Weighted Averages
By the end of this lesson, you’ll be able to:
- Compute weighted averages using weights, percentages, or frequencies.
- Explain how weights affect an overall average.
- Combine groups with different sizes correctly.
- Interpret weighted averages in contexts such as grades, mixtures, and frequency tables.
- Recognize when a simple average of averages is incorrect.
Key Ideas
A weighted average is an average in which some values contribute more than others.
The general formula is:
\[ \boxed{ \text{Weighted Average} = \frac{w_1x_1+w_2x_2+\cdots+w_nx_n} {w_1+w_2+\cdots+w_n} } \]
where:
- \(x_i\) = each value
- \(w_i\) = the weight attached to that value
The larger the weight, the more influence that value has on the final average.
Why Use a Weighted Average?
A regular mean treats every observation equally.
A weighted average is useful when the contributions are not equal.
Common examples include:
- course grades with different category weights
- combining groups with different numbers of people
- frequency tables
- mixtures and concentrations
- average speed over different amounts of time
Percentage Weights
If the weights are percentages that add to:
\[ 100\% \]
you can convert them to decimals.
For example:
\[ 30\%=0.30 \]
and:
\[ 70\%=0.70 \]
Since:
\[ 0.30+0.70=1 \]
the weighted average can be written directly as:
\[ \boxed{ 0.30x_1+0.70x_2 } \]
There is no need to divide again because the weights already total 1.
If the weights add to 1:
\[ \text{Weighted Average} = w_1x_1+w_2x_2+\cdots \]
If the weights do not add to 1:
\[ \text{Weighted Average} = \frac{\sum w_ix_i}{\sum w_i} \]
Frequency Tables
A frequency tells how many times a value occurs.
That frequency acts as the weight.
| Score | Frequency |
|---|---|
| 70 | 2 |
| 80 | 4 |
| 90 | 3 |
| Total | 9 |
Instead of writing all nine scores individually, calculate:
\[ 70(2)+80(4)+90(3) \]
Then divide by the total frequency:
\[ \frac{70(2)+80(4)+90(3)}{9} \]
Compute the numerator:
\[ 140+320+270=730 \]
Therefore:
\[ \text{Weighted Mean} = \frac{730}{9} \approx81.1 \]
So:
\[ \boxed{\text{Weighted Mean}\approx81.1} \]
Common Problem Types
1. Category-Weighted Scores
A course grade may be based on categories with different weights.
Multiply each category average by its weight, then add the weighted parts.
The category with the larger weight has more influence on the final result.
2. Combining Groups With Different Sizes
When groups have different sizes, do not simply average the group means.
Reconstruct each group’s total, add the totals, then divide by the combined number of values.
The combined average will be pulled toward the larger group.
3. Frequency Tables
In a frequency table, each value is weighted by how many times it occurs.
Use:
\[ \text{mean}=\frac{\text{weighted sum}}{\text{total frequency}} \]
4. Mixture Problems
In mixture problems, the amount of each component acts as the weight.
Find the amount of pure substance in each part, add those amounts, then divide by the total mixture amount.
5. Average Speed Over Unequal Time Intervals
When speeds apply over different amounts of time, time acts as the weight.
Use the definition:
\[ \text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}} \]
Do not average the speeds unless the time intervals are equal.
6. Finding a Missing Value From a Weighted Average
Sometimes the weighted average is given and one component is unknown.
Set up the weighted-average equation with a variable for the missing part, then solve.
Strategies
- Identify the value and its corresponding weight.
- Multiply each value by its weight.
- Add the weighted values.
- Divide by the total weight when the weights do not already sum to 1.
- If percentage weights total 100%, convert them to decimals and add the weighted products directly.
- For frequency tables, use frequency as the weight.
- For combined groups, use group size as the weight.
- For mixtures, use amount or volume as the weight.
- Do not average group averages unless the groups are the same size.
- Check whether the final answer lies between the smallest and largest values being averaged.
A weighted average should normally lie between the smallest and largest values being averaged.
Also, it should be pulled toward the value with the larger weight.
Worked Examples
Example 1 — Weighted Course Grade
A student’s grade is based on:
- Homework average: 85, worth 40%
- Test average: 92, worth 60%
Convert the percentages to decimals:
\[ 40\%=0.40 \]
\[ 60\%=0.60 \]
Multiply each score by its weight:
\[ 0.40(85)+0.60(92) \]
\[ =34+55.2 \]
\[ =89.2 \]
Therefore:
\[ \boxed{89.2} \]
Example 2 — Combining Two Groups
Class A has 20 students with mean 78.
Class B has 30 students with mean 84.
Find the combined mean.
First find each class total.
Class A:
\[ 20(78)=1560 \]
Class B:
\[ 30(84)=2520 \]
Add the totals:
\[ 1560+2520=4080 \]
Add the students:
\[ 20+30=50 \]
Therefore:
\[ \text{Combined Mean} = \frac{4080}{50} \]
\[ =81.6 \]
So:
\[ \boxed{81.6} \]
Example 3 — Frequency Table
Suppose the scores are:
- 70 occurs 3 times
- 80 occurs 4 times
- 90 occurs 1 time
Total frequency:
\[ 3+4+1=8 \]
Weighted sum:
\[ 70(3)+80(4)+90(1) \]
\[ =210+320+90 \]
\[ =620 \]
Divide by the total frequency:
\[ \frac{620}{8} = 77.5 \]
Therefore:
\[ \boxed{77.5} \]
Example 4 — Mixture
Mix:
- 2 L of 30% solution
- 3 L of 10% solution
Find the final concentration.
