To find the average of percentages, you cannot simply add the percentages and divide by the number of percentages; this is only correct if the total size of each group is the same. Instead, calculate a weighted average of percentages.
Calculating a Weighted Average of Percentages
Here's a breakdown of how to find the average of percentages, which is more accurately described as a weighted average of percentages:
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Understand Weighted Averages: A weighted average takes into account the size or importance (the weight) of each percentage contributing to the overall average. It's crucial when the total quantities underlying each percentage differ.
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Calculate Total Products:
- For each percentage, you need to know the total quantity or product that the percentage is calculated from.
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Calculate Sum of Actual Values:
- Multiply each percentage by its corresponding total products. This will find the actual quantity represented by each percentage. For example, if you have 50% of 200 items, you would have 100 items.
- Add these actual values together to find a grand total.
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Calculate Total Products:
- Add up all of the total products.
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Calculate the Weighted Average:
- Divide the sum of the actual values (Step 3) by the sum of the total products (Step 4). This produces a decimal number.
- Multiply this decimal by 100 to express the answer as a percentage.
Example
Let's illustrate this with an example based on the reference provided:
| Category | Percentage | Total Products | Actual Value (Percentage x Total) |
|---|---|---|---|
| A | 50% | 300 | 150 |
| B | 70% | 600 | 420 |
| Totals | 900 | 615 |
Using the reference data:
The sum of the actual values is 615.
The total products is 900.
Divide 615 by 900, which is approximately 0.683
Multiply 0.683 by 100, to find the average as a percentage: 68.3%
Therefore, the weighted average percentage is 68.3%, not simply the average of 50% and 70% (which would be 60%). This demonstrates how the size of the sample matters when calculating the average percentage.
Key Takeaways
- Weight matters: The total number of items, the 'weight,' determines a percentage's contribution to the average.
- Avoid simple averages: Directly averaging percentages is incorrect when the totals are different.
- Weighted average is key: Always use the weighted average formula to find the correct average.
