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Mathematics · 04 / 23

Statistics: centre and spread

Middle school · Worked example and 4 exercises with solutions, at your pace.

UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

The mean balances the sum of values; the median divides an ordered series in half. They answer different questions. Range compares extremes, while standard deviation measures spread around the mean. A description does not establish causation or representativeness.

m=(8+10+10+12+15)/5
THE PICTURE TO REMEMBERStatistics: centre and spread
810101215

Middle value: 10. Sum ÷ count: 11.

  1. 01Sort
  2. 02Centre
  3. 03Spread
  4. 04Interpret context

WORKED EXAMPLE

From principle to reasoning

For 8, 10, 10, 12, 15: sum 55, mean 11, median 10, range 7. A very large value moves the mean more than the median.

Solve a problem in this subject

  1. Define the objects

    State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.

  2. Choose a transformation

    Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.

  3. Show the reasoning

    Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.

  4. Test and conclude

    Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.

YOUR TURN

01 · Try before checking

Find the mean and median of 2, 3, 4, 100.

One hint

Sort and average the two middle values.

Reveal and understand the solution

Mean = 109/4 = 27.25; median = (3 + 4)/2 = 3.5. The value 100 strongly affects the mean.

Compare method, conditions and answer; identify the exact part to revisit.

IMPROVE THROUGH PRACTICE

Three additional challenges with solutions

Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.

Reflect after practising
02 · Recognise and understandLink to this exercise

Find the mean and median of 2, 4, 4, 8.

A hint to get started

The median splits sorted data; the mean uses their sum.

Step-by-step solution
  1. Sum 18, count 4: mean 4.5.
  2. The two central values are both 4: median 4.
  3. These measures answer different questions.

Which step did you miss? Retry without help, then explain why the method works.

03 · Apply the methodLink to this exercise

A score of 10 has weight 1 and 16 has weight 2. Find the mean.

A hint to get started

Count the second score twice.

Step-by-step solution
  1. Weighted sum: 10 + 2 × 16 = 42.
  2. Total weight: 3.
  3. Mean: 42/3 = 14, not 13.

Which step did you miss? Retry without help, then explain why the method works.

04 · Go furtherLink to this exercise

Replace 8 with 80 in 2, 4, 4, 8. Which measure resists the outlier?

A hint to get started

Compare the middle positions and the sum.

Step-by-step solution
  1. The median remains 4.
  2. The mean becomes 90/4 = 22.5.
  3. The median resists this outlier, but spread still needs describing.

Which step did you miss? Retry without help, then explain why the method works.

The trap to avoid

Sort before finding the median; a mean does not describe every individual.

Without notes, can you…

  • Explain the principle in your own words?
  • Rebuild the diagram and its connections?
  • Solve the exercise and justify the method?

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