Weighted Averages
By the end of this lesson, you’ll be able to:
- Compute weighted averages using weights or frequencies.
- Interpret weighted averages in contexts like grades, mixtures, and combined test scores.
- Convert between weighted-average formulas and equivalent ratio representations.
Key Ideas
A weighted average accounts for values that contribute unequally.
General formula:
\[ \text{Weighted Average} = \frac{w_1 x_1 + w_2 x_2 + \dots}{w_1 + w_2 + \dots} \]
Common contexts:
- Grades with different category weights
- Average speed with different time intervals
- Mixture problems (combining solutions or populations)
- Frequency tables (weights = frequencies)
| Score | Frequency |
|---|---|
| 70 | 2 |
| 80 | 4 |
| 90 | 3 |
| Total | 9 |
In this table, the frequency of each score acts as its weight.
To compute the weighted mean, multiply each score by the number of times it appears, add those products, and divide by the total of 9 data points:
\[ \frac{70 \cdot 2 + 80 \cdot 4 + 90 \cdot 3}{9}. \]
Common Problem Types
Combining Groups With Different Sizes
When two groups have different numbers of items, the larger group influences the overall average more.
Example:
Class A mean = 70 with 10 students, Class B mean = 90 with 30 students.
The combined average leans toward 90 because class B is larger.
Category-Weighted Scores
Values (like grades) contribute unequally based on given weights.
Example:
Homework = 20%, Tests = 80%.
Test average has four times the influence of homework.
Frequency Tables as Weighted Averages
Each value is multiplied by how many times it appears.
Example:
Score 80 appears 6 times; score 90 appears 2 times.
Weights are 6 and 2.
Mixture or Concentration Problems
Amounts act as weights.
Example:
Mix 2 L of 20% solution with 8 L of 10% solution.
Larger volume dominates the final concentration.
Average Speed Over Unequal Time Intervals
Average speed must be weighted by time, not by averaging the speeds directly.
Example:
30 mph for 1 hour and 60 mph for 1 hour → average = 45 mph.
Strategies
- Multiply each value by its weight (importance, frequency, size).
- Always divide by total weight, not number of categories.
- Convert percentages → decimals when needed.
- For mixtures: total amount = sum of parts; total “value” = sum of weighted parts.
Worked Examples
Example 1 — Category weights
A student’s grade is based on:
- Homework: average 85 (40%)
- Tests: average 92 (60%)
Weighted average:
\[ 0.4(85) + 0.6(92) = 34 + 55.2 = 89.2 \]
Example 2 — Combining groups
Class A: 20 students, mean 78
Class B: 30 students, mean 84
Combined mean:
\[ \frac{20(78) + 30(84)}{20 + 30} = \frac{1560 + 2520}{50} = 81.6 \]
Example 3 — Frequency table
Scores: 70 (3 times), 80 (4 times), 90 (1 time)
\[ \frac{70(3) + 80(4) + 90(1)}{8} = \frac{210 + 320 + 90}{8} = 77.5 \]
- Dividing by the number of categories instead of the total weight.
- Forgetting to multiply values by their weights.
- Mixing up percent weights with numeric weights.
- Averaging averages without accounting for group sizes.
- Ignoring total amount in mixture problems.
Practice Problems
- Weighted test score: Quiz average 80 (30%), Test average 90 (70%).
- Two groups: 10 students with mean 72; 15 students with mean 88.
- Frequency table: values 5 (6 times), 7 (3 times), 10 (1 time).
- Mixture: 2 L of 30% solution with 3 L of 10% solution → final concentration?
1. \(0.3(80) + 0.7(90) = 24 + 63 = 87\)
2.
\(\frac{10(72) + 15(88)}{25} = \frac{720 + 1320}{25} = 81.6\)
3.
Sum = \(5(6)+7(3)+10(1)=30+21+10=61\)
Mean = \(61/10 = 6.1\)
4.
Amount of solute: \(0.30(2) + 0.10(3) = 0.6 + 0.3 = 0.9\)
Total volume = 5 L
Concentration = \(0.9/5 = 18\%\)
Summary
- Weighted average = total weighted value ÷ total weight.
- Use when values contribute unequally (size, frequency, importance).
- Always multiply first, then divide by total weight.
- Don’t average averages — include weights!
- Percent weights → decimals.
- For mixtures: track total amount and total “value.”
- If groups differ greatly in size, the larger group dominates the combined average.