Good Learning Co.
SavannahYear 8
← Maths
MathsShape, probability and data30 min★★★ difficulty

Comparing distributions: averages and spread

An average alone can hide almost everything interesting in a data set. Comparing two groups properly means describing both a typical value and how spread out the values are.

Part of your national curriculum
  • Statistics: Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread

Lesson overview

What you'll learn in this lesson

Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread

Key learning points

  • • Choosing an average
  • • Measuring spread
  • • Mean from grouped data
  • • Writing a comparison

This lesson at a glance

  • 30 minutes
  • 17 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

interpretobserveddistributionssinglevariable

Scroll down — the lesson carries on below

Part 1 of 17 · Discover6%
1

Watch & discover

Part 1 of 17

Watch first

Comparing distributions: averages and spread

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread

2

Learning cycle

Part 2 of 17

Learning cycle 1 of 2

Part 1 · Choosing an average

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 17

Learn

Choosing an average

The mean uses every value but is dragged by extremes. The median is the middle value once ordered and resists outliers, which is why house prices and incomes are reported as medians. The mode is the most common value and is the only average available for categorical data such as favourite subject.

4

Explore the idea

Part 4 of 17

Learn

Measuring spread

The range is the largest value minus the smallest, and is easy but sensitive to a single outlier. The interquartile range, the upper quartile minus the lower quartile, describes the middle half of the data and is far more stable when a data set has an unusual value.

5

Quick check

Part 5 of 17

Quick check

Which average resists outliers best?

6

Quick check

Part 6 of 17

Quick check

The interquartile range describes

7

Learning cycle

Part 7 of 17

Learning cycle 2 of 2

Part 2 · Mean from grouped data

A short piece of teaching, then a check to make sure it has landed.

8

Reset break

Part 8 of 17

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
9

Explore the idea

Part 9 of 17

Learn

Mean from grouped data

With grouped data the exact values are unknown, so use midpoints: for 10 values in 0 < t ≤ 10, 30 in 10 < t ≤ 20 and 10 in 20 < t ≤ 30, the estimated mean is (10×5 + 30×15 + 10×25) ÷ 50 = 15. The answer must be described as an estimate, because the midpoint assumption may not hold.

10

Explore the idea

Part 10 of 17

Learn

Writing a comparison

A comparison needs both measures and a context: 'Class A has a higher median score (32 against 27), so typically performed better, but a larger interquartile range (14 against 6), so its results were less consistent.' Quoting numbers without interpretation, or interpretation without numbers, is a half answer.

11

Quick check

Part 11 of 17

Quick check

Midpoints 5, 15, 25 with frequencies 10, 30, 10 give an estimated mean of

12

Quick check

Part 12 of 17

Quick check

A strong comparison of two data sets includes

13

Explore the idea

Part 13 of 17

Worked example

Worked answer: compare two classes, A (median 32, IQR 14) and B (median 27, IQR 6) (4 marks)

Class A has the higher median, 32 against 27 (1), so a typical student in A scored about five marks more (1). Class A also has the larger interquartile range, 14 against 6 (1), so its middle half of results is far more spread out, meaning B was more consistent even though it was typically lower-scoring (1).

14

Challenge round

Part 14 of 17

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

15

Challenge round

Part 15 of 17

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

The range is the largest value minus the smallest, and is easy but sensitive to a ____ outlier.

16

Mastery quiz

Part 16 of 17

Marked quiz

End of lesson quiz: Comparing distributions: averages and spread

4 questions, marked with the reasoning shown. No timer.

  1. 1. Which average resists outliers best?

  2. 2. The interquartile range describes

  3. 3. Midpoints 5, 15, 25 with frequencies 10, 30, 10 give an estimated mean of

  4. 4. A strong comparison of two data sets includes

17

Lesson round-up

Part 17 of 17

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 17 · Watch & discover

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    Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread

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