๐Ÿ“Š Statistics District ยท Averages

Weighted mean

Calculate and interpret the weighted mean as a useful summary of a real data set. In this lesson, focus on the mean is a fair-share total: add every value, then divide by how many values contributed.

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Understand Weighted mean

Calculate and interpret the weighted mean as a useful summary of a real data set. In this lesson, focus on the mean is a fair-share total: add every value, then divide by how many values contributed.

A weighted mean gives some values more influence because they occur more often or carry a stated weight. Multiply each value by its frequency or weight, add those products, then divide by the total frequency or total weight. This is not the same as finding the mean of the visible category labels.

Start here

Weighted mean: The mean is a fair-share total: add every value, then divide by how many values contributed. Multiply the mean by the number of values and confirm that it recreates the original total. Keep the weighted mean representation visible until the final line.

Picture the idea

Build weighted stacks so repeated values occupy more blocks, then balance the combined blocks to reveal the weighted mean.

Check as you go

Confirm the weighted mean lies between the smallest and largest values, then multiply it by ฮฃf to recover ฮฃfx.

Key vocabulary

meantotalfrequencybalance pointoutlierweighted

Rules and key facts

Given information: Weighted mean โ€” Find the mean of 3, 2, 4, 2. Method choice: Add every value before dividing. Divide by the number of values, not by the largest value. Calculation or reasoning: Add: 3 + 2 + 4 + 2 = 11. Divide by 4: 2.75. Final answer: 2.75. Check: Divide by the number of values, not by the largest value.

  • Multiply each value by its frequency or stated weight.
  • Add the products to find ฮฃfx.
  • Add frequencies or weights to find ฮฃf.
  • Divide ฮฃfx by ฮฃf and check that the result lies within the data range.

Step-by-step method

  1. Multiply each value by its frequency or stated weight.
  2. Add the products to find ฮฃfx.
  3. Add frequencies or weights to find ฮฃf.
  4. Divide ฮฃfx by ฮฃf and check that the result lies within the data range.

What you need first

  • Recognise the vocabulary: mean, total, frequency.
  • Be able to explain the purpose of weighted mean before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Average scores
  • Fair sharing

Visual / interactive

See the idea, then move it around

Skip to Practice

Build weighted stacks so repeated values occupy more blocks, then balance the combined blocks to reveal the weighted mean.

Mean / median / mode / range calculator Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
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Worked examples

Examples, methods and exam thinking

Level 1 ยท Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 ยท Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 ยท Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 ยท Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Weighted mean โ€” Find the mean of 3, 2, 4, 2. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 2 + 4 + 2 = 11. Divide by 4: 2.75. Final answer: 2.75. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Multiply each value by its frequency or stated weight.
  2. Add the products to find ฮฃfx.
  3. Add frequencies or weights to find ฮฃf.
Secure example

Use the normal method

Given information: Weighted mean โ€” Find the mean of 3, 14, 14, 10. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 14 + 14 + 10 = 41. Divide by 4: 10.25. Final answer: 10.25. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Check the weighted mean result against the original information.

Challenge example

Stretch the idea

Given information: Weighted mean โ€” Find the mean of 3, 13, 9, 7. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 13 + 9 + 7 = 32. Divide by 4: 8. Final answer: 8. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Weighted mean โ€” Find the mean of 3, 12, 4, 4. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 12 + 4 + 4 = 23. Divide by 4: 5.75. Final answer: 5.75. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Underline the units and command word. Show the key calculation and write the answer in context.

Common mistakes

  • Dividing by the number of categories instead of the total frequency.
  • Adding values without multiplying by their frequencies.

How to check your answer

Confirm the weighted mean lies between the smallest and largest values, then multiply it by ฮฃf to recover ฮฃfx.

Extension challenge

Create a weighted mean problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Weighted mean challenge

Use mean balance controls to solve three checked weighted mean rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Study cards

Core idea

Weighted mean: The mean is a fair-share total: add every value, then divide by how many values contributed. Multiply the mean by the number of values and confirm that it recreates the original total. Keep the weighted mean representation visible until the final line.

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Key vocabulary

mean ยท total ยท frequency ยท balance point ยท outlier ยท weighted

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Rules

Multiply each value by its frequency or stated weight. Add the products to find ฮฃfx. Add frequencies or weights to find ฮฃf. Divide ฮฃfx by ฮฃf and check that the result lies within the data range.

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Formula / fact

Given information: Weighted mean โ€” Find the mean of 3, 2, 4, 2. Method choice: Add every value before dividing. Divide by the number of values, not by the largest value. Calculation or reasoning: Add: 3 + 2 + 4 + 2 = 11. Divide by 4: 2.75. Final answer: 2.75. Check: Divide by the number of values, not by the largest value.

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Foundation example

Given information: Weighted mean โ€” Find the mean of 3, 2, 4, 2. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 2 + 4 + 2 = 11. Divide by 4: 2.75. Final answer: 2.75. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: Weighted mean โ€” Find the mean of 3, 14, 14, 10. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 14 + 14 + 10 = 41. Divide by 4: 10.25. Final answer: 10.25. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Weighted mean โ€” Find the mean of 3, 13, 9, 7. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 13 + 9 + 7 = 32. Divide by 4: 8. Final answer: 8. Check: substitute or compare with the original information to confirm the result fits the question.

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Exam-style example

Given information: Weighted mean โ€” Find the mean of 3, 12, 4, 4. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 12 + 4 + 4 = 23. Divide by 4: 5.75. Final answer: 5.75. Check: substitute or compare with the original information to confirm the result fits the question.

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Common mistake

Dividing by the number of categories instead of the total frequency.

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Exam tip

For weighted mean, show the key representation before the final calculation. Use this final check: Multiply the mean by the number of values and confirm that it recreates the original total.

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Real-world use

Average scores, Fair sharing

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Checklist

I can explain weighted mean, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for Weighted mean

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Simple explanation

Weighted mean: The mean is a fair-share total: add every value, then divide by how many values contributed. Multiply the mean by the number of values and confirm that it recreates the original total. Keep the weighted mean representation visible until the final line.

Think of weighted mean as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. Multiply each value by its frequency or stated weight.
  2. Add the products to find ฮฃfx.
  3. Add frequencies or weights to find ฮฃf.
  4. Divide ฮฃfx by ฮฃf and check that the result lies within the data range.

Hint 1

Start by naming the given information and the exact result required for weighted mean.

Hint 2

Add all values accurately.

Full worked solution

Given information: Weighted mean โ€” Find the mean of 3, 2, 4, 2. Method choice: use the weighted mean method and show each step with the stated values. Calculation or reasoning: Add: 3 + 2 + 4 + 2 = 11. Divide by 4: 2.75. Final answer: 2.75. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Multiply each value by its frequency or stated weight. โ†’ Add the products to find ฮฃfx. โ†’ Add frequencies or weights to find ฮฃf. โ†’ Divide ฮฃfx by ฮฃf and check that the result lies within the data range.

Common mistake warning

Dividing by the number of categories instead of the total frequency.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain weighted mean in my own words.
  • I can use these words accurately: mean, total, frequency.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Dividing by the largest value instead of the number of values.
  • I can apply this check: Multiply the mean by the number of values and confirm that it recreates the original total.
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