This paper examines the three primary measures of central tendency used in descriptive statistics: the mean, the median, and the mode. Drawing on sources in social research methods, mathematics education, and statistical concepts, the paper defines each measure, outlines its key properties, and identifies its advantages and disadvantages. The discussion covers appropriate measurement scales for each measure, the influence of extreme scores, mathematical tractability, and stability across samples. A summary table recapitulates the definitions and applications of all three measures, and the conclusion highlights which measures are appropriate for nominal, ordinal, interval, and ratio data.
The paper demonstrates effective use of multi-source synthesis. Rather than relying on a single authority, it draws on several peer-reviewed and textbook sources (Lomax, Neuman, Cai et al., Leavy, Zevenbergen et al.) to build a well-rounded, cross-validated account of each statistical concept. Direct quotations are used sparingly and purposefully to anchor key definitions, while paraphrase and enumeration carry the analytical weight.
The paper opens with a contextual introduction that establishes the purpose and scope. Three parallel body sections address the mean, median, and mode respectively, each following the same internal format: a definition, a list of characteristics with explicit advantages and disadvantages noted, and appropriate measurement scale guidance. A summary-and-comparison section consolidates the findings into a table, and a brief conclusion synthesizes the main takeaways regarding scale appropriateness for each measure.
According to Neuman (2003), researchers frequently need to summarize information concerning one variable into a single number, for which they use a measure of central tendency. Measures of central tendency are descriptive statistics that describe the point or points about which a distribution centers. This paper describes the three measures used to describe central tendency, identifies the advantages and disadvantages of each, and describes a situation in which each measure might be used.
According to Zevenbergen, Dole, and Wright (2004), measures of central tendency are a form of data representation that students today need in order to accurately interpret the information with which they are confronted on a daily basis: "For example, to be able to critically appraise reports in the daily paper and to assess the legitimacy of such claims as 'the average number of children per family is 1.7'; 'the average Australian child is overweight'; 'the average number of hours of television that children watch per week is 35'. To do this, understanding the way in which 'average' has been measured is critical" (p. 284).
In his book Social Research Methods, Neuman (2003) reports that "the three measures of central tendency, or measures of the center of the frequency distribution, are mean, median, and mode, which are often called averages (a less precise and less clear way of saying the same thing)" (p. 335). Although other measures exist, these three methods of summarizing a set of scores by constructing a single index or value that can represent the entire collection of scores are the most commonly used (Lomax, 2001).
This measure of central tendency is sometimes referred to as the arithmetic mean or "average" (Lomax, 2001). According to Cai, Lo, and Watanabe (2002), seven properties of the arithmetic average are as follows: (a) the average is located between the extreme values; (b) the sum of the deviations from the average is zero; (c) the average is influenced by values other than the average; (d) the average does not necessarily equal one of the values that was summed; (e) the average can be a fraction that has no counterpart in physical reality; (f) a value of zero, if it appears, must be taken into account when calculating the average; and (g) the average value is representative of the values that were averaged. Lomax (2001) defines the mean statistically as "the sum of all of the scores divided by the number of scores" (p. 45).
The following characteristics of the mean are relevant to this analysis:
First, the mean is a function of every score — a definite advantage in terms of a measure of central tendency representing all of the data. Second, the mean is influenced by extreme scores; because the numerator sum takes all scores into account, it also includes extreme scores, which is a disadvantage. Third, the mean always has a unique value, which is an advantage. Fourth, the mean is easy to deal with mathematically; in fact, it is the most stable measure of central tendency from sample to sample, and because of that it is the measure most often used in inferential statistics. Fifth, the mean is only appropriate for interval and ratio measurement scales, because it implicitly assumes equal intervals — an assumption that nominal and ordinal scales do not satisfy (Lomax, 2001).
According to Lomax (2001), "the median is that score that divides a distribution of scores into two equal parts. In other words, half of the scores fall below the median and half of the scores fall above the median" (p. 44). In instances where there is an odd number of untied scores, the median is the middle-ranked score; where there are an even number of untied scores, the median is the average of the two middle-ranked scores (Lomax, 2001).
The following characteristics of the median are relevant for this discussion:
First, the median is not influenced by extreme scores. Scores far from the middle of the distribution — known as outliers — do not affect it, because the median is defined conceptually as the middle score and the actual size of an extreme score is not relevant. Second, as a consequence, the median is not a function of all of the scores, since it does not take extreme scores into account. Third, the median is difficult to deal with mathematically — a disadvantage it shares with the mode — and it is also somewhat unstable from sample to sample, especially with small samples. Fourth, the median does always have a unique value, which is an advantage and distinguishes it from the mode, which does not always have a unique value. Fifth, the median can be used with all types of measurement scales except the nominal; nominal data cannot be ranked, and therefore percentiles and the median are inappropriate for such data (Lomax, 2001).
The research showed that the three common measures of central tendency are mean, median, and mode, although other indexes also exist. The research also showed that the mode is the only appropriate measure of central tendency for nominal data, but that the median and mode are both appropriate for ordinal data — and conceptually the median fits the ordinal scale particularly well, since both deal with ranked scores. In addition, all three measures of central tendency are appropriate for use with interval and ratio data. As noted by Leavy (2004), when distributions deviate significantly from normal, measures of central tendency alone may be insufficient, and supplemental measures such as variability should be considered alongside them.
Cai, J., Lo, J. J., & Watanabe, T. (2002). Intended treatments of arithmetic average in U.S. and Asian school mathematics textbooks. School Science and Mathematics, 102(8), 391.
Leavy, A. M. (2004). Indexing distributions of data: Preservice teachers' notions of representativeness. School Science and Mathematics, 104(3), 119.
Lomax, R. G. (2001). An introduction to statistical concepts for education and behavioral sciences. Mahwah, NJ: Lawrence Erlbaum Associates.
Neuman, W. L. (2003). Social research methods: Qualitative and quantitative approaches (5th ed.). New York: Allyn & Bacon.
Zevenbergen, R., Dole, S., & Wright, R. J. (2004). Teaching mathematics in primary schools. Crows Nest, N.S.W.: Allen & Unwin.
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