Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Averages & Spread

In one sentence. Mean, median, mode and range — from lists, tables and grouped data — and the combined-mean and reverse-mean questions that catch everyone once.

Statistics — one of 3 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Florence Nightingale. Learn Averages & Spread, teach it, then defend it against Florence Nightingale.

Where this sits in the specification

Specification 4MA1, 6.2 Statistical measures. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 6.2 — Statistical measures

1 · Understand it

What it is

Mean: total ÷ count (uses every value). Median: the middle of the ORDERED list (ignores extremes). Mode: the most frequent. Range: biggest − smallest (spread, not an average). From a frequency table, the mean is Σ(value × frequency) ÷ Σ(frequency); with GROUPED data, midpoints stand in for the unknown values, making the mean an estimate.

Why it works

The mean balances the data — it's the see-saw's pivot, which is why one billionaire in the room drags it wildly (and why house prices are reported as medians). The median only cares about position, so extremes can't bully it. Grouped means are estimates because grouping threw the exact values away; midpoints are the fairest guess.

When a question is asking for it

Choosing an average: skewed data or outliers → median; categorical data → mode; every value should count → mean. Reverse-mean questions ('the mean of 5 numbers is 8…') hinge on one move: rebuild the TOTAL (5 × 8 = 40), because totals — not means — are what add and subtract.

How to hold it in your head

The mean is a shared-out total: 'if everyone had the same, how much would each have?' That's why the total (mean × count) is the mean's real currency — every combined-mean and changed-mean puzzle is totals arithmetic wearing an averages costume.

2 · The mistakes that cost marks

  • Finding the median without ORDERING first. The middle of an unordered list is meaningless.
  • From a frequency table, dividing by the number of ROWS instead of the total frequency Σf.
  • Calling a grouped mean exact. The raw values are gone; midpoints make it an estimate, and saying so is part of the answer.
  • Averaging two groups' means directly. Combine the TOTALS, then divide by the combined count — unless the groups are equal-sized, the plain average of means is wrong.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

Find the median of 4, 7, 2, 9, 5, 5, 8.

  1. Order: 2, 4, 5, 5, 7, 8, 9.
  2. Seven values → the 4th is the middle.

Answer: 5

Class A (12 students) has mean 60. Class B (18 students) has mean 70. Find the combined mean.

  1. Totals: 12 × 60 = 720 and 18 × 70 = 1260.
  2. Combined: 1980 marks over 30 students.
  3. 1980 ÷ 30 = 66.

Answer: 66

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — Work out the median of 4, 7, 2, 9, 5, 5, 8.
  2. Grade 7 — A frequency table shows scores: value 1 (frequency 5), value 2 (frequency 8), value 3 (frequency 4), value 4 (frequency 3). Work out the mean score.
  3. Grade 8 — Times (t minutes) are grouped: 0 ≤ t < 10 (frequency 4), 10 ≤ t < 20 (frequency 11), 20 ≤ t < 30 (frequency 5). Work out an estimate for the mean time. (answer in min)
  4. Grade 9 — Class A has 12 students with a mean mark of 60. Class B has 18 students with a mean mark of 70. Work out the mean mark of all 30 students.
  5. 9+ — The mean of 5 numbers is 8. When one number is removed, the mean of the remaining 4 is 7.5. Work out the number that was removed.

Ask the person who made it

Inside Incandio you can put a question to Florence Nightingale, who answers knowing exactly which module you have just finished: “You used statistics — averages, comparisons, careful counting — to prove that most soldiers were dying from preventable disease, not wounds. How did a well-chosen average change what generals and politicians believed?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Averages & Spread, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Averages & Spread works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Averages & Spread against Florence Nightingale — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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Carry on through the course