Averages of Averages: Why They’re Wrong

Yes, averaging averages is usually wrong because it gives equal weight to unequal groups, distorting the true overall average. The correct method is to compute a weighted average, where each group’s average is multiplied by its group size before summing and dividing by the total. This article explains why the naive approach fails, how to do it right, and when it might actually be acceptable.

Formula

The weighted average of two groups is:

Weighted Average = (A × n + B × m) / (n + m)

• A = average of group 1 (unit depends on the data, e.g., score, dollars, kg)
• B = average of group 2
• n = number of items in group 1 (count)
• m = number of items in group 2 (count)

This extends naturally to more than two groups: multiply each group’s average by its size, sum those products, then divide by the total number of items.

Worked Example

Suppose Group A has an average score of 80 from 10 students, and Group B has an average of 90 from 20 students.

1. Multiply Group A’s average by its size: 80 × 10 = 800
2. Multiply Group B’s average by its size: 90 × 20 = 1800
3. Add those products: 800 + 1800 = 2600
4. Divide by the total number of students: 2600 ÷ (10 + 20) = 2600 ÷ 30 = 86.67

The correct weighted average is 86.67. The naive simple average (80 + 90) ÷ 2 = 85 is lower because it ignores that the larger group (B) pulls the overall average up.

Why the Simple Average Fails

The simple average of group averages treats each group as if it had the same number of members. That is rarely true. In the example above, the 20 students in Group B count twice as much as the 10 in Group A, but the simple average gives them equal weight. This leads to a distorted result whenever group sizes differ.

Another way to see it: the overall average of all raw data points is, by definition, the sum of all values divided by the total count. If you only have group averages and sizes, the weighted average formula reconstructs that exact value. The simple average does not, unless all groups are the same size.

When Is It Okay to Average Averages?

There are a few situations where averaging averages is acceptable:

• Equal group sizes: If every group has exactly the same number of items, the simple average equals the weighted average.
• Rough estimation: If group sizes are close and you only need a ballpark, the error may be small.
• Each average represents a single unit: For example, if you have the average temperature of each day in a week, averaging those seven daily averages gives the weekly average because each day has the same weight (24 hours).

In most real-world data, group sizes vary, so the safe route is to use the weighted formula.

Common Mistakes and Misconceptions

• Ignoring group sizes: The most frequent error is simply adding averages and dividing by the number of groups.
• Confusing weighted and simple averages: A weighted average is not “more complicated for no reason”—it is the only correct way when group sizes differ.
• Using the wrong weights: Sometimes people use a subjective weight (e.g., importance) instead of the actual group size. That is a different concept—a weighted average with chosen weights—which is fine if the weights are intentional and explained.
• Rounding too early: If you round each group average before combining, you lose precision. Keep full precision until the final step.

Real-World Examples

Here are three common scenarios where averaging averages causes trouble:

• Grades: A student scores 90 on a 10-question quiz (average 9 per question) and 80 on a 50-question exam (average 1.6 per question). Averaging 9 and 1.6 gives 5.3, which is meaningless. The correct overall average is (90×10 + 80×50) / 60 = 81.67.
• Customer satisfaction: Store A has 100 ratings averaging 4.5, Store B has 10 ratings averaging 2.0. The simple average is 3.25, but the weighted average is (4.5×100 + 2.0×10) / 110 = 4.27, which better reflects overall satisfaction.
• Speed: If you drive 60 km/h for 1 hour and 30 km/h for 2 hours, your average speed is not (60+30)/2 = 45 km/h. It is total distance (60×1 + 30×2) / total time (1+2) = 120/3 = 40 km/h.

How to Choose the Right Average

Always ask: “What is the unit of observation?” If the unit is an individual item (person, product, transaction), then each item should have equal weight. That means you need the group sizes. If the unit is the group itself (e.g., each day, each store equally), then a simple average of group averages is fine.

When you only have group averages and sizes, use the weighted formula. When you have the raw data, just compute the overall average directly—no need to group at all.

Frequently Asked Questions

Is averaging averages ever correct?

Yes, when all groups have the same size, or when each group average represents a single unit of equal importance (like daily temperatures). Otherwise, it is incorrect.

What is the difference between weighted and simple average?

A simple average treats every value equally. A weighted average assigns a weight (often group size) to each value, so larger groups influence the result more. The weighted average is the true overall average when groups have different sizes.

How do I calculate a weighted average with more than two groups?

Extend the formula: multiply each group’s average by its size, sum all those products, then divide by the total number of items across all groups. For example, with three groups: (A×n + B×m + C×p) / (n + m + p).

Why does my Excel AVERAGE of averages give a different result?

Because Excel’s AVERAGE function simply sums the numbers and divides by the count—it does not know about group sizes. To get the correct overall average, use SUMPRODUCT to multiply each average by its size, then divide by the total size.

Can I average percentages the same way?

Yes, but be careful. If the percentages come from different sample sizes, you must weight by those sizes. For example, a 50% success rate from 100 trials and a 100% success rate from 2 trials: the weighted average is (0.50×100 + 1.00×2) / 102 ≈ 50.98%, not 75%.

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in a | Group A average | - | 80 | 0
in b | Group B average | - | 90 | 0
in n | Group A size | - | 10 | 1
in m | Group B size | - | 20 | 1
out wavg | Weighted average | - | (a*n + b*m) / (n + m) | 2
out naive | Simple average | - | (a + b) / 2 | 2