Unit 6A Characterizing Data

Definition The distribution of a variable (or data set) refers to the way its values are spread over all possible values. A distribution can be shown visually with a table or graph.

Measures of Center in a Distribution The mean is what we most commonly call the average value. It is defined as follows: The median is the middle value in the sorted data set (or halfway between the two middle values if the number of values is even). The mode is the most common value (or group of values) in a distribution.

Example 1 Eight grocery stores sell the PR energy bar for the following prices: $1.09 $1.29 $1.29 $1.35 $1.39 $1.49 $1.59 $1.79 Find the mean, median, and mode for these prices.

Effects of Outliers An outlier is a data value that is much higher or much lower than almost all other values. Consider the following data set of contract offers: $0 $0 $0 $0 $10,000,000 The mean contract offer is As displayed, outliers can pull the mean upward (or downward). The median and mode of the data are not affected.

Example 3 A newspaper surveys wages for assembly workers in regional high-tech companies and reports an average of $22 per hour. The workers at one large firm immediately request a pay raise, claiming that they work as hard as employees at other companies but their average wage is only $19. The management rejects their request, telling them that they are overpaid because their average wage, in fact, is $23. Can both sides be right? Explain.

Shapes of Distributions Two single-peaked (unimodal) distributions A double-peaked (bimodal) distribution

Symmetry A distribution is symmetric if its left half is a mirror image of its right half. Help students make connections between the overall distribution and the mean, median, mode and outliers of a population or data set. You may want to use concrete examples such as physical heights tend to be symmetrically distributed whereas the annual salaries of a medium-sized company may be right-skewed due to high-paying salaries of management. A distribution that is not symmetric must have values that tend to be more spread out on one side than the other. In this case we say the distribution is skewed.

Skewness 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. Help students make connections between the overall distribution and the mean, median, mode and outliers of a population or data set. You may want to use concrete examples such as physical heights tend to be symmetrically distributed whereas the annual salaries of a medium-sized company may be right-skewed due to high-paying salaries of management.

Definition Symmetry and Skewness A single-peaked distribution is symmetric if its left half is a mirror image of its right half. A single-peaked distribution is left-skewed if its values are more spread out on the left side of the mode. A single-peaked distribution is right-skewed if its values are more spread out on the right side of the mode. Help students make connections between the overall distribution and the mean, median, mode and outliers of a population or data set. You may want to use concrete examples such as physical heights tend to be symmetrically distributed whereas the annual salaries of a medium-sized company may be right-skewed due to high-paying salaries of management.

Example 6 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. Number of books read during the school year by fifth graders c. Speeds of cars on a road where a visible patrol car is using radar to detect speeders

Variation Variation describes how widely data values are spread out about the center of a distribution. From left to right, these three distributions have increasing variation.

Example 7 How would you expect the symmetry and variation to differ between times in the Olympic marathon and times in the New York marathon? Explain.