Measures of The Middle: Means, Medians, and Modes
For nominal data, there is no ordering implied in the various categories; we can’t say that a preference for hard ice cream is, in any sense, better than a hatred of all ice cream. The best we can do is to indicate which category was most frequently reported; that is, in our ice cream study, “hard” was indicated most often. This value is called the modal value of the distribution. The modal value can be determined for all types of variables. For example, in Fig.1100, the modal value is 73.
When we turn to ordinal variables, we see now an explicit or implied ordering to the categories of response. However, we don’t know the spacing between categories. In particular, we cannot assume that there is an equal interval between categories, so any determination of an average that uses some measure of distance between categories is not legitimate. However, unlike with nominal variables, we do know that one category is higher or lower than another. For example, if we are talking about a rating of patient disability, it would be legitimate to ask the degree of disability of the average patient. So if we were to rank order 100 patients, with the “no disability” patients at the bottom and the “total disability” patients at the top, where would the dividing line between patient 50 and patient 51 lie? This value, with half the subjects below and half above, is called the median value.
For interval and ratio variables, we can use median and modal values, but we can also use a more common sense approach, simply averaging all the values. The mean is statistical jargon for this straightforward average, which is obtained by adding all the values and then dividing by the total number of subjects.
Note that for a symmetrical distribution, such as in Fig.1100, the mean, median, and mode all occur at the same point on the curve. But this is not the case when the distribution is asymmetrical. For example, the distribution of the income of physicians might look like Fig.1101.
For this distribution, the modal value (the high point of the curve) would be approximately $50,000; the median income of the physician at the 50th percentile would be approximately $60,000; and the few rich specialists may push the mean or average income up to approximately $70,000. (Admittedly, these numbers are woefully behind the times.) This kind of curve is called a “skewed” distribution; in this particular case, positively skewed. In general, if the curve has one tail longer than the other, the mean is always toward the long tail, the mode is nearer the short tail, and the median is somewhere between the two.
Figure 1101 – Figure 2-3: Distribution of physicians’ incomes.
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Figure 1102 – Figure 2-4: Distribution of attitudes of 100 teenagers to parents.
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As a final wrinkle, in some data, there may be two or more high points in the distribution, such as in Fig.1102. In this case, the distribution has two modes, love and hate, and is called “bimodal.”5343
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| 5343. | Norman GR, Streiner DL. PDQ Statistics . 3rd ed. Hamilton, Ontario: BC Decker Inc.; 2003. |
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