Weighted averages when group sizes are different
Combine group averages using the number of observations instead of giving every group equal influence.
Method
A simple average gives each listed value the same weight. When the listed values are themselves group averages, that can give a small group as much influence as a large one. Multiply each group average by its size, add the products, and divide by the total size. The groups must describe the same quantity and measurement period. For grades, an explicit course weighting can replace group size, but it answers a different question.
Worked example
Ten observations average 80 and thirty observations average 90. Their combined mean is (80 × 10 + 90 × 30) ÷ (10 + 30) = 3,500 ÷ 40 = 87.5. Simply averaging 80 and 90 gives 85 and ignores that the second group contains three times as many observations. In Weighted grade, use the two averages as values and 10 and 30 as their weights.
What to check
Check that no observation belongs to both groups. A rounded group average can make the reconstructed total approximate, so retain unrounded totals when available. Weights must be non-negative with a positive combined weight. If both groups have the same size, the weighted and simple means agree. Do not combine averages with different units or definitions merely because both are numbers.