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9.3 Describing Dispersion

9.3 Describing Dispersion
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  • 1. Dispersion of Rainbow Trout Egg Diameters

    The histogram shows the distribution of the diameters of 100 rainbow trout eggs. Describe the dispersion. (See Example 1 and Example 2.)

    • Worked-Out Solution

      The mean is about 4.3 millimeters and the standard deviation is about 0.3 millimeter. Add the number of egg diameters that are in the range from 4.0 millimeters to 4.6 millimeters.

      About 78 egg diameters lie within 1 standard deviation of the mean.

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      ______     ___                 ___       _____  
     /_   _//   / _ \\      ___     / _ \\    / ___// 
     `-| |,-   | / \ ||    /   ||  | / \ ||   \___ \\ 
       | ||    | \_/ ||   | [] ||  | \_/ ||   /    // 
       |_||     \___//     \__ ||   \___//   /____//  
       `-`'     `---`       -|_||   `---`   `-----`   
                             `-`                      
    
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  • 2. Percent of Rainbow Trout Egg Diameters within 2 Standard Deviations

    The histogram shows the distribution of the diameters of 100 rainbow trout eggs. What percent of the diameters lie within 2 standard deviations of the mean? (See Example 1 and Example 2.)

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     ______     _____    _    _      ___      ______  
    |      \\  |  ___|| | \  / ||   / _ \\   /_____// 
    |  --  //  | ||__   |  \/  ||  | / \ ||  `____ `  
    |  --  \\  | ||__   | .  . ||  | \_/ ||  /___//   
    |______//  |_____|| |_|\/|_||   \___//   `__ `    
    `------`   `-----`  `-`  `-`    `---`    /_//     
                                             `-`      
    
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  • 3. Percent of Rainbow Trout Egg Diameters within 3 Standard Deviations

    The histogram shows the distribution of the diameters of 100 rainbow trout eggs. What percent of the diameters lie within 3 standard deviations of the mean? (See Example 1 and Example 2.)

    • Worked-Out Solution

      The mean is about 4.3 millimeters and the standard deviation is about 0.3 millimeter. Data values that lie within 3 standard deviations lie within the range 4.3 - 3(0.3) to 4.3 + 3(0.3), or from 3.4 millimeters to 5.2 millimeters. Because all of the data values lie within that range, you can say that 100% of the data lie within 3 standard deviations of the mean.

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      _  _    __   __   _    _      ___      _____   
     | \| ||  \ \\/ // | \  / ||   / _ \\   |__  //  
     |  ' ||   \ ` //  |  \/  ||  | / \ ||    / //   
     | .  ||    | ||   | .  . ||  | \_/ ||   / //__  
     |_|\_||    |_||   |_|\/|_||   \___//   /_____|| 
     `-` -`     `-`'   `-`  `-`    `---`    `-----`  
                                                     
    
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  • 4. Dispersion of Daily High Temperatures in Pittsburgh

    The histogram shows the distribution of the daily high temperatures of Pittsburgh for 2010. Describe the dispersion. (See Example 1 and Example 2.)

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      ______     ___      ______   __   __   ______   
     /_   _//   / _ \\   /_____//  \ \\/ // |      \\ 
     `-| |,-   | / \ ||  `____ `    \ ` //  |  --  // 
       | ||    | \_/ ||  /___//      | ||   |  --  \\ 
       |_||     \___//   `__ `       |_||   |______// 
       `-`'     `---`    /_//        `-`'   `------`  
                         `-`                          
    
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  • 5. Percent of Daily High Temperatures in Pittsburgh within 1 Standard Deviation

    The histogram shows the distribution of the daily high temperatures of Pittsburgh for 2010. What percent of the temperatures lie within 1 standard deviation of the mean? (See Example 1 and Example 2.)

    • Worked-Out Solution

      The mean is about 60°F and the standard deviation is about 20°F. Add the number of temperatures that are in the range from 40°F to 80°F.

      Then divide by the total number of temperatures. Because 2010 was not a leap year, there were 365 days and thus a total of 365 daily temperatures. (Alternatively, you can add all of the bar heights in the histogram to reach a total number of 365 daily temperatures.)

      So, about 51% of the temperatures lie within 1 standard deviation of the mean.

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     _____      ______    _  __    _____     _____   
    |  __ \\   /_   _//  | |/ //  |  ___||  |__  //  
    | |  \ ||   -| ||-   | ' //   | ||__      / //   
    | |__/ ||   _| ||_   | . \\   | ||__     / //__  
    |_____//   /_____//  |_|\_\\  |_____||  /_____|| 
     -----`    `-----`   `-` --`  `-----`   `-----`  
                                                     
    
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  • 6. Percent of the Daily High Temperatures in Pittsburgh within 2 Standard Deviations

    The histogram shows the distribution of the daily high temperatures of Pittsburgh for 2010. What percent of the temperatures lie within 2 standard deviations of the mean? (See Example 1 and Example 2.)

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      ______     ___     _____       ___     _____    
     /_   _//   / _ \\  |  __ \\    / _ \\  |  __ \\  
     `-| |,-   | / \ || | |  \ ||  | / \ || | |  \ || 
       | ||    | \_/ || | |__/ ||  | \_/ || | |__/ || 
       |_||     \___//  |_____//    \___//  |_____//  
       `-`'     `---`    -----`     `---`    -----`   
                                                      
    
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