# Average Percentage Calculator

By Maciej Kowalski, PhD candidate
Last updated: May 05, 2021

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!

## 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`.

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 `∛75` discount for Black Friday, we'd call them mad.

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:

`average = (a1 + a2 + a3 + ... + an) / n`.

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:

`(80% + 40%) / 2 = 120% / 2 = 60%`.

After all, there were only two results, so we look for the mean of two 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:

`(80% + 80% + 80% + 80% + 40%) / 5 = 360% / 5 = 72%`.

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:

`(80% + 80% + 80% + 80% + 40%) / 5 = 360% / 5 = 72%`.

Equivalently, we could have written:

`(4 * 80% + 1 * 40%) / (4 + 1) = 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 `a1`, `a2`, `a3`, ..., `an` with respective weights `w1`, `w2`, `w3`, ..., `wn`, then:

`weighted average = (a1 * w1 + a2 * w2 + a3 * w3 + ... + an * wn) / (w1 + w2 + w3 + ... + wn)`.

If we translate the notation to our needs (i.e., to explain how to average percentages), `a1`, `a2`, `a3`, ..., `an` will correspond to subsequent percentages, while `w1`, `w2`, `w3`, ..., `wn` 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:

`weighted average = (a1 * w + a2 * w + a3 * w + ... + an * w) / (w + w + w + ... + w)`

`= w * (a1 + a2 + a3 + ... + an) / (n * w)`

`= (a1 + a2 + a3 + ... + an) / n`,

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 a question about the sample sizes. In our case, the groups differ in size, so we choose "No".

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. Note how initially, we can only see two such sections, but new ones show up once you start inputting data (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:

`(64% * 300 + 42% * 450 + 36% * 250) / (300 + 450 + 250)`

`= (19,200% + 18,900% + 9,000%) / 1,000`

`= 47,100% / 1,000`

`= 47.1%`.

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? Make sure to check out Omni's section on descriptive statistics to find the right tools for the job!

## FAQ

### 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 average percentages?

In order to average percentages, you need to:

1. Determine the sample size of each percentage.
2. For each input, multiply the percentage by its sample size.
3. Sum up all the values from step 2.
4. Sum up all the sample sizes.
5. Divide the sum from step 3 by the one from step 4.
6. If you changed percentages to fractions in step 2, change them back.
7. The result gives the average percentage of your dataset.

### 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.

### Is it mathematically correct to average percentages?

Yes, but you should take special care with your dataset. The definition states that percentages are fractions with the denominator equal to `100`, so we can calculate their average the way we usually do. However, in practice, percentages often correspond to other values, i.e., they usually describe what part of some number we should take. Therefore, when we count the mean, we might need to take the percentage together with its corresponding value.

### 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.
Maciej Kowalski, PhD candidate
Are all sample sizes the same?
Yes
Entry #1
Percentage
%
Entry #2
Percentage
%
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