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Weighted Mean

Weighted mean is a type of average where some values count more than others, based on assigned weights. Instead of treating every value equally, you multiply each value by its weight, add those products together, and divide by the total of the weights.

The weighted mean is a measure of central tendency calculated by multiplying each data value by a corresponding weight, summing these products, and dividing by the sum of the weights. It generalizes the arithmetic mean, which is the special case where all weights are equal. The weighted mean is used when certain observations carry greater importance, frequency, or reliability than others.

Key Formula

xˉw=∑i=1nwi⋅xi∑i=1nwi=w1x1+w2x2+⋯+wnxnw1+w2+⋯+wn\bar{x}_w = \frac{\sum_{i=1}^{n} w_i \cdot x_i}{\sum_{i=1}^{n} w_i} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}
Where:
  • xˉwx̄_w = the weighted mean
  • xix_i = each individual data value
  • wiw_i = the weight assigned to each data value
  • nn = the number of data values

Worked Example

Problem: A student's final grade is based on three categories: Homework (weight 20%), Midterm Exam (weight 30%), and Final Exam (weight 50%). The student scored 90 in Homework, 80 in the Midterm, and 70 in the Final Exam. What is the student's weighted mean grade?
Step 1: Multiply each score by its weight.
90×0.20=18,80×0.30=24,70×0.50=3590 \times 0.20 = 18, \quad 80 \times 0.30 = 24, \quad 70 \times 0.50 = 35
Step 2: Add the weighted values together.
18+24+35=7718 + 24 + 35 = 77
Step 3: Add the weights together. Since they are percentages that sum to 100%, the total is 1.
0.20+0.30+0.50=1.000.20 + 0.30 + 0.50 = 1.00
Step 4: Divide the sum of weighted values by the sum of the weights.
xˉw=771.00=77\bar{x}_w = \frac{77}{1.00} = 77
Answer: The student's weighted mean grade is 77. Notice this is lower than the ordinary mean of the three scores (which would be 80), because the lowest score — the Final Exam — carried the heaviest weight.

Visualization

Why It Matters

Weighted means appear whenever not all values should count equally. Course grades are a familiar example — exams typically count more than quizzes. Beyond school, weighted means are used in stock market indices, survey analysis, and any situation where data points differ in importance or frequency.

Common Mistakes

Mistake: Using an ordinary average instead of accounting for weights.
Correction: A regular mean treats every value equally. If weights are given, you must multiply each value by its weight before summing. In the example above, the ordinary mean gives 80, but the correct weighted mean is 77.
Mistake: Forgetting to divide by the sum of the weights.
Correction: After summing the products wi⋅xiw_i \cdot x_i, you must divide by the total of all weights, not by the number of values. When weights are percentages adding to 100%, the divisor happens to be 1, but this isn't always the case.

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