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# Average Percentage Calculator

How to average percentagesThe weighted average of percentagesExample of using the average percentage calculatorFAQs

Welcome to Omni's average percentage calculator, where we'll learn how to average percentages and what it actually means. Truth be told, half the time, the concept boils down to the well-known formula for the mean of a dataset. However, the other half concerns problems when the percentages correspond to samples of different sizes, and we can't apply the same reasoning there. Nevertheless, it still turns out to be a familiar formula, namely the weighted average of the percentages.

No worries: we'll teach you how to differentiate between the two scenarios and how to find the average percentage in each!

With our mean calculator, you can learn more on how to find the mean of a collection of numbers.

## How to average percentages

Let's recall the formal definition of a percentage:

💡 Percentages are fractions with $100$ in the denominator. We represent them using the symbol $\%$, which means that $a\% = a/100$ for any real number $a$.

For more detailed information, check out the percentage calculator.

We're used to the idea that we assign percentages to other numbers in a way we assign discounts to prices. However, mathematically speaking, they can appear on their own.

What is more, the above definition says that $a$ can be any real number. In other words, it can be an integer, a negative number, a decimal, or even a cube root. Mathematically speaking, of course. In real life, if a shop offered a $\sqrt[3]{75}$ discount for Black Friday, we'd call them mad. Similarly, if you report an error of $\sqrt[3]{5}$ in your measurements, your teacher is going to advise you to learn how to calculate percent error.

There is a sense to looking at percentages as regular numbers. After all, when we wonder how to calculate the average percentage, we ask ourselves if we can even average percentages. After all, they're something different, so it may seem unnatural. On the other hand, we know all about averaging numbers, don't we? Just to be on the safe side, let's recall the arithmetic average formula:

$\footnotesize \text{average} = \cfrac{(a_1 + a_2 + a_3 + ... + a_n)}{n}$

You can always take a look at the average calculator if you still need more details on this topic 😉

That being said, we need to be careful here. Wondering how to find the average percentage is often connected with the samples the percentages represent. To understand the difference, let's discuss an example.

Suppose that Amy, Brad, Colin, Debbie, and Edward all had a test in American literature. Some of them got $80\%$, and some got $40\%$. If we blindly apply the reasoning above, we'd say that the average result in percentages was:

$\footnotesize \cfrac{(80\% + 40\%)}{2} = \cfrac{120\%}{2}= 60\%$

After all, there were only two results, so we look for the mean of two values. With our mean, median, mode calculator you can learn how to find the mean of a set of values.

However, it clearly cannot be so. Indeed, five people took part in the test, so we should add five numbers instead of two. If, say, Amy, Brad, Colin, and Debbie got $80\%$, and Edward got $40\%$, then the actual average is:

$\footnotesize \begin{split} &\cfrac{(80\% + 80\% + 80\% + 80\% + 40\%)}{5} \\ &=\cfrac{360\%}{5}= 72\% \end{split}$

Quite a different result, isn't it?

The lesson we learn here is that we must always keep track of the differences in the groups' sizes and the percentages they correspond to. In fact, we can think of those sizes as weights when we look for the weighted average of a dataset. That is, for example, of a sequence of percentages.

## The weighted average of percentages

Recall the example from the end of the above section, where we talked about five people's results on a test. After learning how to find the average percentage, we got:

$\footnotesize \begin{split} &\cfrac{(80\% + 80\% + 80\% + 80\% + 40\%)}{5}\\ &=\cfrac{360\%}{5}= 72\% \end{split}$

Equivalently, we could have written:

$\footnotesize \cfrac{(4 \cdot 80\% + 1 \cdot 40\%)}{4+ 1} = \cfrac{360\%}{5}= 72\%$

Clearly, the new notation is shorter. Moreover, we immediately see how many people got the same result: $4$ got $80\%$, and $1$ got $40\%$. In other words, instead of treating the entries individually, we group them together according to their score.

What we got is the weighted average of the percentages with weights corresponding to how many people obtained the score. Fortunately, the calculations are the same as for the regular weighted average: if we have entries $a_1$, $a_2$, $a_3$, $...$, $a_n$ with respective weights $w_1$, $w_2$, $w_3$, $...$, $w_n$, then:

$\footnotesize \begin{split} &\text{weighted average} = \\ &\cfrac{a_1\!\cdot\! w_1 + a_2\! \cdot\! w_2 + a_3\! \cdot\! w_3 + ... + a_n\! \cdot\! w_n}{w_1 + w_2 + w_3 + ... +w_n} \end{split}$

If we translate the notation to our needs (i.e., to explain how to average percentages), $a_1$, $a_2$, $a_3$, $...$, $a_n$ will correspond to subsequent percentages, while $w_1$, $w_2$, $w_3$, $...$, $w_n$ will be the respective sample sizes of said percentages.

