# Term Paper: Neuman ), Researchers Frequently Need to Summarize

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¶ … 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 those descriptive statistics that describe the point or points about which a distribution centers. This paper provides a description of the three measures which are used to describe central tendency and identify the advantages and disadvantages of each, as well as describing a situation in which each of these measures might be used. A summary of the research and salient findings are presented in the conclusion.

According to Zevenbergen, Dole and Wright (2004) the 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 somehow be used to represent the entire collection of scores are the most commonly used (Lomax, 2001). These three measures are discussed further below, followed by a tabular recapitulation of the findings.

Mean.

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; - 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 one calculates the average; and (g) the average value is representative of the values that were averaged. In order to calculate the mean, Lomax reports that, "Statistically we define the mean as the sum of all of the scores divided by the number of scores" (Lomax, 2001, p. 45). The following characteristics of the mean are considered relevant to the instant analysis:

The mean is a function of every score, a definite advantage in terms of a measure of central tendency representing all of the data.

The second characteristic of the mean is that it is influenced by extreme scores; because the numerator sum takes all of the scores into account, it also includes the extreme scores, which is a disadvantage.

The mean always has a unique value which is an advantage.

The mean is easy to deal with mathematically; in fact, the mean is the most stable measure of central tendency from sample to sample, and because of that is the measure most often used in inferential statistics.

The fifth characteristic of the mean is that it is only appropriate for interval and ratio measurement scales because the mean implicitly assumes equal intervals, which nominal and ordinal scales do not possess (Lomax, 2001).

Median.

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). If those instances where there is an odd number of untied scores, then the median will be the middle-ranked score; in cases where there are an even number of untied scores, then the median is the average of the two middle-ranked scores (Lomax, 2001). The following characteristics of the median are relevant for the purposes of this discussion:

The first characteristic is that the median is not influenced by extreme scores (scores far away from the middle of the distribution are known as outliers); because the median is defined conceptually as the middle score, the actual size of an extreme score is not relevant.

A second characteristic is that the median is not a function of all of the scores because the median is not influenced by extreme scores and therefore does not take such scores into account.

A third characteristic is that the median is difficult to deal with mathematically, a disadvantage as with the mode; in addition, the median is also somewhat unstable from sample to sample, especially with small samples.

As a fourth characteristic, the median does always have a unique value, another advantage. This is unlike the mode, which does not always have a unique value.

The fifth characteristic of the median is that it 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 (Lomax, 2001).

Mode.

According to Lomax (2001), "The simplest method to use for measuring central tendency is the mode. The mode is defined as that value in a distribution of scores that occurs most frequently" (p. 42). The following characteristics about the mode suggest that it is more appropriate for some applications than others:

The first characteristic of the mode is it is simple to obtain and the mode is frequently used as a quick method for identifying central tendency, representing an obvious advantage.

The second characteristic is that the mode does not always have a unique value.

The third characteristic is that the mode is not a function of all of the scores in the distribution which is usually a disadvantage. The mode is strictly determined by which score or interval contains the most frequencies and the location or value of the other scores is not taken into account.

The fourth characteristic of the mode is that it is difficult to deal with mathematically. For instance, the mode tends not to be very stable from one sample to another, particularly with small samples; therefore, it is possible to have two nearly identical samples except for one score, which can alter the mode (Lomax, 2001).

Summary and Recapitulation.

According to Leavy (2004), "The mean and other measures of central tendency are representative when data sets are normal and nonskewed; however, the mean becomes less representative as the distribution deviates from normal. In such situations, using measures of central tendency alone to index a distribution is inadequate, and supplemental indices such as variability may be more appropriate" (p. 119).

Table 1.

Summary of the Three Measures of Central Tendency.

Mean

Median

Mode

Definition measure of central tendency for one variable that indicates the arithmetic average (i.e., the sum of all scores divided by the total number of scores) (Neuman, 2003).

A measure of central tendency for one variable indicating the point or score at which half the cases are higher and half are lower (Neuman, 2003).

A measure of central tendency for one variable that indicate the most frequent or common score (Neuman, 2003).

Applications

Also called the arithmetic average, the mean is the most widely used measure of central tendency; however, unlike mode and media, the mean can only be used with interval- or ratio-level data. This measure is calculated by adding up all the scores in a series, and then dividing by the number of scores (Neuman,…
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