Weighted Average Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Weighted Average.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

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.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Weighted Average asks what single value best stands for the center of the data, then checks whether that value is fair for the situation.

Common stuck point: Students often know a procedure related to weighted average but skip the recognition step: Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice?

Worked Examples

Example 1

medium
A student got 85,92,7885, 92, 78 on three quizzes weighted 2,3,12, 3, 1 respectively. What is the weighted average quiz score?

Answer

5246≈87.33\frac{524}{6} \approx 87.33

First step

1
Weighted sum: 2⋅85+3⋅92+1⋅78=170+276+78=5242 \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%20\%, midterm 30%30\%, final 50%50\%. A student scored 90,75,8290, 75, 82. Compute the final grade.

Example 3

hard
A student needs a course average of 8585. So far: homework 20%20\% at 9090, midterm 30%30\% at 8080. The final exam counts 50%50\%. What score is needed on the final to reach 8585 overall?

Example 4

hard
A weighted average of two values aa and bb with weights waw_a and wbw_b equals aa exactly. What does that tell you about wbw_b?

Example 5

hard
Two groups have means 5050 and 8080. Their combined weighted average is 6565. If the first group has 3030 members, how many are in the second?

Example 6

challenge
Two schools combined have an average test score of 7777. School A averages 8080 with 120120 students. School B averages 7272. How many students does school B have?

Example 7

hard
A teacher wants final grades to weight homework, midterm, and final at 0.2,0.3,0.50.2, 0.3, 0.5. A student got 70,80,x70, 80, x. The teacher wants the final grade to be exactly 8080. What must xx be?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Write the formula for a weighted average.

Example 2

easy
Two values 44 and 1010 have equal weights. What is their weighted average?

Example 3

easy
A grade is 8080 (weight 22) and 9090 (weight 11). Compute the weighted sum ∑wixi\sum w_i x_i.

Example 4

easy
Weights are 22 and 11. What is the total weight ∑wi\sum w_i?

Example 5

easy
Using weighted sum 250250 and total weight 33, find the weighted average.

Example 6

easy
If exams count 60%60\% and homework 40%40\%, do all assignments count equally?

Example 7

easy
A weighted average uses weights that are all the same. What simpler quantity does it equal?

Example 8

easy
Should the weight of a value be confused with the value itself?

Example 9

medium
A course grade: exam 9090 (weight 0.60.6), homework 8080 (weight 0.40.4). Find the weighted average.

Example 10

medium
Test scores 70,80,9070, 80, 90 have weights 1,2,31, 2, 3. Compute the weighted average.

Example 11

medium
A class has 1010 students averaging 8080 and 2020 students averaging 9595. Find the combined (weighted) average.

Example 12

medium
An investment returns 10%10\% on $2000 and 4%4\% on $3000. What is the weighted average return?

Example 13

medium
A student needs a course average of 9090. Exams (weight 0.70.7) average 8888; homework (weight 0.30.3) averages xx. The total is exactly 9090 when... actually compute the homework average needed.

Example 14

medium
A GPA uses grade points 4,3,44, 3, 4 with credit weights 3,4,23, 4, 2. Compute the GPA.

Example 15

medium
Why can't you just average 50%50\% and 90%90\% to get the overall pass rate of two groups of different sizes?

Example 16

medium
Values 2020 and 4040 have weights ww and 11. For what weight ww does the weighted average equal 2525?

Example 17

medium
Quiz scores 6,8,106, 8, 10 have weights 3,1,13, 1, 1. Compute the weighted average.

Example 18

challenge
Two classes: one of 1515 students averaging 7070, another of 2525 students averaging 8686. Find the overall average.

Example 19

challenge
A blend mixes 33 kg of coffee at $8/kg with 22 kg at $13/kg. Find the price per kg of the blend.

Example 20

challenge
Show that if all weights are doubled, the weighted average is unchanged. Give the key reason.

Example 21

easy
Find the weighted average of 5050 (weight 11) and 100100 (weight 11).

Example 22

easy
Compute the weighted sum ∑wixi\sum w_i x_i for x=(8,10)x = (8, 10) with w=(3,2)w = (3, 2).

Example 23

easy
Using weighted sum 4444 and total weight 55, what is the weighted average?

Example 24

easy
Compute the weighted average of 2020 (weight 0.50.5) and 4040 (weight 0.50.5).

Example 25

easy
Two values 6060 and 9090 have weights 11 and 22. Which weighted average is closer to: 6060 or 9090?

Example 26

medium
A test grade is 7070 (counts 25%25\%), midterm 8080 (counts 35%35\%), final 9090 (counts 40%40\%). Find the weighted average.

Example 27

medium
A school has 200200 students at grade-level average 7575 and 300300 students at grade-level average 8585. Find the schoolwide (weighted) average.

Example 28

medium
A car drives 100100 miles at 4040 mph and 100100 miles at 6060 mph. What is the (time-weighted) average speed for the whole trip?

Example 29

medium
A shop sells 5050 shirts at $20 each, 3030 shirts at $30 each, and 2020 shirts at $50 each. What is the average price per shirt sold?

Example 30

medium
A GPA uses credits as weights: A (4.04.0, 33 credits), B (3.03.0, 44 credits), A (4.04.0, 33 credits). Compute the GPA.

Example 31

hard
A student has $5000\$5000 at 3%3\% interest and $2000\$2000 at 7%7\% interest. What is the (dollar-weighted) average interest rate?

Example 32

hard
A truck mixes 400400 kg of alloy at 20%20\% copper with 600600 kg at 35%35\% copper. What is the copper percentage of the mix?

Example 33

hard
In a survey, 60%60\% of 500500 respondents rated a product 4/54/5 stars, and the rest rated it 3/53/5. What is the average rating?

Example 34

hard
A portfolio holds three stocks with values $2000, $3000, $5000 and returns 8%,12%,6%8\%, 12\%, 6\%. What is the portfolio return?

Example 35

medium
A bag has 55 apples averaging $0.40\$0.40 each and 1010 oranges averaging $0.60\$0.60 each. What is the average price per fruit?

Example 36

hard
A class of 2525 students has a mean test score of 8282. A new student joins with a score of 5858. What is the new class mean?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

mean fair sharestat expected value