1. 4.2 Shapes of Distributions
LEARNING GOAL
Be able to describe the general shape of a distribution in terms of its number of modes, skewness, and variation.
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2. Because we are interested primarily in the general shapes of distributions, it’s often easier to examine graphs that show smooth curves rather than the original data sets.
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Figure 4.3 The smooth curves approximate the shapes of the distributions.
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3. Number of Modes
Figure 4.4
Figure 4.4a shows a distribution, called a uniform distribution, that has no mode because all data values have the same frequency.
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Figure 4.4b shows a distribution with a single peak as its mode. It is called a single-peaked, or unimodal, distribution.
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4. Number of Modes
Figure 4.4
By convention, any peak in a distribution is considered a mode, even if not all peaks have the same height.
For example, the distribution in Figure 4.4c is said to have two modes—even though the second peak is lower than the first; it is a bimodal distribution.
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Similarly, the distribution in Figure 4.4d is said to have three modes; it is a trimodal distribution.
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5. Symmetry or Skewness
A distribution is symmetric if its left half is a mirror image of its right half.
Figure 4.6 These distributions are all symmetric because their left halves are mirror images of their right halves. Note that (a) and (b) are single-peaked (unimodal), whereas (c) is triple-peaked (trimodal).
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6. A distribution that is not symmetric must have values that tend to be more spread out on one side than on the other. In this case, we say that the distribution is skewed.
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Figure 4.7 (a) Skewed to the left (left-skewed): The mean and median are less than the mode. (b) Skewed to the right (right-skewed): The mean and median are greater than the mode. (c) Symmetric distribution: The mean, median, and mode are the same.
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7. Definitions
A distribution is symmetric if its left half is a mirror image of its right half.
A distribution is left-skewed if its values are more spread out on the left side.
A distribution is right-skewed if its values are more spread out on the right side.
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8. the median or the mean? Why?
TIME OUT TO THINK
Which is a better measure of “average” (or of the center of the distribution) for a skewed distribution:
the median or the mean? Why?
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9. a. Heights of a sample of 100 women
EXAMPLE 2 Skewness
For each of the following situations, state whether you expect the distribution to be symmetric, left-skewed, or right-skewed. Explain.
a. Heights of a sample of 100 women
b. Family income in the United States
c. Speeds of cars on a road where a visible patrol car is using
radar to detect speeders
Solution:
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a. The distribution of heights of women is symmetric because roughly equal numbers of women are shorter and taller than the mean, and extremes of height are rare on either side of the mean.
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10. EXAMPLE 2 Skewness Solution: (cont.)
b. The distribution of family income is right-skewed. Most families are middle-class, so the mode of this distribution is a middle-class income (somewhere around $50,000 in the United States). But a few very high-income families pull the mean to a considerably higher value, stretching the distribution to the right (high-income) side.
c. Drivers usually slow down when they are aware of a patrol car looking for speeders. Few if any drivers will be exceeding the speed limit, but some drivers tend to slow to well below the speed limit. Thus, the distribution of speeds will be left-skewed, with a mode near the speed limit but a few cars going well below the speed limit.
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11. By the Way ...
Speed kills. On average, in the United States, someone is killed in an auto accident about every 12 minutes. About one-third of these fatalities involve a speeding driver.
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12. TIME OUT TO THINK
In ordinary English, the term skewed is often used to mean something that is distorted or depicted in an unfair way. How is this use of skew related to its meaning in statistics?
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13. Variation
Definition
Variation describes how widely data are spread out about the center of a data set.
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Figure 4.8 From left to right, these three distributions have increasing variation.
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14. EXAMPLE 3 Variation in Marathon Times
How would you expect the variation to differ between times in the Olympic marathon and times in the New York City marathon? Explain.
Solution: The Olympic marathon invites only elite runners, whose times are likely to be clustered relatively near world-record times. The New York City marathon allows runners of all abilities, whose times are spread over a very wide range (from near the world record of just over two hours to many hours). Therefore, the variation among the times should be greater in the New York City marathon than in the Olympic marathon.
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15. The End
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