1.7: Averaging Percents
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1.2: Averaging Percents
We need to be very careful when it comes to averaging percents!
Example 11
A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point 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 35%, but this is likely not correct, since most players make many more 2-point attempts than 3-point attempts. We don't actually have enough information to answer the question. Suppose the player attempted 200 2-point field goals and 100 3-point field goals. Then they made 200(0.40) = 80 2-point shots and 100(0.30) = 30 3-point shots. Overall, they made 110 shots out of 300, for a 110/300 = 0.367 = 36.7% overall field goal percentage.
So, in general, it is not correct to average percents, particularly percent changes, or as shown in the example above, percentages that are calculated relative to different bases.
But it can make sense to average percent scores if all are calculated relative to the same base. Suppose for example that you have the following scores on four tests: 75%, 90%, 95%, and 60%, each of which counts with the same weight. Then it is correct to say that your average test score is:
This is known as the arithmetic mean of the four test scores. Its significance is that the net result from the four scores is equivalent to earning an 80% on each of the tests.
If tests are worth a certain percentage of the overall class grade, then to calculate the final average, we use a weighted average. Suppose a certain class uses the following grade distribution:
- Homework: 30%
- Quizzes: 20%
- Tests: 50%
If your homework average is 87%, your quiz average is 70%, and your test average is 80%, then your final grade is calculated like this:
So, it looks like you may have just squeaked by with a B!
In general, a weighted average is calculated by multiplying each score by its category weight and then adding all of these products.
If n scores x1, x2, x3, …, xn have weights w1, w2, w3, …, wn, respectively, then the weighted average is given by the sum of products:
w1x1 + w2x2 + w3x3 + ⋯ + wnxn
The weights should be written in their decimal form for the calculation and should add to 1. (If the weights do not add to 1, then divide the total by the sum of the weights.)
Example 12
Suppose the final exam in a certain class is worth 40% of the final grade and your class average including everything but the final exam is 83%. What is your final grade if you earn 74% on the final exam? What is the minimum score needed to pass the class with a final average of at least 70%?
If the final exam is worth 40% of the grade, then everything else is worth 60%. So, if you make a 74 on the final exam, then your final class average is:
The final grade is 79.4%. (It looks like you did not keep your B average!)
If your final exam grade is x, then the final class average is 0.6(83) + 0.4x. To determine what score is needed for a final result of 70, we can set the expression equal to 70 and solve for x:
49.8 + 0.4x = 70
0.4x = 20.2
x = 50.5
So, to pass the class with a grade of 70, you need a 50.5% score on the final exam.
Geometric Mean
What if we have a series of percentage increases and decreases? Can we find the average rate of change? Yes, but a new method is required. The arithmetic mean does not make sense for averaging percent changes. If we want to find the average for a set of percent changes, we need a different kind of average, known as the geometric mean.
Suppose a quantity increased by 60% (growth factor 160%), then again by 20% (growth factor 120%). The net increase is found by multiplying the percentage growth factors:
The correct method to find the average rate of increase is to take the square root of the product of the two growth factors:
This is the geometric mean of the two growth factors, showing the average increase is 38.56%.
The geometric mean of the numbers x1, x2, x3, …, xn is the nth root of their product:
n√(x1 · x2 · x3 · ⋯ · xn)
Example 13
The revenue of a certain company increased by 10% in the first quarter, 12% in the second quarter, and 5% in the third quarter. In the fourth quarter, it decreased by 8%. What is the average rate of growth?
An ordinary (arithmetic) average of these rates does not make sense. We need to calculate the geometric mean of the four growth multipliers (1 plus the rate, in decimal form):
The geometric mean is the 4th root of their product.
Try it Now
A store's sales increased by 20% in January and decreased by 10% in February. What is the average monthly growth rate over these two months?
This section is remixed from Quantitative Reasoning (Lachniet et al., 2026), §1.2, licensed CC-BY-SA.