Introduction to Weighted Averages
The multipliers or sets of numbers that express more or less relative importance of various observations (data points) in a data set are called weights.
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The weighted arithmetic mean (simply called weighted average or weighted mean) is similar to an ordinary arithmetic mean except that instead of each data point contributing equally to the final average, some data points contribute more than others. Weighted means are useful in a wide variety of scenarios. Weighted averages are used when there are a bunch of values, but some of those values are more important or contribute more to the overall result.
Example of Weighted Average
For example, a student may use a weighted mean to calculate his/her percentage grade in a course. In such an example, the student would multiply the weight of all assessment items in the course (e.g., assignments, exams, sessionals, quizzes, projects, etc.) by the respective grade that was obtained in each of the categories.
As an example, suppose in a course there are a total of 60 marks, while the distribution of marks is as follows, Assignment-1 has a weightage of 10%, Assignment-2 has a weightage of 10%, the mid-term examination has a weightage of 30% and the final term examination have the weightage of 50%. The scenario is described in the table below:
| Assessment Item | Weight ($w_i$) | Grades ($x_i$) | Marks | Weighted Marks ($w_ix_i$) |
|---|---|---|---|---|
| Midterm | 10 % | 70 % | 6 | 7 % |
| Assignment # 2 | 10 % | 65 % | 6 | 6.5 % |
| Midterm Examination | 30 % | 70 % | 12 | 21 % |
| Final Term Examination | 50 % | 85 % | 30 | 42.5 % |
| 100 % | 290 % | 60 | 77 % |
Weighted Average Formula
Mathematically, the weighted average forma is given as
$$\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i}$$
Another Example
Consider another example: Suppose we have monthly expenditures of a family on different items with their quantity
| Items | Weights ($w_i$) | Expenses ($x_i$) | Weighted Expenses $w_ix_i$ |
|---|---|---|---|
| Food | 7.5 | 290 | 2175 |
| Rent | 2.0 | 54 | 108 |
| Clothing | 1.5 | 96 | 144 |
| Fuel and light | 1.0 | 75 | 75 |
| Misc | 0.5 | 75 | 37.5 |
| Total | 12.5 | 590 | 2539.5 |
The average expenses will be: $AM = \frac{590}{5} = 118$.
However, the weighted average of the scenario will be $\overline{x}_w = \frac{\sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i} = \frac{2539.5}{12.5}=203.16$
Keeping in mind the importance of weight, the average monthly expenses of a family was 203.16, not 118.
Note that in a frequency distribution, the computation of relative frequency (rf) is also related to the concept of weighted averages.
| Classes | Frequency | Mid point ($X$) | rf | Percentage |
|---|---|---|---|---|
| 65-84 | 9 | 74.5 | $\frac{9}{60} = 0.15$ | 15 |
| 85-104 | 10 | 94.5 | $\frac{10}{60} = 0.17$ | 17 |
| 105-124 | 17 | 114.5 | $\frac{10}{60} = 0.28$ | 28 |
| 125-144 | 10 | 134.5 | $\frac{10}{60} = 0.17$ | 17 |
| 145-164 | 5 | 154.5 | $\frac{5}{60} = 0.08$ | 8 |
| 165-184 | 4 | 174.5 | $\frac{4}{60} =0.07$ | 7 |
| 185-204 | 5 | 194.5 | $\frac{5}{60} =0.08$ | 8 |
| Total | 60 |
Some Real-World Examples of Weighted Averages
- Calculating class grade: Different assignments might have different weights (e.g., exams worth more than quizzes). A weighted mean considers these weights to determine the overall grade.
- Stock market performance: A stock index might use a weighted average to reflect the influence of large companies compared to smaller ones.
- Customer Satisfaction: Finding the average customer satisfaction score when some customers’ feedback might hold more weight (e.g., frequent buyers).
- Average Customer Spending: if some customers buy more frequently.
- Expected Value: Determining the expected value of outcomes with different probabilities.
The following are some important questions. What is the importance of weighted mean? Describe its advantages and disadvantages. What is an average? What are the qualities of a good average? What does Arithmetic mean? Describe the advantages and disadvantages of Arithmetic mean. In which situations do we apply arithmetic mean?