In any given situation where change occurs, it’s essential to quantify and understand that change. Absolute and relative changes are two different ways to describe the change in quantity, and each serves a different purpose.
Absolute change measures the actual increase or decrease in a quantity and is expressed in the same units as the quantity itself. For instance, if a population grows from [latex]50,000[/latex] to [latex]55,000[/latex], the absolute change is [latex]5,000[/latex] people.
Relative change, on the other hand, is concerned with the size of the absolute change in relation to the original quantity, providing a proportional perspective of change. It is often expressed as a percentage, showing how large the change is in comparison to the starting point. Continuing with the population example, a growth from [latex]50,000[/latex] to [latex]55,000[/latex] represents a relative change of [latex]10\%[/latex] of the original population.
These concepts are used in a variety of contexts, from the growth rate of investments to the speed of reactions in chemistry. They enable us to track progress, compare changes across different scales, and make predictions.
Absolute change has the same units as the original quantity.
Relative change gives a percent change.
The starting quantity is called the base of the percent change.
The following examples demonstrate how different perspectives of the same information can aid or hinder the understanding of a situation.
There are about [latex]75[/latex] QFC supermarkets in the United States. Albertsons has about [latex]215[/latex] stores. Compare the size of the two companies.
When we make comparisons, we must ask first whether an absolute or relative comparison.
The absolute difference is [latex]215 – 75 = 140[/latex]. From this, we could say “Albertsons has [latex]140[/latex] more stores than QFC.”
However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful.
There are two different relative changes we could calculate, depending on which store we use as the base:
Using QFC as the base, [latex]\displaystyle\frac{140}{75}=1.867[/latex]. This tells us Albertsons is [latex]186.7%[/latex] larger than QFC.
Using Albertsons as the base,[latex]\displaystyle\frac{140}{215}=0.651[/latex]. This tells us QFC is [latex]65.1%[/latex] smaller than Albertsons.
Notice both of these are showing percent differences.
We could also calculate the size of Albertsons relative to QFC:[latex]\displaystyle\frac{215}{75}=2.867[/latex], which tells us Albertsons is [latex]2.867[/latex] times the size of QFC.
Likewise, we could calculate the size of QFC relative to Albertsons:[latex]\displaystyle\frac{75}{215}=0.349[/latex], which tells us that QFC is [latex]34.9%[/latex] of the size of Albertsons.
We’ll wrap up our review of percents with a couple of cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.
A politician’s support increases from [latex]40\%[/latex] of voters to [latex]50\%[/latex] of voters. Describe the change.
We could describe this using an absolute change:
[latex]|50\%-40\%|=10\%[/latex]
Notice that since the original quantities were percents, this change also has the units of percent. In this case, it is best to describe this as an increase of [latex]10[/latex] percentage points.
This is the relative change, and we’d say the politician’s support has increased by [latex]25%[/latex].
Lastly, a caution against averaging percents.
A basketball player scores on [latex]40%[/latex] of [latex]2[/latex]-point field goal attempts, and on [latex]30%[/latex] of [latex]3[/latex]-point of field goal attempts. Find the player’s overall field goal percentage.
It is very tempting to average these values, and claim the overall average is [latex]35%[/latex], but this is likely, not correct, since most players make many more [latex]2[/latex]-point attempts than [latex]3[/latex]-point attempts. We don’t actually have enough information to answer the question.
Suppose the player attempted [latex]200[/latex] [latex]2[/latex]-point field goals and [latex]100[/latex] [latex]3[/latex]-point field goals. Then that player made [latex]200(0.40) = 80[/latex] [latex]2[/latex]-point shots and [latex]100(0.30) = 30[/latex] [latex]3[/latex]-point shots. Overall, they player made [latex]110[/latex] shots out of [latex]300[/latex], for a [latex]\displaystyle\frac{110}{300}=0.367=36.7\%[/latex] overall field goal percentage.
