Pearson Edexcel International GCSE Mathematics A 4MA1: Statistics and probability – Revision Notes

These revision notes cover topic 6, Statistics and probability, of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), first assessed in June 2018 with papers in January and June. They cover sections 6.1 (Graphical representation of data), 6.2 (Statistical measures) and 6.3 (Probability). Foundation content is needed by both tiers; items marked Higher tier only appear only on papers 1H and 2H. For full explanations and worked examples, read the study guide first.

Other links: practice questions with worked answers, Edexcel IGCSE Mathematics hub, printable 4MA1 checklist, free 10-minute diagnostics.

A calculator may be used on every 4MA1 paper, and a calculator with Σx and Σfx functions is on the specification’s minimum list. None of the formulas below is on the formulae sheet.

Formulas and rules

Quantity	Formula or rule	Tier
Pie chart angle	(frequency ÷ total) × 360°	Both
Mean of a list	Σx ÷ n	Both
Mean from a frequency table	Σfx ÷ Σf	Both
Estimated mean (grouped)	Σ(f × midpoint) ÷ Σf	Both
Position of the median in a list	(n + 1)/2	Both
Range	largest − smallest	Both
Frequency density	frequency ÷ class width	Higher tier only
Quartile positions in a list	(n + 1)/4 and 3(n + 1)/4	Higher tier only
Reading a cumulative frequency diagram	median at n/2; quartiles at n/4 and 3n/4	Higher tier only
Interquartile range	upper quartile − lower quartile	Higher tier only
Probability, equally likely outcomes	favourable outcomes ÷ total outcomes	Both
Complement	P(A′) = 1 − P(A)	Both
Mutually exclusive events	P(A or B) = P(A) + P(B)	Both
Relative frequency	times event occurs ÷ number of trials	Both
Expected frequency	probability × number of trials	Both
Independent events	P(A and B) = P(A) × P(B)	Higher tier only

6.1 Graphical representation of data

Key facts

• Pictograms need a key. Bar charts for categories have gaps between bars and equal widths.
• Pie chart sectors: first find the angle per item (360° ÷ total). To go backwards, number = (angle ÷ 360°) × total.
• Two-way tables: every row and column must add to its total. Fill the cell that is missing only one value first.
• Histograms (Higher tier only): no gaps, a continuous scale on the horizontal axis, and frequency density on the vertical axis. Area represents frequency.
• Cumulative frequency (Higher tier only): plot running totals at upper class boundaries; the graph never goes down.

Method: draw a histogram (Higher tier only)

1. Find each class width from the boundaries (for 15 < t ≤ 20 the width is 5).
2. Frequency density = frequency ÷ class width.
3. Label the vertical axis “frequency density” and choose a scale that fits the largest value.
4. Draw each bar between its class boundaries at its frequency density.

To read a frequency back from a histogram, multiply bar height by bar width. For part of a bar, use only the part of the width you need, and assume values are spread evenly across the class.

6.2 Statistical measures

Choosing an average

• Mean: uses every value; affected by extreme values.
• Median: the middle value; not affected by extreme values.
• Mode: the only average for non-numerical data; there can be more than one.

Method: averages from a frequency table

1. Add a column for f × x (or f × midpoint for grouped data).
2. Mean = Σfx ÷ Σf. Check it lies within the range of the data.
3. Median: find the position (Σf + 1)/2, then use running totals to find which value holds it.
4. Mode (or modal class): the value (or class) with the highest frequency.

Small worked reminder: 4, 7, 7, 9, 13 has mean 40 ÷ 5 = 8, median 7, mode 7 and range 9.

Method: quartiles from a cumulative frequency diagram (Higher tier only)

1. Find n (the final cumulative frequency).
2. Read across from n/4, n/2 and 3n/4 to the curve, then down.
3. IQR = upper quartile − lower quartile.
4. “More than x”: read the cumulative frequency at x and subtract it from n.

Must-know distinctions

• Mode vs modal class: a single value for discrete data; a class for grouped data.
• Mean vs estimated mean: grouped data give only an estimate, because midpoints replace the real values.
• Range vs IQR (Higher tier only): the range uses the two extreme values; the IQR measures the spread of the middle half and ignores extremes.
• Frequency vs frequency density (Higher tier only): with unequal widths, the tallest bar is not always the class with the highest frequency.

