Normal distribution appears every S1 — typically 12-15 marks. Most-tested: standardisation (5-7 marks), find μ/σ (7 marks), normal approximation (7 marks).
Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.
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Step-by-step worked examples — The Normal distribution
Step-by-step solutions to past-paper-style questions on the normal distribution , written exactly the way a tutor would explain them at the board.
1Standardise and use table (5 marks)
Extended• Adapted from 9709/52 May/Jun 2024• standardisation
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Question
X∼N(50,16). Find P(X<55). (5 marks)
Step-by-step solution
Step 1
σ=16=4.
Step 2
Standardise.
Z=σX−μ=455−50=1.25
Step 3
Look up Φ(1.25) in table.Φ(1.25)≈0.8944.
Answer
P(X<55)≈0.8944.
2Find μ from probability (7 marks)
Extended• normal
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Question
X∼N(μ,25). Given P(X>60)=0.1. Find μ. (7 marks)
Step-by-step solution
Step 1
P(X>60)=0.1⇒P(X<60)=0.9.
Step 2
From inverse normal:Φ−1(0.9)≈1.282.
Step 3
Standardise.
560−μ=1.282
Step 4
Solve for μ.
60−μ=6.41⇒μ≈53.59
Answer
μ≈53.59.
3Normal approximation to binomial (7 marks)
Extended• approximation
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Question
X∼B(100,0.4). Use normal approximation to find P(X>50). (7 marks)
Continuity correction.P(X>50)≈P(X>50.5) in normal.
Step 4
Standardise.Z=4.89950.5−40≈2.143.
Step 5
Find prob.P(Z>2.143)=1−Φ(2.143)≈1−0.9839=0.0161.
Answer
P(X>50)≈0.0161.
Key Formulae — The Normal distribution
The formulae you need to memorise for the normal distribution on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.
Standardisation
Z=σX−μ
μ
mean
σ
standard deviation
When to use
Convert X∼N(μ,σ2) to Z∼N(0,1) for table lookup.
Normal approximation to binomial
If $X \sim B(n, p)$ with $np \geq 5$ AND $n(1-p) \geq 5$, then $X \approx N(np, np(1-p))$. Use continuity correction $\pm 0.5$.
When to use
Large n. Faster than direct binomial calculation.
Key Definitions and Keywords — The Normal distribution
Definitions to memorise and the exact keywords mark schemes credit for the normal distribution answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.
Normal distribution
Examiner keyword
Continuous symmetric bell curve. X∼N(μ,σ2).
Standard normal
Examiner keyword
Z∼N(0,1). Used with tables. Φ(z)=P(Z<z).
Standardisation
Examiner keyword
Z=(X−μ)/σ. Converts any normal to standard normal.
Continuity correction
Examiner keyword
Adjustment of ±0.5 when approximating discrete (e.g. binomial) with continuous (normal).
Common Mistakes and Misconceptions — The Normal distribution
The traps other students keep falling into on the normal distribution questions — taken from recent Cambridge International A Level 9709 examiner reports and mark schemes — and how to avoid them.
✕Using variance instead of SD in standardisation
9709 Examiner Reports 2022-2024
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Why it happens
Notation N(μ,σ2) — second argument is variance.
How to avoid it
Standardisation uses σ (SD), not σ2. Take SQRT first.
✕Wrong direction of continuity correction
9709 Examiner Reports 2022-2024
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Why it happens
Confusion about inclusion.
How to avoid it
P(X≥50) in binomial → P(X≥49.5) in normal (so include 50). P(X>50) → P(X>50.5) (exclude 50).
Practice questions
Exam-style questions with step-by-step worked solutions. Try one before checking the method.
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The Normal distribution — frequently asked questions
The things students keep getting wrong in this sub-topic, answered.