Understanding the Weighted Mean

Learning Objectives

• Understand the difference between the arithmetic mean and the weighted mean.
• Check the sum of weights and interpret the result.

Key Concepts

The arithmetic mean gives every value equal weight. The weighted mean allows values to have different levels of importance.

Public example: the numbers only demonstrate KaTeX formulas and tables in a note.

Main Content

From Equal to Unequal Weights

The arithmetic mean of $80$ and $90$ is:

$$\bar{x} = \frac{80 + 90}{2} = 85.$$

With corresponding weights $0.4$ and $0.6$, the weighted mean becomes:

$$\bar{x}_w = 0.4 \times 80 + 0.6 \times 90 = 86.$$

Value	Weight	Contribution
80	0.4	32
90	0.6	54
Total	1.0	86

Check the General Form

If the weights have not been normalized, divide by their sum:

$$\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}, \qquad \sum_{i=1}^{n} w_i \ne 0.$$

For nonnegative weights that are not all zero, the result lies between the smallest and largest input values. Here, $86$ lies between $80$ and $90$ and is closer to the more heavily weighted $90$.

Summary

Identify what each weight means, check whether normalization is needed, and interpret the result. Equal weights reduce the weighted mean to the arithmetic mean.