Changes in Percentages
When examining data, it can sometimes be helpful to compare two values by taking their absolute change or their relative change. It 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.
Recall
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 reference value and the second one will be the new value. We’ll use these terms in the skill explanations below.
Core skill:
Compute absolute change between two values
The absolute change between two values indicates the actual increase or decrease from one value to the next.
[latex]\text{absolute change }=\text{new value }-\text{reference value}[/latex]
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.
[latex]12,000\text{ dollars } -15,000\text{ dollars }=-3000\text{ dollars }[/latex]
The value decreased [latex]3,000[/latex] dollars.
When using percentage points as the unit, the absolute change is expressed in percentage points.
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.
[latex]80\text{ percentage points } -50\text{ percentage points }=30\text{ percentage points }[/latex]
The number of dentists who recommend the toothpaste increased [latex]30[/latex] percentage points.
Core skill:
Compute relative change between two values
The relative change between two values is a ratio that expresses the absolute change between two values relative to the reference value.
[latex]\dfrac{\text{absolute change}}{\text{reference value}}=\dfrac{\text{new value }-\text{reference value}}{\text{reference value}}[/latex]
The relative change in two values is unitless, since the units cancel from the numerator to the denominator. We can express relative change as a percentage by multiplying by [latex]100[/latex]%.
Ex. For 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.
[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]
The value decreased [latex]20[/latex]%.
When using percentage points as the unit, the relative change is unitless, but may be expressed as a percentage.
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.
[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]
The number of dentists who would recommend the toothpaste increased [latex]60[/latex]%.
It can be tricky to handle the units when they are in percentage points. Use this rule of thumb when the original units are percentage points:
- The absolute difference will be in percentage points, indicating the change occurred in a number of percentage points.
- The relative difference will be expressed with the [latex]\%[/latex] symbol, indicating that the amount of something has changed by some percent relative to its original amount.
Example
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.
What was the absolute change in the percentage of workers employed at the plant?
Show Solution
[latex]27\text{ percentage points } -60\text{ percentage points }=-33\text{ percentage points }[/latex]
The number of workers from the town who worked at the plant decreased [latex]33[/latex] percentage points.
What was the relative change?
Show Solution
[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]
The number of workers from the town who worked at the plant decreased [latex]55[/latex]%.
Now you try this scenario about graduation rates.
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.
question 10
Hint
Begin with the percentage from 2014.
question 11
Hint
See the recall box above for details. [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 reference value is the starting value.
question 12
Hint
What do you think? Have you been given any information about a number of students in either class?
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.
