Normal distribution — Edexcel A-Level Statistics
Test yourself on Normal distribution with PEARSON EDEXCEL A-Level practice questions.
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Normal distribution explained
The normal distribution is a continuous, symmetric, bell-shaped model defined by its mean μ and standard deviation σ, written N(μ, σ²).
Read the full explanation
Its curve peaks at x = μ, is symmetric about the vertical line x = μ, and approaches the horizontal axis asymptotically at both tails. The total area under the curve is 1, and probabilities correspond to areas, so P(X < μ) = P(X > μ) = 0.5. The mean, median and mode coincide at μ. Roughly 68% of values lie within μ ± σ, about 95% within μ ± 2σ and about 99.7% within μ ± 3σ. Crucially, a real sample drawn from a normal population only approximates these properties: sample means, spreads and shapes vary from sample to sample, especially when n is small. For example, samples of size 5 from N(20, 3²) may show means of 18.4 or 21.7, while samples of size 200 cluster tightly near 20.
Your focus
- State the defining properties of the normal distribution, including symmetry, peak at μ and total area 1.
- Use the notation N(μ, σ²) correctly, identifying the mean and variance.
- Explain why sample data from a normal population only approximate the distribution's properties, with variation between samples.
Normal distribution exam tips
Marking Points
- States that the normal distribution is continuous, symmetric and bell-shaped, with mean, median and mode all equal to μ.
- Identifies the parameters μ (mean) and σ (standard deviation), with variance σ², and notation N(μ, σ²).
- Explains that the curve is symmetric about x = μ, peaks at x = μ and tends asymptotically to the horizontal axis.
- States that total area under the curve is 1 and that P(X < μ) = P(X > μ) = 0.5.
- Recognises that sample data from a normal population only approximate these properties, with variation between samples, particularly for small n.
Examiner Tips
- 💡Learn the notation N(μ, σ²) and be ready to read off μ and σ from a given distribution, converting variance to standard deviation when needed.
- 💡Sketch the curve with the line x = μ marked, and shade the required region before choosing an option in a multiple-choice question.
- 💡When a question mentions samples, check whether it is asking about the population properties or about sample-to-sample variation, as these are different ideas.
Common Mistakes
- Writing the normal distribution as N(μ, σ) instead of N(μ, σ²); the second parameter is the variance, so N(20, 9) has σ = 3.
- Claiming the curve touches the horizontal axis at the tails; it is asymptotic, approaching but never reaching the axis.
- Assuming every sample from a normal population will look perfectly symmetric and bell-shaped; small samples can look skewed or irregular even when the population is normal.
- Thinking the mean, median and mode can differ in a normal distribution; all three equal μ.