Weighted Average Formula

A weighted average is an average in which different values contribute unequally based on their assigned weights, reflecting the relative importance or frequency of each value.

The Formula

{\overset{ˉ}{x}}_{w} = \frac{\sum w_{i} x_{i}}{\sum w_{i}}

When to use: Your final grade: exams count 60%, homework 40% — not every assignment counts equally.

Quick Example

Scores 80 (weight 0.4) and 90 (weight 0.6): weighted average = 80×0.4 + 90×0.6 = 86.

What This Formula Means

A weighted average is an average in which different values contribute unequally based on their assigned weights, reflecting the relative importance or frequency of each value. Unlike a simple average where all values count equally, a weighted average gives more influence to values with larger weights.

Your final grade: exams count 60%, homework 40% — not every assignment counts equally.

Formal View

Given values

x_1, x_2, \ldots, x_n

with corresponding positive weights

w_1, w_2, \ldots, w_n

, the weighted average is

\bar{x}_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

. When all

w_i = 1

, this reduces to the arithmetic mean.

Worked Examples

Example 1

medium

A student got

85, 92, 78

on three quizzes weighted

2, 3, 1

respectively. What is the weighted average quiz score?

Answer

\frac{524}{6} \approx 87.33

First step

Weighted sum:

2 \cdot 85 + 3 \cdot 92 + 1 \cdot 78 = 170 + 276 + 78 = 524

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Example 2

medium

In a course, homework counts

20\%

, midterm

30\%

, final

50\%

. A student scored

90, 75, 82

. Compute the final grade.

Example 3

hard

A student needs a course average of

85

. So far: homework

20\%

90

, midterm

30\%

80

. The final exam counts

50\%

. What score is needed on the final to reach

85

overall?

Common Mistakes

Forgetting to divide by the sum of weights - The safer move is to ask "Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice?" and then state the data source, denominator, or variable before interpreting the result.
Using equal weights when data points have different importance - The safer move is to ask "Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice?" and then state the data source, denominator, or variable before interpreting the result.
Confusing weights with the values themselves - The safer move is to ask "Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice?" and then state the data source, denominator, or variable before interpreting the result.
Choosing weighted average from a keyword alone - Keywords like average, typical, middle are only clues; the data structure must match the concept.

Why This Formula Matters

Weighted Average gives students a disciplined way to summarize where data is centered. It is especially useful when two data sets look different but need a compact comparison, because the center tells where values tend to sit before students discuss spread, shape, or unusual values.

Frequently Asked Questions

What is the Weighted Average formula?

How do you use the Weighted Average formula?