4.2 Shapes of Distributions ! Symmetry " Symmetrical or asymmetrical " If symmetrical, mounded or flat? ! Skew " Right, left ! Peaks or Modes " Unimodal, bimodal, multiple peaks ! Spread " Narrow spread or wide spread 1 Histogram of Octane Histogram of Octane Rating 10 9 8 Symmetrical 7 y c 6 n e 5 One peak u q 4 e r F 3 2 1 0 86 87 88 89 90 91 92 93 94 95 96 Octane A distribution is symmetric if its left half is a mirror image of its right half. 2 Flat or Uniform Figure 4.4 Perfectly flat 3 Flat or Uniform Not perfectly flat, but close. We want to describe the general shape of the distribution. 4 Not symmetrical ! 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. 5 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. 6 Right-skewed pH of Pork Loins 80 70 60 y 50 c n e 40 u q e r 30 F 20 10 0 5.0 5.5 6.0 6.5 7.0 pH 7 Right-skewed ‘ski slope’ to Salary the right 60 40 Frequency 20 0 50000 100000 150000 sarlar(dollars) 8 Left-skewed ‘ski slope’ to Flexibility Index of Young Adult Men the left 20 15 y c n e 10 u q e r F 5 0 1 2 3 4 5 6 7 8 9 10 Flexibility Index 9 Right-skewed or Left-skewed ! 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. 10 Number of Modes ! If there are numerous obvious peaks, we say there are multiple modes. " One peak # unimodal " Two peaks # bimodal " More than two peaks # multiple modes ! Some peaks can be ‘major’ peaks and some can be ‘minor’ peaks 11 Multiple Peaks Major peak Size of Diamonds (carats) 15 Minor peak y 10 c n e u q e r F 5 0 0.1 0.2 0.3 0.4 Size (carats) 12 Time-Series Diagrams – example Homes sold in Iowa City by zip code and month Multiple peaks Year (data by the month) 13 Measures of Center ! These help describe a distribution, too. ! A typical or representative value. " Mean, Median, Mode ! Summary of the whole batch of numbers. ! For symmetric distributions – easy. 14 Histogram of Octane Histogram of Octane Rating 10 9 8 7 y c 6 n e 5 u q 4 e r F 3 2 1 0 86 87 88 89 90 91 92 93 94 95 96 Octane Center 15 Spread ! Variation matters. " Tightly clustered? " Spread out? " Low and high values? Variation describes how widely data are spread out about the center of a data set. 16 Spread ! Variation matters. 0 5 10 0 5 10 0 5 10 17 Comparing Distributions ! How do the distributions compare in terms of… " Shape? " Center? " Spread? 18 Workout Times: Men 5 4 3 Count 2 1 30 40 50 60 70 80 90 100 19 Workout Times: Women 5 4 3 Count 2 1 30 40 50 60 70 80 90 100 20 5 4 3 Count 2 Men 1 30 40 50 60 70 80 90 100 5 4 3 Count 2 1 Women 30 40 50 60 70 80 90 100 21 To describe a distribution, use… ! Shape ! Center ! Spread 22 .