Mathematics
Weighted Average
Quick fact
In a weighted average, changing even a single weight can completely shift the result, making it a powerful tool for prioritizing data.
Why this is interesting
You know the average of your test scores, but what if some tests count more than others? That's where the weighted average comes in—it gives each number a say proportional to its importance.
Read the full explanation
Understanding Weighted Average
Imagine you have three test scores: 80, 90, and 95. A simple average adds them and divides by 3, giving 88.3. But suppose the first test is worth 50% of your grade, the second 30%, and the third 20%. Now each score must be multiplied by its percentage (as a decimal) before summing. You calculate: (80 × 0.50) + (90 × 0.30) + (95 × 0.20) = 40 + 27 + 19 = 86. Then divide by the total weight (0.50 + 0.30 + 0.20 = 1.00) → 86. So your weighted average is 86. The weight reflects how much each score contributes, giving a more accurate picture when items have different importance.
A deeper explanation
The underlying idea is that not all data points are equally representative or important. The formula for a weighted average is: \( \bar{x}w = \frac{\sum{i=1}^n wi xi}{\sum{i=1}^n wi} \), where \( wi \) is the weight assigned to each value \( xi \). The denominator (sum of weights) normalizes the result so it remains a form of average. This concept appears everywhere: calculating GPA (course credits as weights), financial portfolio returns (investment amounts as weights), and opinion polls (weights adjust for demographic representation). The key insight is that weighted averages let you distill a collection of differently important pieces into a single, meaningful number, respecting their relative significance.