Weighted Means and Combined Groups

Objective: Combine two groups by their sizes.

Prerequisite skills: means, totals, and weighted averages. A combined mean equals the sum of group totals divided by total count. Recover each group total by multiplying its mean by its size. Larger groups contribute more weight. Common error: taking the simple average of unequal-sized group means. Restrictions and interpretation: Weights represent counts or proportional importance; do not average subgroup means unless their sizes are equal.

Objective: Use the combined total to recover a subgroup total.

Subtract the known group contribution from the overall total. The remaining total belongs to the other group, so divide by that group count. Keep the subgroup count separate from the overall count. Common error: dividing the residual by the full sample size. Restrictions and interpretation: Weights represent counts or proportional importance; do not average subgroup means unless their sizes are equal.

Objective: Interpret unequal repetition as weights.

A frequency-weighted mean repeats each value according to its count. Counts need not be probabilities, but the denominator must equal their sum. A change in group size can change the combined mean even if subgroup means stay fixed. Common error: dividing by the number of rows in a frequency table. Restrictions and interpretation: Weights represent counts or proportional importance; do not average subgroup means unless their sizes are equal. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.