Calculate Average Percentage

Calculating the average percentage of something may seem like an easy task, but this concept can be more complicated than what may initially be assumed. While in some cases you may be able to simply average two percentages by adding them and dividing them by the number of percentages used, in other instances, you may need to consider factors such as sample size.

The formula to calculate average percentage is \[ \left( \frac{\text{Percentage}_1 + \text{Percentage}_2}{\text{Sample Size}_1 + \text{Sample Size}_2} \right) \times 100 \]
Use an average percentage when determining the average of two or more percentages of a whole.

How to calculate average percentage

Calculating the average of two or more percentages requires a different set of steps than determining the average of two or more regular numbers. There are several things to take into account when calculating average percentage, including the fact that you often must first calculate the real numbers that a percentage represents.

Use the following steps to calculate the average percentage in most cases:

Convert the percentages you wish to average into decimals

The first step in calculating the average of percentages is to turn the percentages into decimals. For example, let's say you want to determine the average of 30% of 150 and 25% of 200. You would turn these two percentages into decimals by dividing them both by 100. This means that 30% is 0.3 as a decimal and 25% is .25 as a decimal.

Determine the number that each decimal represents

Once you have converted the percentages into decimals, you will then need to find the number that each decimal represents. To do this, you will multiply the decimal by the total number in the sample. So, using the prior example, you would multiply 0.3 by 150 to get 45 and multiply .25 by 200 to get 50.

Add the numbers together

The next step is to add the two numbers found in the last step together. So, 45 plus 50 equals 95.

Add the sample sizes together

You also need to add up the sample sizes for each percentage. So, 150 plus 200 equals 350.

Calculate the percentage average

To find the average percentage of the two percentages in this example, you need to first divide the sum of the two percentage numbers by the sum of the two sample sizes. So, 95 divided by 350 equals 0.27. You then multiply this decimal by 100 to get the average percentage. So, 0.27 multiplied by 100 equals 27 or 27%. This means that the average percentage of 30% of 150 and 25% of 200 is 27%.

Example

Calculating an Average Percentage

An average percentage is used to find the mean percentage across multiple values. It is commonly used in finance, academics, and business to analyze trends and compare different sets of data.

The general approach to calculating an average percentage includes:

Summing up all percentage values.
Dividing the sum by the total number of values.
Interpreting the result to understand trends.

Basic Formula for Average Percentage

The formula for calculating the average percentage is:

\[ \text{Average Percentage} = \frac{\sum \text{Percentages}}{\text{Total Count}} \]

Example:

If the percentages of student test scores in three subjects are 80%, 75%, and 90%, the calculation is:

Step 1: Sum the percentages: $ 80 + 75 + 90 = 245 $.
Step 2: Divide by the total count: $ 245 \div 3 = 81.67 $.
Step 3: The average percentage is **81.67%**.

Weighted Average Percentage

In some cases, different values carry different weights. The weighted average percentage formula is:

\[ \text{Weighted Average Percentage} = \frac{\sum (P_i \times W_i)}{\sum W_i} \]

Example:

If a student scores 80% in Math (weight 3), 75% in Science (weight 2), and 90% in English (weight 1), the calculation is:

Step 1: Multiply each percentage by its weight: $ (80 \times 3) + (75 \times 2) + (90 \times 1) = 240 + 150 + 90 = 480 $.
Step 2: Sum the weights: $ 3 + 2 + 1 = 6 $.
Step 3: Divide: $ 480 \div 6 = 80 $.
Step 4: The weighted average percentage is **80%**.

Real-life Applications of Average Percentage

Average percentages are widely used in various fields, such as:

Calculating students' average grades in multiple subjects.
Analyzing business growth rates over different time periods.
Understanding inflation rates over multiple years.
Determining the average discount percentage across different stores.

Common Operations with Average Percentage

Basic Average: $ \frac{\sum P}{n} $

Weighted Average: $ \frac{\sum (P_i \times W_i)}{\sum W_i} $

Adjusting Percentages: If values have different importance, weights must be applied to calculate an accurate representation.

Average Percentage Calculation Examples Table
Problem Type	Description	Steps to Solve	Example
Basic Average Percentage	Calculating the mean percentage of multiple values.	Sum all percentage values. Divide by the total number of values.	For percentages 80%, 75%, and 90%, sum them: $ 80 + 75 + 90 = 245 $, then divide by 3: $ 245 \div 3 = 81.67% $.
Weighted Average Percentage	Calculating the average percentage with different weights.	Multiply each percentage by its respective weight. Sum the weighted values. Divide by the total weight.	For 80% (weight 3), 75% (weight 2), and 90% (weight 1), calculate: $ (80 \times 3) + (75 \times 2) + (90 \times 1) = 480 $. Divide by total weight (6): $ 480 \div 6 = 80% $.
Finding Percentage Change	Determining the percentage increase or decrease between two values.	Find the difference between the new and old value. Divide by the old value and multiply by 100.	If a product price increases from $50 to $60, the change is $ 60 - 50 = 10 $. The percentage increase is $ (10 \div 50) \times 100 = 20% $.
Real-life Applications	Using average percentage in everyday scenarios.	To calculate average exam scores. To determine business performance over multiple periods.	If a student scores 85%, 78%, and 92% in three exams, their average score is $ (85 + 78 + 92) \div 3 = 85% $.