Combined Mean Formula: Why You Cannot Always Average Two Averages

Combined mean = (n₁ × m₁ + n₂ × m₂) ÷ (n₁ + n₂). Here n₁ and n₂ are the numbers of observations in two separate groups, and m₁ and m₂ are their arithmetic means. Multiply each mean by its group size to recover the total, add the totals, then divide by the total number of observations.

Simply calculating (m₁ + m₂)/2 works when the two groups have equal sizes, or when their means are equal. For unequal groups with different means, it gives the wrong answer.

Example 1: two classes with different numbers of students

Class A has 20 students with a mean score of 60. Class B has 30 students with a mean score of 80. Both classes took the same test.

1. Class A total = 20 × 60 = 1,200 marks.
2. Class B total = 30 × 80 = 2,400 marks.
3. Combined total = 3,600 marks; student count = 50.
4. Combined mean = 3,600 ÷ 50 = 72 marks.

The simple average of 60 and 80 is 70. That treats both classes as equally large. The correct answer, 72, is closer to 80 because the higher-scoring class has more students.

Why the formula works

Mean means total divided by count, so total = mean × count. Combining groups means adding both their totals and their counts. This makes the combined mean a weighted average, with group sizes as the weights. For three or more groups, keep adding each size × mean in the numerator and every group size in the denominator.

Yaad rakho: pehle total nikalo, phir total students se divide karo. Sirf averages ko jodkar 2 se divide mat karo.

Example 2: find an unknown group mean

A group of 10 learners has a mean of 50 marks. Another group has 15 learners. The combined mean is 62. What is the second group’s mean?

Combined total = 25 × 62 = 1,550. First group total = 10 × 50 = 500. Second group total = 1,550 − 500 = 1,050. Its mean is 1,050 ÷ 15 = 70 marks.

Example 3: recover the ratio of group sizes

Two groups have means 40 and 70. Their combined mean is 58. If their sizes are a and b, then 40a + 70b = 58(a + b). Rearranging gives 12b = 18a, so a:b = 2:3.

Check with sizes 2 and 3: (2 × 40 + 3 × 70)/5 = 290/5 = 58. More observations belong to the group with mean 70, as expected.

Checks before using the formula

• Use the same measurement and units. Do not combine raw marks from tests with different maximum scores as if they were the same test.
• The groups must not overlap; counting the same learner twice changes the result.
• For positive group sizes, the combined mean lies between the two group means.
• Rounded group means can produce an approximate combined mean. Use original totals when available.
• A median or a mode cannot be combined with this arithmetic-mean formula.

Practice questions with answers

1. Groups of 12 and 8 have means 15 and 25. Combined mean = (180 + 200)/20 = 19.
2. Two groups of 9 have means 20 and 30. Combined mean = (180 + 270)/18 = 25. Equal sizes make the simple average valid.
3. A group of 5 has mean 12; a group of 15 has mean 20. Combined mean = (60 + 300)/20 = 18.
4. Twenty observations have mean 24. Ten of them have mean 18. The remaining ten have total 480 − 180 = 300, so their mean is 30.

Combined mean or corrected mean?

Use combined mean when you join separate groups. If one recorded value was wrong and must be replaced, use the corrected mean formula with worked examples. For exam practice involving mixtures, read our averages and mixtures guide.

Need help checking your method? Bring your group sizes, means and attempted calculation to the DoubtPay study space. AI can make mistakes, so check important steps against your textbook or teacher.