{"id":405,"date":"2022-06-17T16:56:34","date_gmt":"2022-06-17T16:56:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/?post_type=chapter&p=405"},"modified":"2022-06-17T16:56:34","modified_gmt":"2022-06-17T16:56:34","slug":"applications-of-bar-graphs-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/applications-of-bar-graphs-background-youll-need-3\/","title":{"raw":"Applications of Bar Graphs: Background You'll Need 3","rendered":"Applications of Bar Graphs: Background You’ll Need 3"},"content":{"raw":"

Changes in Percentages<\/strong><\/h3>\r\nWhen examining data, it can sometimes be helpful to compare two values by taking their absolute change or their relative change.\u00a0It can be challenging, though, to compare relative change when data are presented as percentages. Before we try to understand how to apply these comparisons to percentages, please refresh your understanding of absolute and relative difference in the Recall box below.\r\n
\r\n

Recall<\/h3>\r\nWhen computing absolute change and relative change between two values, we need one value that we think of as the starting point and the other that represents the value after some change. We'll call the starting point value the\u00a0reference value<\/strong> and the second one will be the\u00a0new value<\/strong>. We'll use these terms in the skill explanations below.\r\n\r\nCore skill:<\/span>\r\n[reveal-answer q=\"754599\"]Compute absolute change between two values[\/reveal-answer]\r\n[hidden-answer a=\"754599\"]\r\n\r\nThe\u00a0absolute change<\/strong> between two values indicates the actual increase or decrease from one value to the next.\r\n\r\n \r\n

[latex]\\text{absolute change }=\\text{new value }-\\text{reference value}[\/latex]<\/p>\r\n \r\n\r\nAbsolute change retains the units on the two values. For example a decrease in value from $[latex]15,000[\/latex] to $[latex]12,000[\/latex] is expressed in dollars.\r\n\r\n \r\n

[latex]12,000\\text{ dollars } -15,000\\text{ dollars }=-3000\\text{ dollars }[\/latex]<\/p>\r\n \r\n

The value decreased [latex]3,000[\/latex] dollars.\u00a0<\/em><\/strong><\/p>\r\n \r\n\r\n

\r\n\r\nWhen using percentage points as the unit, the absolute change is expressed in percentage points.<\/strong>\r\n\r\nEx. A survey indicates that [latex]50[\/latex]% of dentists polled would recommend a toothpaste. After the company improves the toothpaste, a new survey indicates that [latex]80[\/latex]% of dentists polled would recommend it.\r\n\r\n \r\n

[latex]80\\text{ percentage points } -50\\text{ percentage points }=30\\text{ percentage points }[\/latex]<\/p>\r\n \r\n

The number of dentists who recommend the toothpaste increased [latex]30[\/latex] percentage points.<\/em><\/strong><\/p>\r\n \r\n\r\n

\r\n\r\n[\/hidden-answer]\r\n\r\nCore skill:<\/span>\r\n[reveal-answer q=\"434757\"]Compute relative change between two values[\/reveal-answer]\r\n[hidden-answer a=\"434757\"]\r\n\r\nThe\u00a0relative change<\/strong> between two values is a ratio that expresses the absolute change between two values relative to the reference value.\r\n\r\n \r\n

[latex]\\dfrac{\\text{absolute change}}{\\text{reference value}}=\\dfrac{\\text{new value }-\\text{reference value}}{\\text{reference value}}[\/latex]<\/p>\r\n \r\n\r\nThe relative change in two values is unitless, since the units cancel from the numerator to the denominator.\u00a0We can express relative change as a percentage by multiplying by [latex]100[\/latex]%.\r\n\r\n \r\n\r\nEx.\u00a0For example, the relative change in value from $[latex]15,000[\/latex] to $[latex]12,000[\/latex] is unitless, but may be expressed as a percentage.\r\n\r\n \r\n

[latex]\\dfrac{\\left(12,000-15,000\\right)\\text{ dollars }}{15,000\\text{ dollars }}=\\dfrac{-3,000 \\cancel{\\text{ dollars} }}{15,000\\cancel{\\text{ dollars} }}=-\\dfrac{1}{5}=-0.20\\times100\\%=-20\\%[\/latex]<\/p>\r\n \r\n

The value decreased [latex]20[\/latex]%.\u00a0<\/em><\/strong><\/p>\r\n \r\n\r\n

\r\n\r\nWhen using percentage points as the unit, the relative change is unitless, but may be expressed as a percentage.<\/strong>\r\n\r\n \r\n\r\nEx. A survey indicated that [latex]50[\/latex]% of dentists polled would recommend a toothpaste. After the company improved the toothpaste, a new survey indicated that [latex]80[\/latex]% of dentists polled would recommend it.\r\n\r\n \r\n

[latex]\\dfrac{\\left(80-50\\right)\\text{ percentage points }}{50\\text{ percentage points }}=\\dfrac{30\\cancel{\\text{ percentage points} } }{50\\cancel{\\text{ percentage points} } }=\\dfrac{3}{5}=0.6\\times100\\%=60\\%[\/latex]<\/p>\r\n \r\n

