Combined mean (or pooled mean) is the overall average when two or more groups with known individual means and sizes are merged into a single dataset.
## Core concept
When you have separate groups—each with its own mean and number of observations—the combined mean is not simply the average of the individual means. It must be weighted by group size.
The combined mean accounts for: - The mean value of each group - The number of observations (or frequency) in each group - Smaller groups contribute less to the overall average than larger groups
Formula for combined mean:
$$\bar{X}_{combined} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2 + n_3\bar{x}_3 + \ldots}{n_1 + n_2 + n_3 + \ldots}$$
Where: - $\bar{x}_1, \bar{x}_2, \bar{x}_3$ = mean of each group - $n_1, n_2, n_3$ = number of observations in each group - Numerator = sum of (group size × group mean) for all groups - Denominator = total number of observations
In words: Multiply each group's mean by its size, sum those products, then divide by the total number of observations.
## Common exam applications
Scenario 1: Merging two class datasets - Class A has 40 students with average marks 65 - Class B has 60 students with average marks 70 - Find the combined mean
$$\bar{X}_{combined} = \frac{40 \times 65 + 60 \times 70}{40 + 60} = \frac{2600 + 4200}{100} = \frac{6800}{100} = 68$$
The combined mean is 68 marks (closer to Class B's mean because Class B is larger).
Scenario 2: Finding missing data If combined mean is given and you need to find a missing group mean or size, rearrange the formula: $$n_1\bar{x}_1 + n_2\bar{x}_2 = \bar{X}_{combined} \times (n_1 + n_2)$$
Scenario 3: Departmental or regional consolidation - Sales teams, branches, or departments have separate performance averages - Use combined mean to report organizational overall average
## Common mistakes
| Mistake | Correction | |---------|-----------| | Adding the two means and dividing by 2: $\frac{65 + 70}{2} = 67.5$ | Always weight by group size; answer is 68, not 67.5 | | Forgetting that larger group pulls the combined mean toward its own mean | The weight matters; heavier groups dominate the result | | Using unequal group sizes without adjusting | Each observation counts once; larger $n$ means more weight | | Confusing with simple average of means | Combined mean ≠ average of the individual means (unless all groups are equal size) |
## Key takeaway for exam strategy
When a question provides two or more group means and their frequencies/sizes, immediately recognize this as a combined mean problem. Set up the weighted average formula systematically: 1. Identify all group means and sizes 2. Multiply mean × size for each group 3. Sum all products 4. Divide by total observations 5. State the result with units
Combined mean questions often appear as 2–4 mark problems requiring calculation and interpretation. Practice rearranging the formula to solve for unknown group means or sizes.