Weighted Average — Definition, Formula & Examples

Weighted Average

A method of computing a kind of arithmetic mean of a set of numbers in which some elements of the set carry more importance (weight) than others.

Example:

Grades are often computed using a weighted average. Suppose that homework counts 10%, quizzes 20%, and tests 70%.

If Pat has a homework grade of 92, a quiz grade of 68, and a test grade of 81, then

Pat's overall grade = (0.10)(92) + (0.20)(68) + (0.70)(81)
= 79.5

See also

Average

Key Formula

\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n} = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

Where:

$x_i$ = The individual data values
$w_i$ = The weight assigned to each value, representing its relative importance
$n$ = The number of data values
$\bar{x}_w$ = The weighted average (result)

Worked Example

Problem: A student's final grade is based on three categories: Homework (weight 10%), Quizzes (weight 20%), and Tests (weight 70%). The student scored 92 in Homework, 68 in Quizzes, and 81 in Tests. Find the final grade.

Step 1: Identify each value and its corresponding weight. The weights are 0.10, 0.20, and 0.70.

x_1 = 92,\; w_1 = 0.10 \quad x_2 = 68,\; w_2 = 0.20 \quad x_3 = 81,\; w_3 = 0.70

Step 2: Multiply each value by its weight.

0.10 \times 92 = 9.2 \qquad 0.20 \times 68 = 13.6 \qquad 0.70 \times 81 = 56.7

Step 3: Add the weighted products together.

9.2 + 13.6 + 56.7 = 79.5

Step 4: Since the weights are percentages that already sum to 1.00, the denominator is 1. The weighted average is simply the sum from Step 3.

\bar{x}_w = \frac{79.5}{1.00} = 79.5

Answer: The student's final grade is 79.5.

Another Example

Problem: Three boxes of apples weigh 10 kg, 20 kg, and 30 kg. The apples in each box cost

2,

3, and $5 per kilogram, respectively. What is the weighted average price per kilogram across all the apples?

Step 1: Identify the values (prices) and their weights (kilograms). Here the weights do not sum to 1, so you must divide by their total.

x_1 = 2,\; w_1 = 10 \qquad x_2 = 3,\; w_2 = 20 \qquad x_3 = 5,\; w_3 = 30

Step 2: Compute each weighted product.

10 \times 2 = 20 \qquad 20 \times 3 = 60 \qquad 30 \times 5 = 150

Step 3: Sum the weighted products and sum the weights.

\text{Numerator} = 20 + 60 + 150 = 230 \qquad \text{Denominator} = 10 + 20 + 30 = 60

Step 4: Divide to get the weighted average.

\bar{x}_w = \frac{230}{60} \approx 3.83

Answer: The weighted average price is approximately

3.83 per kilogram. Notice this is higher than the simple average of the three prices (

3.33) because the most expensive apples make up the largest portion of the total weight.

Frequently Asked Questions

What is the difference between a weighted average and a regular average?

A regular (arithmetic) average adds all values and divides by the count, treating every value equally. A weighted average multiplies each value by a weight before summing, then divides by the total weight. Use a weighted average whenever some values matter more than others—like when a final exam counts more than homework.

How do I find the weights for a weighted average?

The weights are given by the context of the problem. In a course syllabus, the weights are the percentage each category counts toward your grade. In finance, the weights might be the amount of money invested in each asset. If all weights are equal, the weighted average simplifies to the ordinary arithmetic mean.

Weighted Average vs. Arithmetic Mean

The arithmetic mean assigns equal importance to every value: you add them up and divide by the count. The weighted average assigns different importance to each value through weights. If every weight is the same, the weighted average equals the arithmetic mean. The distinction matters whenever data points represent different quantities, frequencies, or levels of significance.

Why It Matters

Weighted averages appear constantly in real life. Course grades almost always use weighted averages to reflect how much exams, homework, and quizzes count. In finance, a stock portfolio's return is the weighted average of each stock's return, weighted by the amount invested. Understanding weighted averages helps you accurately combine information where not all pieces carry the same significance.

Common Mistakes

Mistake: Forgetting to divide by the sum of the weights when the weights do not add up to 1.

Correction: Always check whether your weights sum to 1 (as percentages often do). If they do not, you must divide the sum of the weighted products by the total of the weights. Skipping this step gives an answer that is too large.

Mistake: Using the simple arithmetic mean when the problem requires a weighted average.

Correction: If different data points represent different quantities or importance levels, a simple mean will give the wrong result. Look for clues like percentages, frequencies, or amounts—these signal that you need a weighted average.

Related Terms

Average — General term; weighted average is a specific type
Arithmetic Mean — Equal-weight special case of weighted average
Mean — Broader concept encompassing several types of averages
Set — Collection of values being averaged
Sum — Core operation used in computing weighted averages
Median — Alternative measure of central tendency