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. Heights of 30 people
a) How many people have heights between and cm?  b) How many people have heights less than cm?  c) How many people have heights more than cm?
d) What percentage of people have heights between and cm? 
e) Where is the mode of the heights in the histogram? g) Where is the median of the heights in the histogram?
f) Where is the mean of the height in the histogram?

3. 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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4. 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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5. 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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6. 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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7. 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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8. 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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9. 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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10. 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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11. 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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12. 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.
Hint: The variation among the times should be greater in ______________ than in ____________________________.
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