In the last section, you calculated the mean of a data set. In this section, you will use the mean to calculate the mean absolute deviation of a set of data.

Mean absolute deviation is the mean of the absolute values of the deviations, or differences, between each number in the data set and the mean of the data set.

Mean is a useful measure for identifying a "typical" value in a data set, but sometimes, you also need to know how far the data set is spread out.

Consider the data table below, which shows the charge times, in hours, of two different computer batteries.

• Calculate the mean battery charge time for each battery.

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  Battery A: 5 hours
  Battery B: 5 hours

Construct a dot plot for each of the batteries. To do so, click and drag the point from the number onto the number line.

• What do you notice about the variability, or spread, of Battery A compared to Battery B when you study the dot plots?

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  The data values for Battery A are more spread out, and the data values for Battery B are more closely clustered.

• How does the variability of the data for each battery compare to the mean of the data values for each battery?

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  The mean of the data values for both batteries was five hours. However, the data values for Battery A are spread out farther from the mean, and the data values for Battery B are closer to the mean.

You can use mean absolute deviation to determine exactly how spread out a set of data is from the mean.

Click on the Begin button to begin the animation below to see how Marley calculated the mean absolute deviation for Battery A.

• Repeat the process that Marley used to calculate the mean absolute deviation for the data set from Battery B.
  
  

  Battery B

  4

  4

  4

  5

  5

  5

  5

  6

  6

  6

  
  
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  See a correct dot plot.

  
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  The absolute deviations from the mean, 5, for Battery B are shown in the table.

  

  The mean absolute deviation is (1 + 1 + 1 + 0 + 0 + 0 + 0 + 1 + 1 + 1) ÷ 10 = (6) ÷ 10 = 0.6.

  

  Pause and Reflect

  How does the mean absolute deviation for Battery A and Battery B compare to what you observed in the dot plots for Battery A and Battery B?

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  The dot plots reveal that the data values for Battery A are more spread out than those for Battery B. The data set for Battery A also has a greater mean absolute deviation than the data set for Battery B. The greater mean absolute deviation matches the greater spread of the data.

  How does mean absolute deviation describe the spread of a data set from the mean of the data set?

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  Mean absolute deviation describes how far a typical data value is from the mean of the data set. The lesser the mean absolute deviation, the closer the data values are to the mean of the data set. The greater the mean absolute deviation, the farther the data values are from the mean of the data set.

  Practice

  1. Devon recorded the maximum speeds of eight popular roller coasters. In miles per hour, the speeds were: 45, 48, 55, 60, 72, 74, 76, and 80. What is the mean absolute deviation for the speeds of this set of roller coasters?

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     To calculate the mean absolute deviation, first calculate the mean of the data set. Then, subtract each data value from the mean, and take the absolute value of the difference. These are the absolute deviations of each number in the data set. Finally, calculate the mean of the absolute deviations.
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     11.75 miles per hour
  2. The top five and bottom five salaries for a particular professional baseball team in a recent season are shown in the table below.
     

     Baseball Team Salaries (millions of dollars)
     Top Five Salaries	Bottom Five Salaries
     33.00    24.35     22.80     20.33     16.45	0.47     0.45    0.43    0.44     0.41

     
     

     What is the approximate mean absolute deviation of the top five salaries for this baseball team?

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     To calculate the mean absolute deviation, first calculate the mean of the data set. Then, subtract each data value from the mean, and take the absolute value of the difference. These are the absolute deviations of each number in the data set. Finally, calculate the mean of the absolute deviations.
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     About 4.2312 million dollars
  3. The table below shows the average low temperatures for six Texas cities for the month of January.
     

     City	Average January Low Temperature (°C)
     Amarillo	-5
     Laredo	8
     Houston	6
     El Paso	0
     Texarkana	1
     San Antonio	5

     
     

     What is the approximate mean absolute deviation of the average low temperatures for this set of Texas cities for the month of January?

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     Need a hint?

     To calculate the mean absolute deviation, first calculate the mean of the data set. Then, subtract each data value from the mean, and take the absolute value of the difference. These are the absolute deviations of each number in the data set. Finally, calculate the mean of the absolute deviations.
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     About 3.83°C