6.3 Probability

Key terms

• Outcome: one possible result. Event: one or more outcomes. Sample space: the list of all outcomes.
• Random / equally likely: each outcome has the same chance.
• Probability scale: 0 (impossible) to 1 (certain).
• Theoretical probability comes from a model, such as a fair dice. Relative frequency estimates probability from data.
• Mutually exclusive: cannot happen together. Independent (Higher tier only): one does not affect the other.

Method: Venn diagram questions

1. Fill the overlap (both) first.
2. Subtract it from each set total to get the “only” regions.
3. Subtract everything from the grand total for the region outside both circles.
4. Read probabilities as region total ÷ grand total. If a question restricts the group (for example “given that the person is in A”), divide by that group’s total instead.

Method: tree diagrams (Higher tier only)

1. Draw one set of branches per stage and write a probability on every branch.
2. Branches from one point add to 1.
3. Without replacement, the second-stage fractions have denominators one less, and the chosen colour’s numerator goes down by one.
4. Multiply along each route; add the routes you need.
5. For “at least one”, use 1 − P(none).

Small worked reminder: if P(A) = 0.4 and P(B) = 0.25, then P(A or B) = 0.65 if A and B are mutually exclusive, and P(A and B) = 0.1 if they are independent.

Must-know distinctions

• Mutually exclusive vs independent: add for “or” with mutually exclusive events; multiply for “and” with independent events.
• With vs without replacement: with replacement the probabilities stay the same at every stage; without replacement they change.
• Relative frequency vs probability: relative frequency is an estimate that improves with more trials.
• Expected frequency vs actual result: expected frequency is a long-run average; the actual count will usually differ a little.

Quick self-test

1. Find the mean, median, mode and range of 4, 7, 7, 9, 13.
2. The mean of five numbers is 12. Four of them are 10, 14, 9 and 15. Find the fifth.
3. In a pie chart for 45 people, 12 chose “walk”. Find the sector angle.
4. Estimate the mean: 0 < x ≤ 4 (frequency 3), 4 < x ≤ 8 (frequency 5), 8 < x ≤ 12 (frequency 2).
5. (Higher tier only) The class 20 < x ≤ 35 has frequency 24. Find its frequency density.
6. (Higher tier only) A histogram bar has frequency density 2.5 and class width 8. Find the frequency.
7. (Higher tier only) Find the interquartile range of 2, 4, 5, 7, 8, 10, 13.
8. P(A) = 0.27. Find P(A′).
9. P(win) = 0.15. How many wins would you expect in 240 games?
10. Two fair coins are thrown. Find P(at least one head).
11. (Higher tier only) A and B are independent with P(A) = 0.6 and P(B) = 0.3. Find P(A and B).
12. (Higher tier only) A bag has 4 green and 6 yellow sweets. Two are taken without replacement. Find P(both green).

Answers

1. Mean 8, median 7, mode 7, range 9
2. Total = 5 × 12 = 60; 60 − 48 = 12
3. 12/45 × 360° = 96°
4. (3 × 2 + 5 × 6 + 2 × 10) ÷ 10 = 56 ÷ 10 = 5.6
5. 24 ÷ 15 = 1.6
6. 2.5 × 8 = 20
7. n = 7: lower quartile = 2nd value = 4; upper quartile = 6th value = 10; IQR = 6
8. 1 − 0.27 = 0.73
9. 0.15 × 240 = 36
10. Outcomes HH, HT, TH, TT; three contain a head: 3/4
11. 0.6 × 0.3 = 0.18
12. 4/10 × 3/9 = 12/90 = 2/15

Where marks are usually lost

• Using class width 10 for a class such as 15 < t ≤ 20; read widths from the boundaries every time.
• Labelling a histogram’s vertical axis “frequency” when the heights are frequency densities.
• Plotting cumulative frequencies at midpoints, or not starting the curve at the lower boundary with zero.
• Reading quartiles at the wrong height, for example at 25 and 75 when the total is 80.
• Dividing Σfx by the number of classes, or by Σx, instead of by Σf.
• Giving the modal class’s frequency instead of the class itself.
• Writing a probability as a ratio (3 : 4) or as “3 out of 4” instead of 3/4.
• On a tree diagram without replacement, keeping the same denominator for the second pick.
• Adding the probabilities along a branch instead of multiplying them.
• In a Venn diagram, putting the full set total inside the “only” region and counting the overlap twice.

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification Issue 2, November 2017, Pearson Education Limited. Topic 6, Statistics and probability: 6.1 Graphical representation of data, 6.2 Statistical measures and 6.3 Probability, with Foundation and Higher tier content.