Copyright © 2014 Pearson Education. All rights reserved What Is Average? LEARNING GOAL Understand the difference between a mean, median, and mode and how each is affected by outliers. Also understand how these different types of “average” can lead to confusion and when it is appropriate to use a weighted mean.
Copyright © 2014 Pearson Education. All rights reserved Slide Mean, Median, and Mode Definitions—Measures of Center in a Distribution The mean is what we most commonly call the average value. It is found 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 data set. sum of all values total number of values mean =
Copyright © 2014 Pearson Education. All rights reserved Slide Figure 4.1 A histogram made from blocks would balance at the position of its mean.
Copyright © 2014 Pearson Education. All rights reserved Slide EXAMPLE 1 Price Data 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.
Copyright © 2014 Pearson Education. All rights reserved Solution: (cont.) To find the median, we first sort the data in ascending order: Because there are eight prices (an even number), there are two values in the middle of the list: $1.35 and $1.39. Therefore the median lies halfway between these two values, which we calculate by adding them and dividing by 2: Using the rounding rule, we could express the mean and median as $1.410 and $1.370 respectively. Slide EXAMPLE 1 Price Data $ $ median = = $ values below 2 middle values 3 values above
Copyright © 2014 Pearson Education. All rights reserved To explore the differences among the mean, median, and mode, imagine that the five graduating seniors on a college basketball team receive the following first-year contract offers to play in the National Basketball Association (zero indicates that the player did not receive a contract offer): 0 $2 $6 $8 $3,500,000 (1)What is the mean contract offer? (2) Is it fair to say the average senior on this basketball them received a $700,000 contract? Slide Effects of Outliers
Copyright © 2014 Pearson Education. All rights reserved Slide Definition An outlier in a data set is a value that is much higher or much lower than almost all others. A.Is the outlier affect the mean? B.Is the outlier affect the median? C.Is the outlier affect the mode?
Copyright © 2014 Pearson Education. All rights reserved Slide
Copyright © 2014 Pearson Education. All rights reserved Slide Confusion About “Average” A newspaper surveys wages for 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. EXAMPLE 4 Wage Dispute
Copyright © 2014 Pearson Education. All rights reserved Slide Solution: EXAMPLE 4 Wage Dispute Both sides can be right if they are using different definitions of average. In this case, the workers may be using the median while the management uses the mean. For example, imagine that there are only five workers at the company and their wages are $19, $19, $19, $19, and $39. The median of these five wages is $19 (as the workers claimed), but the mean is $23 (as management claimed).
Copyright © 2014 Pearson Education. All rights reserved Slide Weighted Mean Example: Suppose your course grade is based on four quizzes and one final exam. Each quiz counts as 15% of your final grade, and the final counts as 40%. Your quiz scores are 75, 80, 84, and 88, and your final exam score is 96. What is your overall score?
Copyright © 2014 Pearson Education. All rights reserved Slide Definition A weighted mean accounts for variations in the relative importance of data values. Each data value is assigned a weight and the weighted mean is weighted mean = sum of (each data value x its weight) sum of all weights
Copyright © 2014 Pearson Education. All rights reserved Slide Means with Summation Notation (Optional Section) The symbol Σ (the Greek capital letter sigma) is called the summation sign and indicates that a set of numbers should be added. We use the symbol x to represent each value in a data set, so we write the sum of all the data values as sum of all values = Σx
Copyright © 2014 Pearson Education. All rights reserved Slide We use n to represent the total number of values in the sample. Thus, the general formula for the mean is The symbol x is the standard symbol for the mean of a sample. When dealing with the mean of a population rather than a sample, statisticians instead use the Greek letter μ (mu). x = sample mean = = sum of all values total number of values Σx n
Copyright © 2014 Pearson Education. All rights reserved Slide Summation notation also makes it easy to express a general formula for the weighted mean. weighted mean = Σ(x × w) Σw