The number of dentists who would recommend the toothpaste increased [latex]60[\/latex]%.<\/strong>\r\n<\/em><\/p>\r\n \r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIt can be tricky to handle the units when they are in percentage points. Use this rule of thumb when\u00a0the original units are percentage points:\r\n

    \r\n \t
  • The absolute difference<\/strong> will be in percentage points, indicating the change occurred in a number of percentage points.<\/li>\r\n \t
  • The relative difference<\/strong> will be expressed with the [latex]\\%[\/latex] symbol, indicating that the amount of something has changed by some percent relative to its original amount.<\/li>\r\n<\/ul>\r\n
    \r\n

    Example<\/h3>\r\nConsider a small town that suffered the downsizing of a large manufacturing plant. In 2010, [latex]60[\/latex]% of the town's workers were employed in the plant. But in 2011, only [latex]27[\/latex]% of the town's workers remained.\r\n\r\nWhat was the absolute change in the percentage of workers employed at the plant?\r\n\r\n[reveal-answer q=\"583613\"]Show Solution[\/reveal-answer][hidden-answer a=\"583613\"]\r\n

    [latex]27\\text{ percentage points } -60\\text{ percentage points }=-33\\text{ percentage points }[\/latex]<\/p>\r\n

    The number of workers from the town who worked at the plant decreased [latex]33[\/latex] percentage points.<\/em><\/strong><\/p>\r\n[\/hidden-answer]\r\n\r\nWhat was the relative change?\r\n\r\n[reveal-answer q=\"646464\"]Show Solution[\/reveal-answer]\r\n

    [hidden-answer a=\"646464\"]<\/p>\r\n

    [latex]\\dfrac{\\left(27-60\\right)\\text{ percentage points }}{60\\text{ percentage points }}=\\dfrac{-33\\cancel{\\text{ percentage points} } }{60\\cancel{\\text{ percentage points} } }=-\\dfrac{11}{20}=-0.55\\times100\\%=-55\\%[\/latex]<\/p>\r\n \r\n

    The number of workers from the town who worked at the plant decreased [latex]55[\/latex]%.<\/strong><\/em><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try this scenario about graduation rates.\r\n\r\nIn 2013, [latex]80[\/latex]% of the original class of 2013 (who started in 2009) graduated from Valley High School, and in 2014, [latex]73[\/latex]% of the original class of 2014 graduated.\r\n

    \r\n

    question 10<\/h3>\r\n[ohm_question hide_question_numbers=1]240797[\/ohm_question]\r\n\r\n[reveal-answer q=\"818757\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"818757\"]Begin with the percentage from 2014.[\/hidden-answer]\r\n\r\n<\/div>\r\n
    \r\n

    question 11<\/h3>\r\n[ohm_question hide_question_numbers=1]240798[\/ohm_question]\r\n\r\n[reveal-answer q=\"220308\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"220308\"]See the recall box above for details.\u00a0 [latex]\\text{relative change }=\\dfrac{\\text{absolute change}}{\\text{reference value}}=\\dfrac{\\text{new value }-\\text{reference value}}{\\text{reference value}}\\times100\\%[\/latex], where the\u00a0reference value<\/em> is the starting value.[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n
    \r\n

    question 12<\/h3>\r\n[ohm_question hide_question_numbers=1]240800[\/ohm_question]\r\n\r\n[reveal-answer q=\"55079\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"55079\"]What do you<\/em> think? Have you been given any information about a number of students in either class?[\/hidden-answer]\r\n\r\n<\/div>\r\n
    Now that you've gained some experience reading and interpreting graphs that represent percentages, it's time to move on to the next section and activity.<\/div>","rendered":"

    Changes in Percentages<\/strong><\/h3>\n

    When examining data, it can sometimes be helpful to compare two values by taking their absolute change or their relative change.\u00a0It can be challenging, though, to compare relative change when data are presented as percentages. Before we try to understand how to apply these comparisons to percentages, please refresh your understanding of absolute and relative difference in the Recall box below.<\/p>\n

    \n

    Recall<\/h3>\n

    When computing absolute change and relative change between two values, we need one value that we think of as the starting point and the other that represents the value after some change. We’ll call the starting point value the\u00a0reference value<\/strong> and the second one will be the\u00a0new value<\/strong>. We’ll use these terms in the skill explanations below.<\/p>\n

    Core skill:<\/span><\/p>\n

    Compute absolute change between two values<\/span><\/p>\n
    \n

    The\u00a0absolute change<\/strong> between two values indicates the actual increase or decrease from one value to the next.<\/p>\n

     <\/p>\n

    [latex]\\text{absolute change }=\\text{new value }-\\text{reference value}[\/latex]<\/p>\n

     <\/p>\n

    Absolute change retains the units on the two values. For example a decrease in value from $[latex]15,000[\/latex] to $[latex]12,000[\/latex] is expressed in dollars.<\/p>\n

     <\/p>\n

    [latex]12,000\\text{ dollars } -15,000\\text{ dollars }=-3000\\text{ dollars }[\/latex]<\/p>\n

     <\/p>\n

    The value decreased [latex]3,000[\/latex] dollars.\u00a0<\/em><\/strong><\/p>\n

     <\/p>\n

    \n

    When using percentage points as the unit, the absolute change is expressed in percentage points.<\/strong><\/p>\n

