Measures of central tendency

            Measures of central tendency are numerical descriptors that locate the center of a distribution. The measures of central tendency that we will consider are the mean, mode, and median. The mean is the most widely used measure of central tendency. The mean of a sample and a population are defined in the same way but require different symbols. The mean for a sample is denoted with ª (bar x), the arithmetic mean and this is an estimate of the mean of the entire population (signified by the Greek letter, mu, µ).

            The mean (ª) is one measure of the central value of the distribution of the sample. The sample mean, (ª, the arithmetic mean) is calculated by adding up the values and dividing by the number of observations, n. for example, suppose an ecologist measures the weights of 5 spiders and obtains the following data: 13, 40, 27, 18, 32 mg. For this data,

            ª = 13 + 40 + 27 + 18 + 32  = 130 = 26mg/spider

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            It should be clear that any value measuring central tendency does not in itself give a clear picture of the distribution. As the diagram in figure 1 shows, many types of distributions can have the same mean, and yet differ greatly in appearance.

   

Figure 1. Examples of distributions having the same mean but different variances

            Another measure of central tendency is the median. The median is computed the same way for either a sample or a population. The median for an even number of measurements is the mean of the middle two observations (measurements) when the measurements are arranged in order of magnitude, or rank. For example, to find the median of the following ten achievement scores: 62, 95, 91, 73, 73, 75, 89, 86, 78, 90 rank them: 62, 73, 73, 75, 78, 86, 89, 90, 91, 95, since there are an even number of observations, the sample is the mean of the two middle scores when the scores are arranged in numerical order. The two midpoint scores are 78 and 86, therefore the median is

                                    Median = 78 + 86 = 82

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            The median for an odd number of measurements is the middle measurement when the measurements are arranged in order of magnitude or ranked. For example, to find the median of the following seven test scores: 95, 86, 78, 90, 62, 73, 89, rank them: 62, 73, 78, 86, 89, 90, 95. Since there are an odd number of measurements, the median score is the middle score, that is 86.

            The median seems to be the preferred measure of central tendency to describe sociological and educational data.  Newspaper reports and magazines frequently refer to the median wage increase won by unions, the median income of families in the U.S., and the median of persons receiving Social Security.  There are three important characteristics of the median: 1.) The mean and the median are not necessarily equal in value, 2.) The median of a set of measurements is a number such that half the measurements lie below and above it, 3.) The median has most relevance when used to describe large sets of data.

            The mode is the measurement that occurs with the greatest frequency within a set of data or it is the highest point or peak on a histogram. The mode is especially useful summary for data sets that consist not of numbers but of names or categories. For example, consider the birthplaces of 300 students at a small school in Kalamazoo. Most likely the mode for birth place is Kalamazoo, that is to say, that the most common birth site out of the 300 students is Kalamazoo. This is the least common of the three measurements of central tendency used in science.