Weighted AM
Weighted Arithmetic Mean (Weighted AM)
The concept of the weighted arithmetic mean (Weighted AM) is an extension of the arithmetic mean (AM), which is used when different values in a data set have different levels of importance, or weights. This is particularly useful in situations where certain data points contribute more to the overall average than others.
Understanding Weighted AM
The weighted AM is calculated by multiplying each value in the data set by its corresponding weight, summing these products, and then dividing by the sum of the weights. The formula for the weighted AM is:
[ \text{Weighted AM} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i} ]
where:
- $x_i$ represents the $i^{th}$ value in the data set,
- $w_i$ represents the weight corresponding to the $i^{th}$ value,
- $n$ is the number of values in the data set.
Example of Weighted AM
Suppose we have a set of exam scores with different weights due to their varying importance:
| Exam | Score ($x_i$) | Weight ($w_i$) |
|---|---|---|
| 1 | 85 | 2 |
| 2 | 90 | 1 |
| 3 | 78 | 3 |
To calculate the weighted AM, we would do the following:
[ \text{Weighted AM} = \frac{(85 \times 2) + (90 \times 1) + (78 \times 3)}{2 + 1 + 3} = \frac{170 + 90 + 234}{6} = \frac{494}{6} \approx 82.33 ]
So, the weighted AM of the exam scores is approximately 82.33.
Differences Between AM and Weighted AM
| Aspect | Arithmetic Mean (AM) | Weighted Arithmetic Mean (Weighted AM) |
|---|---|---|
| Definition | The sum of values divided by the number of values. | The sum of the products of values and their weights divided by the sum of the weights. |
| Formula | $\text{AM} = \frac{\sum_{i=1}^{n} x_i}{n}$ | $\text{Weighted AM} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}$ |
| Weights | All values have equal weight (implicitly 1). | Values have different weights, which can vary. |
| Application | Used when all data points are equally important. | Used when some data points are more important than others. |
| Sensitivity | Not sensitive to the importance of individual data points. | Sensitive to the importance of individual data points through weights. |
Important Points to Remember
- The weights should be chosen carefully to reflect the importance of each data point accurately.
- If all weights are equal, the weighted AM reduces to the regular AM.
- The weighted AM can be influenced significantly by a data point with a very high weight.
- In some cases, the weights might be subjective and could affect the outcome of the weighted AM.
Example to Illustrate Important Points
Consider a student's overall course grade, which is determined by several types of assessments with different weights:
| Assessment Type | Grade ($x_i$) | Weight ($w_i$) |
|---|---|---|
| Homework | 80 | 0.2 |
| Quizzes | 85 | 0.3 |
| Midterm Exam | 75 | 0.2 |
| Final Exam | 90 | 0.3 |
The weighted AM for the student's overall grade is:
[ \text{Weighted AM} = \frac{(80 \times 0.2) + (85 \times 0.3) + (75 \times 0.2) + (90 \times 0.3)}{0.2 + 0.3 + 0.2 + 0.3} = \frac{16 + 25.5 + 15 + 27}{1} = 83.5 ]
The student's overall grade is 83.5, which reflects the different importance of each assessment type.
In conclusion, the weighted AM is a valuable tool for calculating averages when different values have different levels of significance. It is widely used in academics, finance, and other fields where weighting factors are essential for a more accurate representation of data.