1. 4.1 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.
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2. 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 =
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3. Page 146
Figure 4.1 A histogram made from blocks would balance at the position of its mean.
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4. Rounding Rule for Statistical Calculations
State your answers with one more decimal place of precision than is found in the raw data.
Example: The mean of 2, 3, and 5 is , which we round to 3.3. Because the raw data are whole numbers, we round to the nearest tenth. As always, round only the final answer and not any intermediate values used in your calculations.
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5. Eight grocery stores sell the PR energy bar for the following prices:
EXAMPLE 1 Price Data
Eight grocery stores sell the PR energy bar for the following prices:
$ $ $ $ $ $ $ $1.79
Find the mean, median, and mode for these prices.
Solution:
The mean price is $1.41:
mean =
= $1.41
$ $ $ $ $ $ $ $1.79 8
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6. To find the median, we first sort the data in ascending order:
EXAMPLE 1 Price Data
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 $1.370 respectively.
3 values below middle values values above
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$ $1.39 2
median = = $1.37
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7. EXAMPLE 1 Price Data Solution: (cont.)
The mode is $1.29 because this price occurs more times than any other price.
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8. TECHNICAL NOTE
If the measure of center has the same number of significant digits as the original data, you can either include an extra zero or use the exact result without the extra decimal place. For example, the mean of 2 and 4 can be expressed as 3 or 3.0.
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9. Effects of Outliers
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):
$3,500,000
The mean contract offer is
mean = = $700,000
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$3,500,000 5
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10. Effects of Outliers
Is it therefore fair to say that the average senior on this basketball team received a $700,000 contract offer?
Not really. The problem is that the single player receiving the large offer makes the mean much larger than it would be otherwise.
If we ignore this one player and look only at the other four, the mean contract offer is zero.
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11. Definition
An outlier in a data set is a value that is much higher or much lower than almost all others.
In general, the value of an outlier has no effect on the median, because outliers don’t lie in the middle of a data set. Outliers do not affect the mode either. (However, the median may change if we delete an outlier, because we are changing the number of values in the data set.)
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12. Page 149
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13. Confusion About “Average”
EXAMPLE 4 Wage Dispute
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.
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14. 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).
Solution:
Page 150. Example 4 also illustrates confusion about “average”.
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15. Weighted Mean
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?
weighted mean =
= = 87.45
Following the rounding rule, we round this score to 87.5.
(75 × 15) + (80 × 15) + (84 × 15) + (88 × 15) + (96 × 40)
8745
100
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16. 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
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17. TIME OUT TO THINK
Because the weights are percentages in the course grade example, we could think of the weights as 0.15 and 0.40 rather than 15 and 40. Calculate the weighted mean by using the weights of 0.15 and Do you still find the same answer? Why or why not?
Page 152. There are two additional examples of weighted means on page 152.
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18. 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
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19. 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).
sum of all values
total number of values Σx n
x = sample mean = =
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20. Summation notation also makes it easy to express a general formula for the weighted mean.
Σ(x × w) Σw
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21. Means and Medians with Binned Data (Optional Section)
The ideas of this section can be extended to binned data simply by assuming that the middle value in the bin represents all the data values in the bin. For example, consider the following
table of 50 binned data values:
Bin
Frequency
0-6 10
7-13
14-20
21-27 20
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22. Bin
Frequency
0-6 10
7-13
14-20
21-27 20
The middle value of the first bin is 3, so we assume that the value of 3 occurs 10 times. Continuing this way, we have for the total of the 50 values in the table
(3 × 10) + (10 × 10) + (17 × 10) + (24 × 20) = 780
Thus, the mean is 780/50 = 15.6.
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23. Bin
Frequency
0-6 10
7-13
14-20
21-27 20
With 50 values, the median is between the 25th and 26th values. These values fall within the bin 14–20, so we call this bin the median class for the data.
The mode is the bin with the highest frequency—the bin in this case.
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24. The End
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