Measures of Centrality Essay ⋆ Sociology Essay Examples ⋆ EssayEmpire

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Measures of centrality (or central tendency) are statistical indices of the “typical” or “average” score. They constitute one of three key characteristics of a set of scores: center, shape, and spread. There are three common measures of centrality: mode, median, and mean. Their applicability to a set of scores depends on the scale of measurement of the scores, as explained below.

The simplest index of centrality is the mode, or most frequently occurring score. Since the mode is found by counting the number of occurrences of each score, it can used for the categorical data of nominal scales, the lowest level of measurement, for which it is the only applicable index (Stevens 1946). Nominal measurement sorts things into different categories, such as Republican, Democrat, or Libertarian. If more voters were registered as Republicans than any other party, then Republican would be the modal party, even if Republicans did not constitute a majority of voters.

The median is the score that occurs in the middle of the set of scores when they are ranked from smallest to largest. It is the score at the fiftieth percentile, for which half of the scores are smaller and half larger. Identification of the median requires at least ordinal data (i.e., data that can be ranked).

The most statistically sophisticated measure of centrality is the mean: the sum of the scores divided by the number of scores. Calculation of a mean is appropriate only for interval or ratio scales (e.g., Fahrenheit vs. Kelvin temperatures, respectively), the common feature of which is that the differences between scores (e.g., 38° — 33° = 76° — 71°) are meaningful and consistent for all scores. The mean is used to calculate the variance and standard deviation. These measures of variability, along with the mean, are the key ingredients of statistical analyses such as analysis of variance, correlation and regression, hierarchical linear modeling, and structural equation modeling.

For interval and ratio data, the shape of the distribution of scores influences relationships among the three measures of centrality. For some distributions, such as the bell curve (i.e., normal distribution), the mean, median, and mode all have the same value. However, for skewed distributions, their values differ. For example, in positively skewed distributions, where the scores pile up at the lower end of the scale and tail off to the upper end, the mean will have the highest value, followed by the median and mode, respectively. In negatively skewed distributions, the order is reversed. Thus, for example, if most household incomes in a community were under $30,000 but a few were $100,000 or higher, the mean income would be highest, and the mode would be lowest.

Outliers, or scores that fall well outside the range of the rest of the distribution, also differentially affect measures of centrality. Since the mean is the only measure of centrality that reflects the exact value of every score, it is the only one affected by outliers. For example, it would not affect the modal or median income in the community described above if the highest income was $300,000 or $300,000,000, but it would affect the mean. The impact of outliers on the mean is greatest when the number of scores in the distribution (e.g., households in the community) is small.

Thus, despite its utility in statistical analyses, the mean can be a misleading indicator of central tendency. For this reason, the median typically is used to depict the average” score in skewed distributions such as personal income and cost of houses. In addition, outliers sometimes are excluded to avoid distortion of the mean (as well as standard deviation and variance) for higher level statistical analyses. When this is done, the researcher should report that fact, providing information about the number of outliers discarded and the rationale and rules for exclusion, so that readers can evaluate whether eliminating outliers biased the analyses in favor of confirming the researchers’ hypotheses.

Bibliography:

Glass, G. V. & Hopkins, K. D. (1996) Statistical Methods in Education and Psychology, 3rd edn. Allyn & Bacon, Boston, MA.
Shavelson, R. J. (1996). Statistical Reasoning for the Behavioral Sciences, 3rd edn. Allyn & Bacon, Boston, MA.
Stevens, S. S. (1946) On the theory of levels of measurement. Science 103: 677-80.