Weighted Averages

A weighted average accounts for the fact that not all values in a dataset are equally important. In a regular (simple) average, every value counts the same. In a weighted average, each value is multiplied by a weight that reflects its relative importance or frequency.

The Formula

$\text{Weighted Average} = \frac{\sum (value \times weight)}{\sum weights} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}$

Where $x_i$ is each value and $w_i$ is the corresponding weight.

When Do You Need a Weighted Average?

Use a weighted average instead of a simple average when:

• Categories have different importance — exam scores count more than homework
• Groups have different sizes — a class of 30 students should count more than a class of 10
• Items have different frequencies — a product sold 500 times matters more than one sold 5 times

If all weights are equal, a weighted average gives the same result as a simple average.

Worked Examples

Example 1: Course Grade Calculation

A student’s course grade is calculated with these weights:

Step 1: Multiply each score by its weight.

$92 \times 0.20 = 18.4$

$78 \times 0.30 = 23.4$

$85 \times 0.50 = 42.5$

Step 2: Add the weighted values.

$18.4 + 23.4 + 42.5 = 84.3$

Step 3: Confirm the weights sum to 1 (or 100%).

$0.20 + 0.30 + 0.50 = 1.00 \checkmark$

Answer: The weighted course grade is 84.3.

Notice that the simple average would be $\frac{92 + 78 + 85}{3} = 85.0$. The weighted average is lower because the midterm (the student’s weakest score) carries more weight than the homework (the strongest score).

Example 2: Average Price per Unit

A retailer buys t-shirts from three suppliers:

What is the average cost per shirt?

Step 1: Multiply each price by its quantity (the weight).

$8.00 \times 200 = 1{,}600$

$6.50 \times 500 = 3{,}250$

$9.25 \times 100 = 925$

Step 2: Add the total costs and total quantities.

$\text{Total cost} = 1600 + 3250 + 925 = \$5{,}775$

$\text{Total quantity} = 200 + 500 + 100 = 800$

Step 3: Divide.

$\text{Weighted average price} = \frac{5775}{800} \approx \$7.22$

Answer: The average cost per shirt is $7.22.

A simple average of the three prices would be $\frac{8.00 + 6.50 + 9.25}{3} \approx 7.92$. The weighted average is lower because the cheapest supplier (B at $6.50) provided the most shirts.

Example 3: GPA Calculation

GPA is one of the most common weighted averages. Each course grade is weighted by the number of credit hours.

Step 1: Multiply grade points by credits.

$3.3 \times 3 = 9.9$

$4.0 \times 4 = 16.0$

$3.0 \times 3 = 9.0$

$3.7 \times 4 = 14.8$

Step 2: Sum the products and total credits.

$\text{Total weighted points} = 9.9 + 16.0 + 9.0 + 14.8 = 49.7$

$\text{Total credits} = 3 + 4 + 3 + 4 = 14$

Step 3: Divide.

$\text{GPA} = \frac{49.7}{14} = 3.55$

Answer: The semester GPA is 3.55. The A in Biology and A- in Math pull the average up because those 4-credit courses carry more weight than the 3-credit courses.

Real-World Application: Nursing Certification Exam

A nursing certification exam has four sections, each weighted differently:

The passing score is 80. Did the candidate pass?

Step 1: Multiply each score by its weight.

$82 \times 0.30 = 24.6$

$91 \times 0.35 = 31.85$

$88 \times 0.20 = 17.6$

$76 \times 0.15 = 11.4$

Step 2: Add the weighted scores.

$24.6 + 31.85 + 17.6 + 11.4 = 85.45$

Step 3: Compare to the passing threshold.

$85.45 \geq 80 \checkmark$

Answer: The weighted score is 85.45 — the candidate passes. Even though the Health Promotion score (76) is below 80, its low weight of 15% prevents it from dragging the overall score below passing. The strong Patient Care score (91) at 35% weight has the largest positive impact.

What if the weights were equal? The simple average would be $\frac{82 + 91 + 88 + 76}{4} = 84.25$. The weighted average is actually higher (85.45) because the candidate scored highest on the most heavily weighted section.

Weighted Average Reference

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A weighted average multiplies each value by its weight before averaging: $\frac{\sum(value \times weight)}{\sum weights}$
• Use weighted averages when values have different importance, frequency, or quantity
• If all weights are equal, the weighted average equals the simple average
• Weighted averages appear everywhere: GPA, course grades, average prices, survey ratings, and exam scores
• Always check that your weights sum correctly (to 100%, to 1.0, or to the total count) as a way to verify your setup

Return to Statistics for more topics in this section.