Weighted Averages and Combined Means

A weighted average gives different importance (weights) to each data value.

\text{Weighted mean} = \frac{w_1x_1 + w_2x_2 + \cdots + w_nx_n}{w_1 + w_2 + \cdots + w_n}.

Weights may represent frequency, percentage, or relative importance.

When combining groups with different averages, use the formula:

\text{Combined mean} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}.

Core Skills

Multiply each value by its corresponding weight.
Divide by the sum of weights to find the weighted mean.
For group combinations, use total sums or averages with counts.
Recognize that a weighted average leans toward the group with the larger weight.

A student has grades: Homework 90 (weight 40%), Tests 80 (weight 60%).

\text{Average} = \frac{0.4(90)+0.6(80)}{1} = 36 + 48 = \boxed{84}.

Group A: average 70, 10 people. Group B: average 90, 20 people.

\text{Combined mean} = \frac{10(70)+20(90)}{30} = \frac{700+1800}{30} = \boxed{83.3}.

\begin{array}{c|ccc} \text{Score} & 60 & 80 & 100\\ \hline \text{Frequency} & 3 & 4 & 3 \end{array}

Weighted mean:

\frac{3(60)+4(80)+3(100)}{3+4+3} = \frac{180+320+300}{10} = \boxed{80}.

A student’s grade is 85 on homework (20%), 90 on quizzes (30%), and 80 on tests (50%).

\text{Average} = 0.2(85) + 0.3(90) + 0.5(80) = 17 + 27 + 40 = \boxed{84}.

Two factories produce widgets. Factory A makes 500 widgets with average cost $10. Factory B makes 200 widgets with average cost $12.

\text{Combined cost} = \frac{500(10)+200(12)}{700} = \frac{7400}{700} = \boxed{10.57}.