley achieved a good result by consistent good marks, whereas Jenny achieved the same result with a much greater variation in her performance. On the other hand, Jenny achieved an exceptional result for Project 1, achieving 17 points above the mean and 19 above the median, whereas all students did similarly well on Project 2. If the apparently greater difficulty of Project 1 resulted in differential WEIGHTING of the scores, giving a higher weight to Project 1, then Jenny might have fared better than Dudley. The table provides a simple example of how the median can vary in relation to the mean. Of course, what would be the fairest outcome is a continuous matter of debate in educational institutions.

Historical Development

The use of combining repeat observations to provide a check on their accuracy has been dated to Tycho Brahe's astronomical observations of the 16th century. This application of the arithmetic mean as a form of verification continued in the field of astronomy, but the development of the average for aggregation and estimation in the 18th and 19th centuries was intimately connected with philosophical debates around the meaning of nature and the place of mankind. Quetelet, the pioneer of analysis of social scientific data, developed the notion of the “average man” based on his measurements and used this as an ideal reference point for comparing other men and women. Galton, who owed much to Quetelet, was concerned rather with deviation from the average, with his stress [Page 49]on “hereditary genius” and eugenics, from which he derived his understanding of regression. Since then, “average” qualities have tended to be associated more with mediocrity than with perfection.

AVERAGES AND LOW-INCOME MEASUREMENT

A common use of averages is to summarize income and income distributions. Fractions of average resources are commonly used as de facto poverty lines, especially in countries that lack an official poverty line. Which fraction, which average, and which measure of resources (e.g., income or expenditure) varies with the source and the country and with those carrying out the measurement. Thus, in Britain, for example, the annual Households Below Average Income series operationalizes what has become a semi-official definition of low income as 50% of mean income. The series does also, however, quote results for other fractions of the mean and for fractions of the median. By comparison, low-income information for the European Union uses 60% of the median as the threshold, and it is the median of expenditure that is being considered. In both cases, incomes have been weighted according to household size by equivalence scales. As has been pointed out, the fact that income distributions are skewed to the right means that the mean will come substantially higher up the income distribution than the median. It is therefore often considered that the median better summarizes the center of an income distribution. However, when fractions of the average are being employed, the issue changes somewhat, both conceptually and practically. Conceptually, a low-income measure is attempting to grasp a point at which people are divorced from participation in expected or widespread standards and practices of living. Whether this is best expressed in relation to the median or the mean will depend in part on how the relationship of participation and exclusion is constructed. Similarly, at a practical level, a low-income measure is attempting to quantify the bottom end of the income distribution. Whether this is best achieved with an upper limit related to the mean or the median is a different question from whether the mean and the median themselves are the better summaries of the income distribution. In practice, in income distributions, 50% of the mean and 60% of the median typically fall at a very similar point. These two measures are consequently those that are most commonly used as they become effectively interchangeable.

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