    Ex. A survey indicates that [latex]50[\/latex]% of dentists polled would recommend a toothpaste. After the company improves the toothpaste, a new survey indicates that [latex]80[\/latex]% of dentists polled would recommend it.<\/p>\n

     <\/p>\n

    [latex]80\\text{ percentage points } -50\\text{ percentage points }=30\\text{ percentage points }[\/latex]<\/p>\n

     <\/p>\n

    The number of dentists who recommend the toothpaste increased [latex]30[\/latex] percentage points.<\/em><\/strong><\/p>\n

     <\/p>\n

    \n<\/div>\n<\/div>\n

    Core skill:<\/span><\/p>\n

    Compute relative change between two values<\/span><\/p>\n
    \n

    The\u00a0relative change<\/strong> between two values is a ratio that expresses the absolute change between two values relative to the reference value.<\/p>\n

     <\/p>\n

    [latex]\\dfrac{\\text{absolute change}}{\\text{reference value}}=\\dfrac{\\text{new value }-\\text{reference value}}{\\text{reference value}}[\/latex]<\/p>\n

     <\/p>\n

    The relative change in two values is unitless, since the units cancel from the numerator to the denominator.\u00a0We can express relative change as a percentage by multiplying by [latex]100[\/latex]%.<\/p>\n

     <\/p>\n

    Ex.\u00a0For example, the relative change in value from $[latex]15,000[\/latex] to $[latex]12,000[\/latex] is unitless, but may be expressed as a percentage.<\/p>\n

     <\/p>\n

    [latex]\\dfrac{\\left(12,000-15,000\\right)\\text{ dollars }}{15,000\\text{ dollars }}=\\dfrac{-3,000 \\cancel{\\text{ dollars} }}{15,000\\cancel{\\text{ dollars} }}=-\\dfrac{1}{5}=-0.20\\times100\\%=-20\\%[\/latex]<\/p>\n

     <\/p>\n

    The value decreased [latex]20[\/latex]%.\u00a0<\/em><\/strong><\/p>\n

     <\/p>\n

    \n

    When using percentage points as the unit, the relative change is unitless, but may be expressed as a percentage.<\/strong><\/p>\n

     <\/p>\n

    Ex. A survey indicated that [latex]50[\/latex]% of dentists polled would recommend a toothpaste. After the company improved the toothpaste, a new survey indicated that [latex]80[\/latex]% of dentists polled would recommend it.<\/p>\n

     <\/p>\n

    [latex]\\dfrac{\\left(80-50\\right)\\text{ percentage points }}{50\\text{ percentage points }}=\\dfrac{30\\cancel{\\text{ percentage points} } }{50\\cancel{\\text{ percentage points} } }=\\dfrac{3}{5}=0.6\\times100\\%=60\\%[\/latex]<\/p>\n

     <\/p>\n

    The number of dentists who would recommend the toothpaste increased [latex]60[\/latex]%.<\/strong>
    \n<\/em><\/p>\n

     <\/p>\n<\/div>\n<\/div>\n<\/div>\n

    It can be tricky to handle the units when they are in percentage points. Use this rule of thumb when\u00a0the original units are percentage points:<\/p>\n

      \n
    • The absolute difference<\/strong> will be in percentage points, indicating the change occurred in a number of percentage points.<\/li>\n
    • The relative difference<\/strong> will be expressed with the [latex]\\%[\/latex] symbol, indicating that the amount of something has changed by some percent relative to its original amount.<\/li>\n<\/ul>\n
      \n

      Example<\/h3>\n

      Consider a small town that suffered the downsizing of a large manufacturing plant. In 2010, [latex]60[\/latex]% of the town’s workers were employed in the plant. But in 2011, only [latex]27[\/latex]% of the town’s workers remained.<\/p>\n

      What was the absolute change in the percentage of workers employed at the plant?<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      [latex]27\\text{ percentage points } -60\\text{ percentage points }=-33\\text{ percentage points }[\/latex]<\/p>\n

      The number of workers from the town who worked at the plant decreased [latex]33[\/latex] percentage points.<\/em><\/strong><\/p>\n<\/div>\n<\/div>\n

      What was the relative change?<\/p>\n

      Show Solution<\/span><\/p>\n

      \n

      \n

      [latex]\\dfrac{\\left(27-60\\right)\\text{ percentage points }}{60\\text{ percentage points }}=\\dfrac{-33\\cancel{\\text{ percentage points} } }{60\\cancel{\\text{ percentage points} } }=-\\dfrac{11}{20}=-0.55\\times100\\%=-55\\%[\/latex]<\/p>\n

       <\/p>\n

      The number of workers from the town who worked at the plant decreased [latex]55[\/latex]%.<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n

      Now you try this scenario about graduation rates.<\/p>\n

      In 2013, [latex]80[\/latex]% of the original class of 2013 (who started in 2009) graduated from Valley High School, and in 2014, [latex]73[\/latex]% of the original class of 2014 graduated.<\/p>\n

      \n

      question 10<\/h3>\n