A simple average treats every value as equally important. That is wrong when some values count more than others: a final exam worth 40% of a course matters more than a homework set worth 10%, and a 4-credit class moves your GPA more than a 1-credit one. A weighted average gives each value a weight that says how much it counts.
The formula
Multiply each value by its weight, add the results, and divide by the sum of the weights:
$$ \bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n} $$
The weights can be percents, credits, hours, kilograms or any positive numbers. If they are percents that add up to 100, the division by 100 is built in and you can just add the products.
Example 1: a course grade. Homework is 10% of the grade, quizzes 20%, the midterm 30% and the project 40%. Your marks are 95, 80, 62 and 88.
$$ \frac{95(10) + 80(20) + 62(30) + 88(40)}{10 + 20 + 30 + 40} $$
$$ = \frac{7930}{100} = 79.3 $$
The simple average of the four marks is 81.25, which overstates the result because the weak midterm carries a lot of weight.
A weighted average is a seesaw
Picture a beam marked from 0 to 100. Put a circle at each grade, with a size that matches its weight. The weighted average is the point where the beam balances on a pivot. Heavy circles pull the balance point toward themselves. Change a grade or slide a weight, then tick the box to move the pivot to the simple average and watch the beam tip.
Each circle is a grade, and its size is its weight. The weighted average is the point where the beam balances. Change a grade or slide a weight.
| Part | Grade | Weight | Points |
|---|
Credits and GPA
Credits are weights. A semester has Calculus (4 credits, grade point 3.3), Physics (3 credits, 4.0), History (3 credits, 3.0) and Gym (1 credit, 4.0).
$$ \text{GPA} = \frac{4(3.3) + 3(4.0) + 3(3.0) + 1(4.0)}{4 + 3 + 3 + 1} $$
$$ = \frac{38.2}{11} \approx 3.47 $$
The plain average of the four grade points is 3.58. The GPA is lower because the two classes below 4.0, Calculus and History, carry 7 of the 11 credits.
What do I need on the final?
Say your graded work so far averages c and the final counts for a fraction w of the course. The other work counts for $latex 1 - w$. To end with a target grade T:
$$ c(1 - w) + x\,w = T \quad\Longrightarrow\quad x = \frac{T - c(1 - w)}{w} $$
Example 2. You average 82 on everything but the final, which is worth 30%. You want 85 in the course.
$$ x = \frac{85 - 82(0.70)}{0.30} = \frac{85 - 57.4}{0.30} = 92 $$
You need a 92 on the final. Now try a target of 95: $latex x = \frac{95 - 57.4}{0.30} \approx 125.33$. That is over 100, so 95 is out of reach. Even a perfect final gives $latex 57.4 + 30 = 87.4$.
Example 3: when you have the pieces instead of an average. Homework (15%) 90, midterm (35%) 76 and project (20%) 84, with a final worth 30%. Points banked so far:
$$ 0.15(90) + 0.35(76) + 0.20(84) = 56.9 $$
For an 80:
$$ x = \frac{80 - 56.9}{0.30} = 77 $$
You need 77 on the final. Subtracting the points you already have, then dividing by the weight of the final, works every time.
Weighted averages outside the classroom
Any time quantities of different size are combined, the average is weighted. A shop mixes 3 kg of coffee at $10 per kg with 2 kg at $15 per kg. The price of the mix is not $12.50, the plain average of 10 and 15. It is
$$ \frac{3(10) + 2(15)}{3 + 2} = \frac{60}{5} = 12 $$
dollars per kilogram, closer to 10 because there is more of the cheaper coffee. The same idea gives the average cost of shares bought at different prices and the average concentration of a mixture.
Common mistakes
- Using the simple average. Adding the grades and dividing by how many there are ignores the weights. (A simple mean is fine when every value counts the same: see mean, median and mode.)
- Forgetting to divide by the sum of the weights. The numerator alone is not an average, unless the weights already add to 1.
- Mixing percents and decimals. Use 30 with a total of 100, or 0.30 with a total of 1, but not 30 with a total of 1.
- Dividing by the wrong weight on the final. To find the final score you divide by the weight of the final only, after subtracting the points already earned.
- Averaging percentages of different totals. A 90% on a 10-point quiz and a 60% on a 100-point exam are not equal contributions unless the course says so.
Practice problems
Problem 1
Exams score 70, 85 and 90, with weights 25%, 35% and 40%. What is the weighted average?
Answer
$$ 0.25(70) + 0.35(85) + 0.40(90) = 17.5 + 29.75 + 36 = 83.25 $$
Answer: 83.25.
Problem 2
A student takes a 3-credit course with a 90, a 4-credit course with an 80 and a 1-credit course with a 100. What is the weighted average?
Answer
$$ \frac{3(90) + 4(80) + 1(100)}{3 + 4 + 1} = \frac{690}{8} = 86.25 $$
Answer: 86.25.
Problem 3
Your average before the final is 78. The final is worth 40%. What score gives you an 80 in the course?
Answer
$$ x = \frac{80 - 78(0.60)}{0.40} = \frac{80 - 46.8}{0.40} = 83 $$
Answer: 83.
Problem 4
Your average before the final is 70 and the final is worth 20%. Can you still finish with an 85?
Answer
$$ x = \frac{85 - 70(0.80)}{0.20} = \frac{85 - 56}{0.20} = 145 $$
Answer: no. It would take a 145 on the final. A perfect final gives at most $latex 56 + 20 = 76$.
Problem 5
A baker uses 6 kg of flour at $4.50 per kg and 4 kg at $7.25 per kg. What is the average price per kilogram?
Answer
$$ \frac{6(4.50) + 4(7.25)}{10} = \frac{27 + 29}{10} = 5.6 $$
Answer: $5.6 per kilogram.
Problem 6
Grades 60, 80 and 100 have weights 2, 3 and 5 (not percents). What is the weighted average?
Answer
$$ \frac{2(60) + 3(80) + 5(100)}{2 + 3 + 5} = \frac{860}{10} = 86 $$
Answer: 86.
Frequently asked questions
Do the weights have to add up to 100?
No. Any positive weights work, because the formula divides by their sum. Weights of 2, 3 and 5 behave exactly like 20%, 30% and 50%.
When is a weighted average the same as a simple average?
When all the weights are equal. Then the weights cancel and you are left with the plain mean.
Can I get a weighted average above the highest grade?
No. It always falls between the lowest and highest value, because it is a blend of them.
How do I include a missing assignment?
A zero counts like any other grade with its weight. If the course drops it, remove it and divide by the remaining weights only.
Related calculators
- Weighted average calculator and final grade needed
- Mean, median and mode calculator
- Percentage calculator
Browse all Neurochispas calculators.