Combined (Weighted) Mean

Statement

The combined (weighted) mean is used when two or more groups with different means and sizes are merged into one. The formula for combining two groups is:

\[\bar{x} = \frac{n_{1}\bar{x}_{1} + n_{2}\bar{x}_{2}}{n_{1}+n_{2}}\]

Here, \(\bar{x}\) is the combined mean, \(n_{1}\) and \(n_{2}\) are the group sizes, and \(\bar{x}_{1}, \bar{x}_{2}\) are the individual group means. This can be extended to more than two groups.

Why it’s true (short reason)

• The mean of a group is its total sum of values divided by the number of values.
• The total sum of all data when groups are merged is the sum of each group’s total (\(n_{1}\bar{x}_{1} + n_{2}\bar{x}_{2}\)).
• Dividing this total by the overall number of items, \(n_{1}+n_{2}\), gives the combined mean.

Recipe (how to use it)

1. Find the total contribution of each group: \(n_{1}\bar{x}_{1}\), \(n_{2}\bar{x}_{2}\), etc.
2. Add these contributions to get the total sum of all data values.
3. Divide by the total number of values across all groups.
4. If more than two groups exist, extend the formula accordingly.

Spotting it

Look for questions involving “combined average”, “overall mean”, or situations where two or more sets of averages need merging (e.g., two classes of students, two surveys, or two batches of production).

Common pairings

• Frequency tables (where each class mean contributes with a weight equal to its frequency).
• Data from multiple groups or trials combined into one statistic.
• Weighted averages in statistics and probability problems.

Mini examples

1. Group A: 10 students, mean = 60. Group B: 20 students, mean = 70. Answer: Combined mean = \(\dfrac{10\cdot60+20\cdot70}{30}=\dfrac{2000}{30}\approx 66.7\).
2. Two factories: Output 50 items at mean weight 2.4 kg, and 30 items at mean weight 2.8 kg. Answer: Combined mean = \(\dfrac{50\cdot2.4+30\cdot2.8}{80}=2.55 \, kg\).

Pitfalls

• Averaging the means directly: You cannot simply take \((\bar{x}_{1}+\bar{x}_{2})/2\) unless the groups are the same size.
• Forgetting weights: Always multiply each mean by its group size before summing.
• Mixing totals and means: Keep track of whether you are working with sums or averages.

Exam strategy

• Underline whether the question asks for the “combined mean” or just an “average of averages”.
• Write down the full formula and substitute group sizes and means carefully.
• Check units (marks, kg, etc.) and include them in your final answer.
• Estimate: The combined mean must lie between the two group means.

Summary

The combined mean gives the overall average when data from different groups are pooled. The formula weights each group’s mean by its size, ensuring larger groups have more influence. This prevents incorrect results from simply averaging group means. Always check that your combined mean lies between the smallest and largest group mean.