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Problem-Solving & Data

SAT combined averages: why you can’t just average the averages

A class of 10 averages 90 and a class of 30 averages 70. The combined average is not 80. Move the sliders in an interactive lab and learn the sum trick that solves every average question.

SPSAT Practice Editorial6 min read

The short version

What you'll take away

  • Mean × count = sum. Convert averages to sums before combining groups.
  • A combined mean lies closer to the mean of the larger group.
  • To find a needed score, subtract the current total from the target total.
In this article

The trap: averaging the averages

Class A has 10 students with an average score of 90. Class B has 30 students with an average of 70. It is tempting to say the combined average is 80, halfway between. But Class B has three times as many students, so it pulls the combined average toward 70.
Try it below. Change the group sizes and watch the combined mean slide toward the bigger group, while the “average of the averages” stays put.

Try it yourself · interactive

Two classes, one combined average

Class A
Class B
Combined mean (correct)
(10×90 + 30×70) ÷ 40 = 75.0
Average of the averages (trap)
(90 + 70) ÷ 2 = 80.0
Dot size shows group size. The green line is the true combined mean; the red diamond marks the trap answer. They match only when the groups are the same size.

The sum trick

Turn every average into a total. Class A’s scores add to \(\displaystyle 10 \times 90 = 900\), and Class B’s add to \(\displaystyle 30 \times 70 = 2{,}100\). Together that is 3,000 points across 40 students, so the combined mean is \(\displaystyle 3{,}000 \div 40 = 75\).
The same idea handles changes to a data set. If one value is added, the new mean is the old sum plus the new value, divided by the new count.

What score do I need?

Suppose your first four test scores average 85, and you want an average of 88 after five tests. The target total is \(\displaystyle 5 \times 88 = 440\). Your current total is \(\displaystyle 4 \times 85 = 340\). You need \(\displaystyle 440 - 340 = 100\) on the fifth test.

Your turn

Quick practice

One group of 20 students has a mean score of 80, and another group of 30 students has a mean score of 90. What is the mean score of all 50 students?

Hint: Convert each mean to a sum first.

Choose one answer

Practice: try these before checking the answers

Work with sums, not averages.
  • 1. The mean of 6 numbers is 12. What is their sum?
  • 2. Three tests average 80. A fourth test score is 92. What is the new average?
  • 3. Group A has 5 people with a mean of 30; group B has 15 people with a mean of 50. What is the combined mean?
  • 4. Four tests average 88. What score on a fifth test gives an average of 90?

Sources & further reading

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