Can you just add up all of those percentages and divide by the total number to get an average?

This is only possible if each percentage comes from a population of the same size. If populations are different sizes, then individual percentages must be weighted according to sample size. Otherwise small and large populations will have equal effect, leading to an incorrect result.

How do you calculate the average of percentages with different sample sizes?

You multiply each percentage by the corresponding sample size, add up the products and divide the result by the sum of the sample sizes. For example, if there are 20 people with a percentage of 80% and 80 people with a percentage of 90%, you calculate ((80*20) + (90*80))/100. The result is 88%.

What is the difference between a simple average and a weighted average?

A simple average treats each value equally and divides the sum by the number of values. A weighted average multiplies each value by a weight. In this case the weights are the sample sizes, so values with larger samples have more influence on the result. If all the weights are equal then you get the same answer from both the simple and weighted averages.

Can average percentage go over 100?

Yes, it is possible if the values entered can exceed 100% themselves. This applies for example to growth rates or efficiency data. For results where there is normally a maximum of 100, such as test results or survey results, the average will always be in the range from 0 to 100.

How many numbers can you average at once?

You can average up to six numbers and each number can have a different sample size. When you select the number of items above, only those rows will be used in the calculation so additional rows will never affect the result.

Average Percentage Calculator

Your values

Weight each value by its sample size

Turn this on when the percentages came from groups of different sizes.

Your percentages, one per line

3 values

Average percentage

Every value counts equally right now (a simple average). If each percentage came from a different-sized group, switch on sample-size weighting.

Average percentage

Charts and breakdown

Calculating the average of percentages seems simple. You add them all up and divide by the number of values. However, this method is only correct if each percentage comes from a group of equal size. If the group sizes are different, then a simple average will give small samples and large samples equal weight, which can lead to an incorrect result. This calculator handles the data in the right way. It uses a weighting based on the number of samples for each percentage, so larger groups, more items or higher scores have a greater impact on the result.

Why you shouldn't average percentages:

Suppose five students take a literature test. Four of them get an 80 (percent), while one gets a 40 (percent). If you average these two different values, you get a 60 (percent). However, since there were five students who took the test, you need to add up five values, not just two.

\frac{80 + 80 + 80 + 80 + 40}{5} = \frac{360}{5} = 72\%

The actual average is 72 (percent), not 60 (percent). Averaging these two values would mean that the single value of 40 (percent) and the four values of 80 (percent) each are given equal weight. This is exactly the problem a weighted average avoids.

The formula for a weighted average is:

Each percentage is given a weight corresponding to its sample size. Each percentage is multiplied by its respective weight, the products are added and then divided by the sum of the weights.

\bar{p} = \frac{p_1 w_1 + p_2 w_2 + \cdots + p_n w_n}{w_1 + w_2 + \cdots + w_n}

In this case each "p" value represents a percentage and each "w" is the number of subjects in that group. For example, suppose a survey asked people in three different age groups if they liked pancakes. 64% of 300 teenagers, 42% of 450 adults, and 36% of 250 seniors said they like pancakes.

\bar{p} = \frac{64 \times 300 + 42 \times 450 + 36 \times 250}{300 + 450 + 250} = \frac{47{,}100}{1{,}000} = 47.1\%

If the sample sizes are equal then the weights will be equal and the formula reduces to the usual average in which the sum is divided by the number. So the simple average is just a special case that occurs when all groups have the same size.

How to use this calculator:

Select the number of values to enter for calculating an average and then enter a percentage for each value. The result of this entry is a simple average. If these values are from groups of different sizes, enable weighting by sample size. A field will appear for each percentage where you can enter the sample size. After entering the sample sizes, the results will be updated to a weighted average so that you can compare the effects of the weighting and see how much it changed the result. You can also enter values greater than 100%. This is appropriate for growth rates or efficiency data with no upper limit.

Here's how to read the numbers:

The number prominently displayed is the average in percentage terms. In weighted mode, a simple average is also shown for comparison purposes along with the total sample size used to produce the result. A detailed table shows each element's contribution, i.e. its value multiplied by its size, so that you can see exactly how the result was produced.

Symbol	Meaning	Example
p	A percentage value	64 percent
w	Its sample size (weight)	300
n	Number of values	3
p-bar	The average percentage	result

Common errors:

The most common problem is that percentages are simply averaged without taking the sample sizes into account. Always make sure first that the size of the populations represented by these numbers is equal. Another related mistake is to round off each percentage before summing them up. This will lead to a slightly biased final result. Maintain full accuracy until the last step. Finally, check if the weighting is appropriate. The sample sizes must be stated in terms of number and not percentages.

Areas of application:

Teachers can use this to combine class grading results for classes of different sizes. Analysts can merge multiple surveys with differing response numbers or weight growth rates based on income. When there are several percentages and the data amounts behind each one differ, a weighted average is a number that fairly represents the entire set of data.

Frequently asked questions

Can you just add up all of those percentages and divide by the total number to get an average?: This is only possible if each percentage comes from a population of the same size. If populations are different sizes, then individual percentages must be weighted according to sample size. Otherwise small and large populations will have equal effect, leading to an incorrect result.
How do you calculate the average of percentages with different sample sizes?: You multiply each percentage by the corresponding sample size, add up the products and divide the result by the sum of the sample sizes. For example, if there are 20 people with a percentage of 80% and 80 people with a percentage of 90%, you calculate ((80*20) + (90*80))/100. The result is 88%.
What is the difference between a simple average and a weighted average?: A simple average treats each value equally and divides the sum by the number of values. A weighted average multiplies each value by a weight. In this case the weights are the sample sizes, so values with larger samples have more influence on the result. If all the weights are equal then you get the same answer from both the simple and weighted averages.
Can average percentage go over 100?: Yes, it is possible if the values entered can exceed 100% themselves. This applies for example to growth rates or efficiency data. For results where there is normally a maximum of 100, such as test results or survey results, the average will always be in the range from 0 to 100.
How many numbers can you average at once?: You can average up to six numbers and each number can have a different sample size. When you select the number of items above, only those rows will be used in the calculation so additional rows will never affect the result.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Weighted arithmetic mean (Wikipedia)
Definition and properties of the weighted mean.
Percentage (Wikipedia)
What a percentage is and how it relates to fractions.
Investopedia: Weighted Average
Worked explanation of weighting values by importance.