Sample averages for both net profit and sales revenue are slightly biased or skewed upward because sample mean values are somewhat above median levels. This reflects the fact that a few very large regional markets can cause sample average values to be greater than the typically observed level. As discussed earlier, differences between sample means and medians are to be expected for much economic data given the long upward "tail" provided by the giants of industry. However, there is no necessary reason to suspect any relation between profit margins and firm size. Profit margins are net profit as a percentage of sales revenue. Because sales revenue is a commonly used measure of firm size, profit margin data are an example of "normalized" or size-adjusted data. The sample average profit margin of 14.8 percent is very close to the sample median of 14.9 percent. This indicates that the distribution of profit margin data is fairly centered around the sample mean observation, as is often the case when "normalized" or size-adjusted data are considered. There is, however, substantial variation around the sample averages for net profit, profit margin, and sales revenues, and the chance of atypical sample values is correspondingly high.

Mode mode Another commonly employed measure of central tendency is the mode, or the most frequently

Most common vdue encountered value in the sample. The mode is not often relied on in cases where continuous data are employed. Continuous data are numbers, such as net profit, profit margin, or sales revenue data, that can vary by small amounts—or continuously. For example, it is quite rare to find instances where several firms in an industry have exactly the same levels of net profits in dollars, whereas many firms might report the same profit level in millions of dollars. In the regional telecommunications services markets example, three regional markets generate exactly the same $4.7 million profit level. This modal profit level is slightly below the mean profit level, but exactly equals the median profit level. Thus, these net profit data are reasonably well centered in the sense that the mean, median, and mode measures of central tendency converge on a narrow range of values. By way of comparison, three markets each have a net profit margin of 14.4 percent while three others have a net profit margin of 14.3 percent. Given the very small difference between these modal profit margin levels, the sample median of 14.9 percent, and the sample average of 14.8 percent, it appears reasonable to conclude that profit margins are also centered in a very narrow range. However, no two markets have exactly the same level of revenue when sales is measured in millions of dollars—so there is no modal level for this series of sales data.

The mode is most attractive as a measure of central tendency in instances when only a modest amount of variation in continuous data is observed or when grouped data are being analyzed. For example, if only a limited variety of colors and sizes are offered to customers, identification of the modal or most popular color and size class is important for both marketing and production purposes. If customer classes are analyzed in terms of age groupings, identifying important characteristics of the modal age group becomes similarly important.

If a sample of observations has more than one mode, it is called multimodal; a bimodal distribution, for example, has two modes. Samples with more than one mode often include groups of data that are quite different on some important dimension. The distribution of customer weight and height is likely to be bimodal because both weight or height tend to vary by sex. The mode weight and height of women is far less than that for men, so any analysis of customer weight and height that does not control for sex is likely to be bimodal. In instances where measurements of sample groups have a multimodal distribution, it is often appropriate to construct separate frequency distributions for each sample subgroup, rather than to ignore the important underlying causes of modal differences.

Comparing Measures of Central Tendency

The mean, median, and mode are all useful measures of central tendency, but their value can be limited by unique characteristics of the underlying data. A comparison across alternate measures is useful for determining the extent to which a consistent pattern of central tendency emerges. If the mean, median, and mode all coincide at a single sample observation, the sample data are said symmetrical to be symmetrical. If the data are perfectly symmetrical, then the distribution of data above the a tailed d^rtofon mean is a perfect mirror image of the data distribution below the mean. A perfectly symmetrical distribution is illustrated in Figure 3.1(b). Whereas a symmetrical distribution implies balance in skewness sample dispersion, skewness implies a lack of balance. If the greater bulk of sample observa-

Lack of balance tions are found to the left of the sample mean, then the sample is said to be skewed downward or to the left as in Figure 3.1(a). If the greater bulk of sample observations are found to the right of the mean, then the sample is said to be skewed upward or to the right as in Figure 3.1(c).

When alternate measures of central tendency converge on a single value or narrow range of values, managers can be confident that an important characteristic of a fairly homogeneous sample of observations has been discovered. When alternate measures of central tendency fail to converge on a single value or range of values, then it is likely that underlying data comprise a heterogeneous sample of observations with important subsample differences. A comparison of alternate measures of central tendency is usually an important first step to determining whether a more detailed analysis of subsample differences is necessary.

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