Calculating the average of two percentages depends on whether the percentages are based on the same total or different totals. The method varies significantly based on the context.
Scenario 1: Percentages Based on the Same Total
If both percentages represent portions of the same total value, you can simply average them arithmetically.
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Formula: (Percentage 1 + Percentage 2) / 2
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Example: A store offers a 10% discount on Monday and a 20% discount on Tuesday for all items. The average discount for those two days is (10% + 20%) / 2 = 15%.
Scenario 2: Percentages Based on Different Totals (Weighted Average)
If the percentages represent portions of different total values, you need to calculate a weighted average. This is the more common and accurate approach. The reference mentions this scenario.
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Formula: [(Percentage 1 Sample Size 1) + (Percentage 2 Sample Size 2)] / (Sample Size 1 + Sample Size 2) This results in a decimal. Multiply by 100 to express as a percentage.
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Where:
- Percentage 1 = First percentage (expressed as a decimal)
- Sample Size 1 = Total value for the first percentage
- Percentage 2 = Second percentage (expressed as a decimal)
- Sample Size 2 = Total value for the second percentage
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Example:
- 20 out of 100 students (20%) passed an exam in Class A.
- 40 out of 50 students (80%) passed the same exam in Class B.
To find the average passing rate across both classes, you can't simply average 20% and 80%. You need a weighted average:
- [(0.20 100) + (0.80 50)] / (100 + 50) = (20 + 40) / 150 = 60 / 150 = 0.40
- 0.40 * 100 = 40%
The average passing rate across both classes is 40%.
Why Simple Averaging Can Be Misleading
Simple averaging when totals differ can lead to inaccurate results. In the example above, simply averaging 20% and 80% would give you 50%, which doesn't accurately reflect the overall passing rate when considering the different class sizes. The weighted average accounts for these differences.
Summary
When averaging percentages, first determine if they are based on the same or different totals. If the totals are the same, a simple average will work. If the totals are different, use the weighted average formula for accurate results.