So what happens if all weights are the same (i.e., if all samples have the same size)? Well, if we denote the mutual weight by $w$, then:

$\footnotesize \begin{split} &\text{weighted average} = \\ &\cfrac{a_1 \cdot w + a_2 \cdot w + a_3 \cdot w + ... + a_n \cdot w}{w + w + w + ... +w} \\[1.2em] &=\cfrac{w \cdot \left( a_1 + a_2 + a_3 + ... + a_n \right)}{nw} \\[1em] &=\cfrac{ a_1 + a_2 + a_3 + ... + a_n}{n}, \end{split}$

by the rules of fraction simplification. In other words, the weight doesn't matter, and the weighted average of the percentages turns out to be the regular (non-weighted) average.

All in all, we see that learning how to calculate the average percentage boils down to learning about the usual weighted average. Still, let's go through one more example to show how it applies to real-life statistics. And we'll take the opportunity to do it using Omni's average percentage calculator.

## Example of using the average percentage calculator

Suppose that we've asked a thousand people whether they eat pancakes at least once a week. There were $300$ teenagers, $450$ people aged 20-49, and $250$ aged 50 and above. In the first group, $64\%$ said they ate pancakes every week. In the second, it was $42\%$, and in the third, $36\%$. Let's see how to calculate the average percentage of pancake-eaters in our thousand-strong group.

However, before we do the calculations ourselves, let's see how easy the task is with the average percentage calculator at hand. In the tool, at the top, we see an option for sample sizes. In our case, the groups differ in size, so we tick the checkbox.

That will trigger additional variable fields underneath corresponding to the dataset's percentages and sample sizes. They appear in pairs, each dedicated to one group. Select three in the number of entries (you can have up to ten entries in Omni's average percentage calculator). Looking back at our example, we input subsequently:

• Entry #1: $64\%$, $300$;

• Entry #2: $42\%$, $450$; and

• Entry #3: $36\%$, $250$.

Once you give the last value, the average percentage calculator will spit out the answer underneath together with the intermediate steps.

Now, let's see how to find the average percentage ourselves. First of all, we identify our dataset according to what we've learned in the above section: our subsequent percentages are $64\%$, $42\%$, and $36\%$, while the respective sample sizes are $300$, $450$, and $250$ people. Next, we use the weighted average of percentages formula:

$\footnotesize \begin{split} &\cfrac{64\% \cdot 300 + 42\% \cdot 450 + 36\% \cdot 250}{300 + 450 + 250}\\[1em] &=\cfrac{19,200\% + 18,900\% + 9,000\%}{1000} \\ &=\cfrac{47,100 \%}{1000} \\[1em] &=47.1 \% \end{split}$

It turns out that, on average, $47.1\%$ of people eat pancakes every week. But do they have them once a week or every day? Maybe we could introduce some new questions to the survey and have a more detailed study?

FAQs

### What is the average percentage?

The average percentage is simply an average over different percentages. You should, however, take into account the sample size each percentage represents and be mindful of whether you're writing each percentage as a fraction or as a decimal number.

### How do I calculate the average percentage?

To calculate the average percentage, you need to:

1. Determine the sample sizes corresponding to each percentage.
2. For each percentage, multiply it by its sample size.
3. Add all the numbers obtained in step 2.
4. Add all the sample sizes.
5. Divide the number from step 3 by that from step 4.
6. If you converted percentages to fractions in step 2, convert back.
7. The calculated result is the average percentage.

### Can I average percentages?

Yes, but you need to be careful. By definition, percentages are fractions with 100 in the denominator, so we can calculate their average as we do with any number. However, in practice, percentages rarely come alone, i.e., they usually describe how much of some value we should take. As such, when counting the mean, we might need to take the two together and not just the percentage itself.

### How do I add percentages together to get an average?

To add percentages together to get an average, you need to:

1. Determine the sample sizes corresponding to each percentage.
2. For each percentage, multiply it by its sample size.
3. Only now can you add the values.
4. If you need the average percentage,
• Add all the sample sizes;
• Divide the value from step 3 by that sum; and
• If you converted percentages to fractions in step 2, convert back.

### How can I calculate the average percentage in Excel?

To calculate the average percentage of some datasets in Excel:

1. Enter the list of percentages into column B.
2. Choose the built-in function AVERAGE.
3. Enter the starting parentheses “(“ and specify the range of cells in which the percentages are located.
4. Enter the closing parentheses and hit ENTER.

### How do I find the average of 4 percentages?

To find the average of four percentages, you need to:

1. Determine the sample sizes corresponding to each percentage.
2. For each percentage, multiply it by its sample size.
3. Add the four numbers obtained in step 2.
4. Add the four sample sizes.
5. Divide the number from step 3 by that from step 4.
6. If you converted percentages to fractions in step 2, convert back.
7. The calculated result is the average percentage.