12.5 The Normal Curve 807 per week by students may have a distribution like that in Fig. 12.18(b). The bars might represent (from left to right) 0–5 hours, 6–10 hours, 11–15 hours, and so on. A bimodal distribution (Fig. 12.19) is one in which two nonadjacent values occur more frequently than any other values in a set of data. For example, if an equal number of men and women were weighed, the distribution of their weights would probably be bimodal, with one mode for the women’s weights and the second for the men’s weights. For a distribution to be considered bimodal, both modes need not have the same frequency but they must both have a frequency greater than the frequency of each of the other values in the distribution. The life expectancy of lightbulbs has a bimodal distribution: a small peak very near 0 hours of life, resulting from the bulbs that burned out very quickly because of a manufacturing defect, and a much higher peak representing the nondefective bulbs. A bimodal frequency distribution generally means that you are dealing with two distinct populations, in this case, defective and nondefective lightbulbs. Another distribution, called a skewed distribution, has more of a “tail” on one side than the other. A skewed distribution with a tail on the right (Fig. 12.20(a)) is said to be skewed to the right. If the tail is on the left (Fig. 12.20(b)), the distribution is referred to as skewed to the left. (a) (b) Skewed to the left Skewed to the right Figure 12.20 The number of children per family might be a distribution skewed to the right. Some families have no children, more families may have one child, the greatest percentage may have two children, fewer may have three children, still fewer may have four children, and so on. Since few families have high incomes, distributions of family incomes might also be skewed to the right. Smoothing the histograms of the skewed distributions shown in Fig. 12.20 to form curves gives the curves illustrated in Fig. 12.21. Skewed to the right Skewed to the left (a) (b) Figure 12.21 In Fig. 12.21(a), the greatest frequency appears on the left side of the curve and the frequency decreases from left to right. Since the mode is the value with the greatest frequency, the mode would appear on the left side of the curve. Every value in the set of data is considered in determining the mean. The values on the far right side of the curve in Fig. 12.21(a) would tend to increase the value of the mean. Thus, the value of the mean would be farther to the right than the mode. The median would be between the mode and the mean. The relationship between the Bimodal distribution Figure 12.19
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