Weighted Average

Measures Of Center
definition

Also known as: weighted mean

Grade 6-8

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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. Weighted averages are essential in GPA calculation, financial portfolio returns, polling aggregation, and any analysis where data points differ in importance or reliability.

Definition

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.

💡 Intuition

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

🎯 Core Idea

Multiply each value by its weight, sum the products, then divide by the total weight.

Example

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

Formula

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

🌟 Why It Matters

Weighted averages are essential in GPA calculation, financial portfolio returns, polling aggregation, and any analysis where data points differ in importance or reliability.

💭 Hint When Stuck

To compute a weighted average, multiply each value by its weight, add up all the products, then divide by the sum of the weights. For example, if your exam (weight 3) score is 90 and quiz (weight 1) score is 70, the weighted average is (90 \times 3 + 70 \times 1) / (3 + 1) = 340 / 4 = 85.

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.

🚧 Common Stuck Point

Weights don't have to be percentages — any positive numbers work as long as you divide by their sum.

⚠️ Common Mistakes

  • Forgetting to divide by the sum of weights
  • Using equal weights when data points have different importance
  • Confusing weights with the values themselves

Frequently Asked Questions

What is Weighted Average in Statistics?

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.

What is the Weighted Average formula?

\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}

When do you use Weighted Average?

To compute a weighted average, multiply each value by its weight, add up all the products, then divide by the sum of the weights. For example, if your exam (weight 3) score is 90 and quiz (weight 1) score is 70, the weighted average is (90 \times 3 + 70 \times 1) / (3 + 1) = 340 / 4 = 85.

How Weighted Average Connects to Other Ideas

To understand weighted average, you should first be comfortable with mean fair share and stat expected value. Once you have a solid grasp of weighted average, you can move on to linear regression.

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