Putting Statistics to Work Copyright © 2011 Pearson Education, Inc.

Characterizing Data Unit 6A Activity: Bankrupting the Auto Companies 4 groups Discuss and be ready to share info from your group. Copyright © 2011 Pearson Education, Inc.

Definition The distribution of a variable (or data set) describes the values taken on by the variable and the frequency (or relative frequency) of these values. Copyright © 2011 Pearson Education, Inc.

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. Copyright © 2011 Pearson Education, Inc.

Price Data CN (1a-c) 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 1.Find the a)mean, b)median and c)mode for these prices Copyright © 2011 Pearson Education, Inc.

Mean vs. Average Copyright © 2011 Pearson Education, Inc.

Finding the Median for an Odd Number of Values Example: Find the median of the data set below. 6.72 3.46 3.60 6.44 26.70 (data set) 3.46 3.60 6.44 6.72 26.70 (sorted list) (odd number of values) median is 6.44 exact middle Copyright © 2011 Pearson Education, Inc.

Finding the Median for an Even Number of Values Example: Find the median of the data set below. 6.72 3.46 3.60 6.44 (data set) 3.46 3.60 6.44 6.72 (sorted list) (even number of values) 3.60 + 6.44 2 median is 5.02 Copyright © 2011 Pearson Education, Inc.

Finding the Mode Example: Find the mode of each data set below. Mode is 5 Bimodal (2 and 6) No Mode a. 5 5 5 3 1 5 1 4 3 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 c. 1 2 3 6 7 8 9 10 Copyright © 2011 Pearson Education, Inc.

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 $2,500,000 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. Copyright © 2011 Pearson Education, Inc.

Mistake CN (2) A track coach wants to determine an appropriate heart rate for her athletes during their workouts. She chooses five of her best runners and asks them to wear heart rate monitors during a workout. In the middle of the workout, she reads the following heart rates for five athletes: 130, 135, 140, 145, 325. 2. Which is a better measure of the average in this case: the mean or the median? Why? Copyright © 2011 Pearson Education, Inc.

Wage Dispute CN (3) A newspaper surveys wages for assembly workers in regional high-tech companies and reports and 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 $29. The management rejects their request, telling them that they are overpaid because their average wage, in fact, is $23. 3. Can both sides be right? Explain. Copyright © 2011 Pearson Education, Inc.

Which Mean? CN (4) All 100 first year students at a small college take three courses in the Core Studies Program. The first two courses are taught in large lectures, with all 100 students in a single class. The third course is taught in ten classes of 10 students each. Students and administrators get into an argument about whether classes are too large. The students claim that the mean size of their Core Studies classes is 70. The administrators claim that the mean class size is only 25 students. 4. Can both sides be right? Explain Copyright © 2011 Pearson Education, Inc.

Shapes of Distributions Two single-peaked (unimodal) distributions A double-peaked (bimodal) distribution Copyright © 2011 Pearson Education, Inc.

Number of Peaks CN (5a-c) 5. How many peaks would you expect for each of the following distributions and why? a) Heights of all women at a college b) heights of all students at a college c) The numbers of people with particular last digits (0 through 9) in their Social Security numbers. Copyright © 2011 Pearson Education, Inc.

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. Copyright © 2011 Pearson Education, Inc.

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. Copyright © 2011 Pearson Education, Inc.

Skewness CN (6a-c) 6. For each of the following situations, state whether you expect the distribution to be symmetric, left skewed, or right skewed and 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 Copyright © 2011 Pearson Education, Inc.

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. Copyright © 2011 Pearson Education, Inc.

Variation in Marathon Times CN (7) 7. How would you expect the variation to differ between times in the Olympic marathon and the times in the New York Marathon? Explain. Copyright © 2011 Pearson Education, Inc.

Quick Quiz CN (8) 8. Choose the best answer to the ten multiple choice questions. Copyright © 2011 Pearson Education, Inc.

Homework 6A p. 379:1-12 1 web (write at least 5 sentences) Salary Data New York Marathon Tax Statistics Education Statistics 1 world (write at least 5 sentences) Averages in the News Daily Averages Distributions in the News Class Notes 1-8 Copyright © 2011 Pearson Education, Inc.