Why you can’t just average percentages

You can only average percentages directly when each is out of the same total. Otherwise the percentage from the bigger total must count for more. Add up the amounts and the totals, then divide: 9 out of 10 and 60 out of 100 is 69 out of 110, or 62.7%, not the 75% a simple average gives.

The mistake in one example

You score 90% on a 10-mark quiz and 60% on a 100-mark exam. The simple average of 90 and 60 is 75. But the quiz gave you 9 marks and the exam 60, so you earned 69 marks out of 110 available:

• Simple average: (90 + 60) ÷ 2 = 75%
• Real result: (9 + 60) ÷ (10 + 100) = 69 ÷ 110 = 62.7%

The simple average treats a 10-mark quiz as if it mattered as much as a 100-mark exam. A percentage hides the size of the thing it came from, and averaging throws that information away.

When a simple average is fine

• Every percentage has the same base. Four tests each out of 50: averaging the percentages gives exactly the same answer as pooling the marks.
• The rules say so. Some courses deliberately give every assessment equal weight regardless of marks. Then the simple average is the official answer, even if it differs from pooled marks. Check your course handbook.

The two correct methods

1. Pool the raw numbers

Add all the parts, add all the totals, divide once. This works for marks, sales, survey responses — anything where you know the underlying counts.

2. Use a weighted average

When you only know the percentages and their weights, multiply each percentage by its weight, add the results, and divide by the total weight. If coursework is worth 40% of the grade and you scored 80%, and the exam is worth 60% and you scored 55%:

• (80 × 40 + 55 × 60) ÷ (40 + 60) = (3,200 + 3,300) ÷ 100 = 65%

The Grade Calculator does exactly this: enter each assessment with its weight and it returns the overall percentage, even if the weights do not add up to 100. For credit-weighted grade point averages, the GPA Calculator weights each course by its credits for the same reason.

The same trap outside school

• Conversion or success rates. A shop converts 10% of 1,000 visitors on Monday and 2% of 100 visitors on Tuesday. The two-day rate is 102 ÷ 1,100 = 9.3%, not the 6% average.
• Discounts across a basket. 50% off a 2.00 item and 10% off a 50.00 item is not an average 30% saving. You saved 1.00 + 5.00 = 6.00 on 52.00, which is 11.5%.
• Growth rates over several years. +50% then −50% does not average to 0%: 100 becomes 150, then 75. Successive percentage changes multiply; to find the average yearly rate, use a compound (geometric) average instead.
• Averages of averages. Averaging the class averages of a class of 10 and a class of 40 overweights the small class. Pool the students instead.

Why the simple average feels right

Percentages look like they are already on the same scale — everything out of 100 — so averaging them seems natural. But the “out of 100” is a presentation, not a fact about the data. 90% of a 10-mark quiz represents 9 real marks; 60% of a 100-mark exam represents 60. Once you picture the marks rather than the percentages, the weighting becomes obvious.

Quick check before you average

Ask: “Are all these percentages out of the same thing?” If the answer is no, find the counts or the weights. If you just need a single “what percent is X of Y” after pooling, the Percentage Calculator shows the working.

Mistakes to avoid

• Averaging your module percentages when modules carry different credits.
• Reporting the average of monthly percentages as the yearly figure.
• Rounding each percentage before combining them; round only the final answer.

Checklist

• Same base for every percentage? A simple average is fine.
• Different bases? Pool the raw numbers, or weight each percentage.
• Changes over time? Multiply, do not average.