๐ What is the Weighted Mean?
The weighted mean, also known as the weighted average, is a type of mean where some data points contribute more than others. It's used when some values are more significant or have a greater influence than others in a dataset. Unlike a regular arithmetic mean where all values are treated equally, the weighted mean assigns a weight to each value, reflecting its relative importance.
๐ A Brief History
The concept of weighted averages has been around for centuries, quietly used in various fields. Its formalization in statistics grew alongside the need for more nuanced analysis, especially in economics and finance where the relative importance of data points often varies greatly.
๐ Key Principles of Weighted Mean
- โ๏ธ Weights Represent Importance: Each weight signifies the importance or contribution of its corresponding value. Higher weights mean greater influence on the final result.
- โ Sum of Weights: The sum of all weights is generally equal to 1 (if expressed as proportions) or 100% (if expressed as percentages). If not normalized, the weighted mean formula accounts for this.
- ๐ข Calculation Formula: The weighted mean is calculated by multiplying each value by its weight, summing these products, and then dividing by the sum of the weights.
๐ The Formula
The formula for calculating the weighted mean is as follows:
$\text{Weighted Mean} = \frac{\sum_{i=1}^{n} (w_i * x_i)}{\sum_{i=1}^{n} w_i}$
Where:
- ๐ $x_i$ represents the individual data values.
- ๐ $w_i$ represents the weight assigned to each corresponding data value.
- โ represents the summation across all data values.
โ๏ธ Steps to Calculate the Weighted Mean
- ๐ท๏ธ Identify Data Values and Weights: Clearly list all data values ($x_i$) and their corresponding weights ($w_i$).
- โ๏ธ Multiply Each Value by its Weight: For each data point, multiply the value by its assigned weight ($x_i * w_i$).
- โ Sum the Weighted Values: Add up all the products obtained in the previous step. This gives you the numerator of the weighted mean formula.
- โ Sum the Weights: Add up all the weights. This gives you the denominator of the weighted mean formula.
- โ Divide: Divide the sum of the weighted values (numerator) by the sum of the weights (denominator).
๐ Real-world Examples
Example 1: Calculating a Grade Point Average (GPA)
A student's grades are weighted based on the credit hours of each course:
| Course |
Credits (Weight) |
Grade (Value) |
| Math |
4 |
3.0 |
| Science |
3 |
4.0 |
| English |
3 |
3.7 |
| History |
2 |
3.3 |
Weighted Mean GPA = $\frac{(4 * 3.0) + (3 * 4.0) + (3 * 3.7) + (2 * 3.3)}{4 + 3 + 3 + 2} = \frac{12 + 12 + 11.1 + 6.6}{12} = \frac{41.7}{12} = 3.475$
Example 2: Portfolio Returns
An investor has a portfolio with the following investments:
| Investment |
Value (Weight) |
Return (Value) |
| Stock A |
$50,000 |
10% |
| Stock B |
$30,000 |
15% |
| Bond C |
$20,000 |
5% |
Weighted Mean Return = $\frac{(50000 * 0.10) + (30000 * 0.15) + (20000 * 0.05)}{50000 + 30000 + 20000} = \frac{5000 + 4500 + 1000}{100000} = \frac{10500}{100000} = 0.105 = 10.5\%$
โ
Conclusion
The weighted mean is a powerful tool for calculating averages when some values are more important than others. It is widely used in various fields like finance, academics, and statistics to provide a more accurate and relevant measure of central tendency. By understanding the principles and steps involved, you can effectively apply the weighted mean to analyze data and make informed decisions.