While both arithmetic mean and weighted average are used to find central values in data sets, they serve different purposes and produce different results when the importance of data points varies.
Arithmetic Mean (Simple Average)
The arithmetic mean treats all values equally. You add all values together and divide by the number of values. For example: (5 + 4 + 3) / 3 = 4. This method is appropriate when all data points have equal importance.
Arithmetic Mean = (value₁ + value₂ + ... + valueₙ) / n
Weighted Average
The weighted average takes into account the importance (weight) of each value. Values with higher weights have a greater impact on the result. Formula: Σ(value × weight) / Σ(weight).
Weighted Average = Σ(value × weight) / Σ(weight)
Key Differences
- Arithmetic mean treats all values equally; weighted average assigns different importance to values
- Arithmetic mean is a special case of weighted average where all weights are equal
- Weighted average is more accurate when data points have varying reliability or importance
Practical Example
Consider three test scores: 80, 90, and 70. With arithmetic mean: (80 + 90 + 70) / 3 = 80. With weights of 1, 2, and 3 respectively: (80×1 + 90×2 + 70×3) / 6 = 78.33.
The weighted average (78.33) differs from the arithmetic mean (80) because the value 70 with weight 3 has more influence on the final result.
When to Use Each Method
Use arithmetic mean when all values are equally important (e.g., daily temperature averages). Use weighted average when values have different levels of importance (e.g., university GPA with credits, portfolio returns with different investment amounts).