EQN-05-05

Weighted Average

Average groups of different sizes correctly — the simple average is a trap when the groups are unequal.

~5 min read•Difficulty 3/5

When the simple average fails

If two groups have different sizes, you cannot just average their means. Each mean must be weighted by its group size.

weighted average = (sum of all values) / (total count)

= Σ (group mean \times group size) /Σ (group size)

Worked logic

A class of 30 averages 80; another class of 20 averages 70. The combined average is:

(30 \times 80 + 20 \times 70) / (30 + 20) = (2400 + 1400) /50 = 3800/50 = 76

Note it is not $(80 + 70) /2 = 75$ — it leans toward the larger group.

The lever intuition

The weighted average always sits closer to the mean of the bigger group. Use this to sanity-check: with more high-scorers, the average must be above the midpoint.

Same idea everywhere

Weighted averages also model average price across different quantities, average speed across different times, and grade point averages across different credit hours.

Diagram

Weight each average by its group size, then divide by the total.

Worked examples

Example 1 (difficulty 2/5)

A test has 4 questions worth 10 points each and 1 question worth 60 points. A student scores 8 on each small question and 30 on the big one. What is the score as a percentage of 100?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Group A: 10 people averaging 90. Group B: 30 people averaging 50. What is the combined average?

◎ Predict the answer first:

Tips

Add ALL the values and divide by the TOTAL count — never average the averages.
Sanity check: the result sits closer to the larger group's mean.
Weights can be counts, points, hours, or quantities — same formula.

Common mistakes

Averaging two group means directly when the groups differ in size

Why: The midpoint feels fair.

Fix: Weight each mean by its group size before combining.

Averaging percentages of parts with different point values

Why: Percentages look comparable.

Fix: Convert to actual points, sum, then divide by the total possible.

← Back to Data Analysis & Statistics Practice this lesson →