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Bimodal Distributions

Obviously, if we calculate the median or mean for a bimodal variable, we won’t get a realistic picture of the central tendency in the data. To understand TFR better, I divided the data into cases at or above the median of 2.26 and cases below it. Then I calculated the means and medians for each of the two sets of data and ran the frequency polygons.

Figure 20.7a shows the frequency polygon for the cases up to the median; figure 20.7b shows the polygon for the cases above the median. Now it looks as though we may have

FIGURE 20.6.

Histograms (left) and frequency polygons (right) for MFRATIO, TFR, and PCGDP in table 20.8.

FIGURE 20.7.

Frequency polygons for TFR (in table 20.8) above the median (a) and below the median (b).

a multimodal distribution. This might be the result of sampling error. I tested this by running the same analysis on the 195 countries of the world for which we have data on TFR. The frequency polygon for this analysis is shown in figure 20.8. Compare the shape of this graph to the one for TFR on the lower right in figure 20.6. This tells us that (1) the results we got by analyzing the 50-country sample are a good reflection of the data from which the sample was drawn, and (2) TFR really is bimodal.

Frequency polygon for TFR for 195 countries of the world, 2010.

SOURCE: UN Department of Economic and Social Affairs, Population Division (2007). United Nations World Population Prospects: 2006 Revision, table A15. http://www.un.org/esa/population/publications/wpp2006/ WPP2006_Highlights_rev.pdf.

FIGURE 20.8.

Bimodal and multimodal distributions are everywhere. Figure 20.9 shows the frequency polygon for LEXFEM, female life expectancy in table 20.8. This is no sampling

FIGURE 20.9.

Frequency polygon for LEXFEM in table 20.8.

artifact, either. The bulge on the right covers countries in which women can expect to live, on average, about 80 years. The bulge on the left covers countries in which women can expect to live, on average, just 61 years.

The moral is: Examine the frequency distribution for each of your variables. For interval and ordinal variables, find out if the distributions around the mean or median are symmetrical. If the distributions are symmetrical, then the mean is the measure of choice for central tendency. If the distributions are skewed, then the median is the measure of choice. And if the distributions are bimodal (or multimodal), then do a much closer examination and find out what’s going on.

 
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