1. Learning Goal: To be able to describe the general shape of a distribution in terms of its number of modes, skewness, and variation. 4.2 Shapes of Distributions

2. Number of Modes One way to describe the shape of a distribution is by its number of peaks, or modes. Uniform distribution—has no mode because all data values have the same frequency.

3. Any peak is considered a mode, even if all peaks do not have the same height. A distribution with a single peak is called a single-peaked, or unimodal, distribution. A distribution with two peaks, even though not the same size, is a bimodal distribution. What is the following distribution?

4. How many modes would you expect for each of the following distributions? Why? Make a rough sketch with clearly labeled axes? The body temperature of 2000 randomly selected college students The attendance at Disney World during a year The last digit of your phone number

5. Symmetry or Skewness A distribution is symmetric if its left half is a mirror image of its right half. A symmetric distribution with a single peak and a bell shape is known as a normal distribution.

6. Symmetry or Skewness A distribution is left-skewed (or negatively skewed) if the values are more spread out on the left, meaning that some low values are likely to be outliers. A distribution is right skewed or positively skewed if the values are more spread out on the right. It has a tail pulled toward the right.

7. What is the relationship between mean, median and mode for a normal distribution? Find the mean median and mode of: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7 Mean is 4. Median is 4. Mode is 4.

9. What is the relationship between mean, median and mode of a left-skewed distribution? Find the mean, median and mode of: 0, 5, 10, 20, 40, 45, 45, 50, 50, 50, 60, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70, 70, 70, 70 The mean is 51.5. The median is 60. The mode is 70.

11. What is the relationship between mean, median and mode of a right-skewed distribution? Find the mean, median, and mode of: 20, 20, 20, 20, 20, 20, 20, 20, 30, 30, 30, 30, 30, 30, 45, 45, 45, 50, 50, 60, 70, 90 The mean is 36.1. The median is 30. The mode is 20.

13. For each of the following situations, state whether you expect the distributions to be symmetric, left-skewed or right-skewed. House prices in the United States. Weight in a sample of 30 year old men. The heights of all players in the NBA.

14. Copyright © 2009 Pearson Education, Inc. Which is a better measure of “average” (or of the center of the distribution) for a skewed distribution: the median or the mean? Why?

15. Variation Variation describes how widely data are spread out about the center of a data set. How would you expect the variation to differ between times in a 5K city run and a 5K run in a state meet?

16. To summarize-- The general shape of a distribution can be discussed using: 1. The number of modes 2. Symmetry or skewness 3. Variation.