First find the amount of pure substance:
\[ 0.30(2)=0.60 \]
\[ 0.10(3)=0.30 \]
Add:
\[ 0.60+0.30=0.90 \]
Total volume:
\[ 2+3=5 \]
Final concentration:
\[ \frac{0.90}{5} = 0.18 \]
Convert to a percent:
\[ \boxed{18\%} \]
Example 5 — Missing Weighted Score
A student’s final grade is based on:
- Projects: 25%, average 88
- Tests: 75%, average \(x\)
The final grade is 91.
Write the weighted-average equation:
\[ 0.25(88)+0.75x=91 \]
Multiply:
\[ 22+0.75x=91 \]
Subtract 22:
\[ 0.75x=69 \]
Divide:
\[ x=92 \]
Therefore:
\[ \boxed{92} \]
- Dividing by the number of categories instead of the total weight.
- Forgetting to multiply each value by its weight.
- Averaging two group means when the groups have different sizes.
- Forgetting to convert percentage weights to decimals.
- Dividing again when percentage weights already add to 1.
- Using frequency values as data values instead of as weights.
- Ignoring total amount in a mixture problem.
- Averaging speeds directly when the time intervals are unequal.
- Pairing a value with the wrong weight.
- Getting an answer outside the range of the original values without checking the work.
Practice Problems
- A course grade consists of:
- Quiz average: 80, worth 30%
- Test average: 90, worth 70%
Find the weighted grade.
- Two groups have:
- 10 students with mean 72
- 15 students with mean 88
Find the combined mean.
- A frequency table contains:
- 5 occurring 6 times
- 7 occurring 3 times
- 10 occurring 1 time
Find the mean.
- Mix 2 L of a 30% solution with 3 L of a 10% solution.
Find the final concentration.
- A final grade is based on:
- Homework: 20%, average 90
- Tests: 80%, average \(x\)
If the final grade is 86, find \(x\).
- A car travels 40 mph for 2 hours and 70 mph for 3 hours. Find its average speed over the entire trip.
1. Convert the weights to decimals:
\[ 30\%=0.30 \]
\[ 70\%=0.70 \]
Calculate:
\[ 0.30(80)+0.70(90) \]
\[ =24+63 \]
\[ =87 \]
Therefore:
\[ \boxed{87} \]
2. Find the total represented by each group.
First group:
\[ 10(72)=720 \]
Second group:
\[ 15(88)=1320 \]
Combined total:
\[ 720+1320=2040 \]
Total students:
\[ 10+15=25 \]
Combined mean:
\[ \frac{2040}{25} = 81.6 \]
Therefore:
\[ \boxed{81.6} \]
3. Find the total frequency:
\[ 6+3+1=10 \]
Find the weighted sum:
\[ 5(6)+7(3)+10(1) \]
\[ =30+21+10 \]
\[ =61 \]
Divide by the total frequency:
\[ \frac{61}{10} = 6.1 \]
Therefore:
\[ \boxed{6.1} \]
4. Find the amount of pure substance in each solution:
\[ 0.30(2)=0.60 \]
\[ 0.10(3)=0.30 \]
Add:
\[ 0.60+0.30=0.90 \]
Total volume:
\[ 2+3=5 \]
Final concentration:
\[ \frac{0.90}{5} = 0.18 \]
Convert to a percent:
\[ \boxed{18\%} \]
5. Write the weighted-average equation:
\[ 0.20(90)+0.80x=86 \]
Multiply:
\[ 18+0.80x=86 \]
Subtract 18:
\[ 0.80x=68 \]
Divide:
\[ x=85 \]
Therefore:
\[ \boxed{85} \]
6. Find the distance traveled during each interval.
At 40 mph for 2 hours:
\[ 40(2)=80 \]
miles.
At 70 mph for 3 hours:
\[ 70(3)=210 \]
miles.
Total distance:
\[ 80+210=290 \]
Total time:
\[ 2+3=5 \]
Average speed:
\[ \frac{290}{5} = 58 \]
Therefore:
\[ \boxed{58\text{ mph}} \]
Summary
A weighted average gives different amounts of influence to different values.
The general formula is:
\[ \boxed{ \text{Weighted Average} = \frac{\sum w_ix_i}{\sum w_i} } \]
When percentage weights add to 100%, or decimal weights add to 1:
\[ \boxed{ \text{Weighted Average} = \sum w_ix_i } \]
Common weights include:
- percentages
- frequencies
- group sizes
- amounts or volumes
- time intervals
The value with the larger weight has more influence on the final average.
- Multiply value × weight.
- Then add.
- Divide by total weight when needed.
- Percentage weights totaling 100% → no extra division.
- Frequency = weight.
- Group size = weight.
- Don’t average averages unless group sizes are equal.
- Mixtures → track total amount of substance.
- Average speed → total distance ÷ total time.
- Larger weight → stronger pull on the final average.