Weighted mean is an average where each value is multiplied by its assigned weight before calculating the overall mean.
## Core concept
The weighted mean accounts for the relative importance (weight) of different values in a dataset. Unlike a simple arithmetic mean where all values are equally important, the weighted mean gives more influence to values with higher weights.
Key principle: Weight represents frequency, importance, or relative contribution of each observation.
Formula: $$\bar{x}_w = \frac{\sum (x_i \times w_i)}{\sum w_i}$$
Where: - $x_i$ = individual values - $w_i$ = weight assigned to each value - $\sum (x_i \times w_i)$ = sum of (value × weight) products - $\sum w_i$ = sum of all weights
## Step-by-step calculation
- Identify each value and its corresponding weight
- Multiply each value by its weight
- Sum all the (value × weight) products
- Sum all the weights
- Divide total weighted sum by total weights
## Common exam applications
Subject-wise performance: A student scores: English 75 (weight 2), Math 85 (weight 3), Science 80 (weight 2).
$$\bar{x}_w = \frac{(75 \times 2) + (85 \times 3) + (80 \times 2)}{2 + 3 + 2}$$
$$= \frac{150 + 255 + 160}{7} = \frac{565}{7} = 80.71$$
The weighted average is 80.71, reflecting higher importance of Math (weight 3).
Price index calculation: Weighted average is essential for computing price indices where different commodities have different weightages in the basket.
Portfolio returns: Different investments with different weights contribute to overall portfolio return.
Quality assessment: Products or batches of different sizes (weights) contribute to average quality assessment.
## Relationship to arithmetic mean
- If all weights are equal, weighted mean equals arithmetic mean
- Weighted mean shifts toward values with higher weights
- Weighted mean lies between the minimum and maximum values in the dataset
## Common mistakes
| Mistake | Why it's wrong | Correct approach | |---------|---|---| | Forgetting to sum weights in denominator | Leads to incorrect scaling | Always calculate $\sum w_i$ separately | | Using arithmetic mean formula on weighted data | Ignores importance differential | Apply weighted formula explicitly | | Treating frequency as weight without clarification | Context matters for interpretation | Confirm weight definition in problem | | Decimal weight handling errors | Rounding errors accumulate | Maintain precision until final answer |
## Exam strategy points
- Always show weights: Clearly identify and list all weights before calculation
- Label units: Maintain units throughout calculation (if marks, currency, etc.)
- Verify reasonableness: Final answer should lie between minimum and maximum values
- Check denominator: Sum of weights must appear in denominator—this is the defining feature
- Compare with arithmetic mean: Mention how weighted differs from simple mean to show understanding
Related concepts: Weighted mean connects to combined mean (when combining groups with different sizes) and partition values, as weights determine how data is partitioned by importance rather than just position.