How can I tell whether a displayed average is weighted or unweighted?

Look for an explanation of how the group figures were combined. If each group average is given equal influence, regardless of how many observations it represents, the result is an unweighted average of group averages. If groups with more observations contribute more, it is weighted by group size. A weighted mean uses values multiplied by their weights, then divides by the sum of the weights; for group averages, the observation counts can serve as those weights. [S1]

The displayed number alone usually cannot tell you which method was used. Ask for the group averages, the count behind each one, and the stated method. If the report gives each group’s total and count instead, those can also be used to reconstruct the combined mean.

Suppose two groups have these invented results:

• Group A: average 80 from 2 observations.
• Group B: average 60 from 8 observations.

The unweighted average of the two group averages is (80 + 60) ÷ 2 = 70. Each group average gets equal influence.

For an overall average across all 10 observations, use the group sizes as weights: (80 × 2 + 60 × 8) ÷ (2 + 8) = 64. Group B has more observations, so its lower average contributes more. These calculations assume the group figures are means for the same measure and that the counts refer to the observations included in those means. The example illustrates the calculation; it does not establish which method a particular report used.

Ask for enough detail to identify both the inputs and the rule for combining them. You could write:

“Could you show how the combined average was calculated? Please provide each group’s average and number of observations, or each group’s total and count, and say whether the combined figure weights groups by their observation counts or gives each group equal weight.”

If some groups may have been left out, ask which groups and observations were included. The available figures may let you check the arithmetic, but they do not by themselves explain why the report chose that method.

You can calculate the unweighted result if the groups to include are known: add their averages and divide by the number of groups. But without group sizes, you generally cannot reconstruct a combined average weighted by observation count from group averages alone. Ask for the counts, or for each group’s total and count, and confirm which calculation the displayed figure represents. If the report does not provide that information, describe your result as an unweighted calculation rather than assuming it matches the reported method. [S1]