📚 Normal Distribution in A-Level Edexcel Maths: Key Exam Topics | A-Level Edexcel 数学：正态分布 考点精讲
The normal distribution is one of the most heavily examined topics across the Edexcel A-Level Mathematics specification, appearing both in pure statistical methods and as a foundation for hypothesis testing. This article draws together every essential skill you need: from standardising and reading tables to approximating the binomial and working with sample means. Each section is written to mirror the style of real exam questions while clarifying common mistakes.
正态分布是 Edexcel A-Level 数学考试中出现频率最高的专题之一，贯穿统计方法以及假设检验的基础。本文汇总了所有必会技能：从标准化和查表，到二项分布的正态近似和样本均值分布。每一节都贴近真实考题风格，同时澄清常见错误。
1. The Normal Probability Density Function | 正态概率密度函数
ous random variable X follows a normal distribution with mean μ and variance σ², written X ~ N(μ, σ²). Its probability density function is f(x) = 1/(σ√2π) · e^(–½((x–μ)/σ)²). The total area under the curve equals 1, and the curve is perfectly bell‑shaped and symmetric about μ. The points of inflection occur at μ ± σ.
连续型随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。其概率密度函数为 f(x) = 1/(σ√2π) · e^(–½((x–μ)/σ)²)。曲线下总面积等于 1，图像呈完美的钟形且关于 μ 对称；拐点位于 μ ± σ 处。
dardised scores and distribution tables. Remember that P(X = a) = 0 for any single value a — only intervals have non‑zero probability.
考试中绝不会要求对该函数积分，而是借助标准化值和分布表。务必记住，对任何一个点 a，P(X = a) = 0 ——只有区间上的概率才非零。
Key proportions: P(μ – σ < X < μ + σ) ≈ 0.683, P(μ – 2σ < X < μ + 2σ) ≈ 0.954, P(μ – 3σ < X < μ + 3σ) ≈ 0.997.
关键比例：P(μ – σ < X < μ + σ) ≈ 0.683，P(μ – 2σ < X < μ + 2σ) ≈ 0.954，P(μ – 3σ < X < μ + 3σ) ≈ 0.997。
2. The Standard Normal Distribution and Z‑Scores | 标准正态分布与 Z 值
The standard normal distribution has mean 0 and variance 1, written Z ~ N(0, 1). Any normal variable X ~ N(μ, σ²) can be converted to a standard score using z = (x – s, called standardisation, shifts the mean to zero and scales the spread to unit variance.
标准正态分布的均值为 0、方差为 1，记作 Z ~ N(0, 1)。任何正态变量 X ~ N(μ, σ²) 都可以通过 z = (x – μ) / σ 转化为标准值。这个过程称为标准化，将均值平移到零，并把离散度缩放至单位方差。
. You must be able to read these tables accurately and handle both positive and negative z‑values using symmetry.
Edexcel 公式手册给出了正 z 值的累积分布函数 Φ(z) = P(Z < z)。你必须能准确读表，并利用对称性处理正负 z 值。
For example, if X ~ N(100, 15²) and you need P(X < 115), compute z = (115 – 100)/15 = 1.00, then look up Φ(1.00) = 0.8413.
例如，若 X ~ N(100, 15²)，求 P(X < 115)，则计算 z = (115 – 100)/15 = 1.00，查表得 Φ(1.00) = 0.8413。
3. Using the Standard Normal Table to Find Probabilities | 用标准正态表求概率
e standard normal table gives Φ(z) = P(Z < z) for 0 ≤ z ≤ 3.99. When z is negative, use Φ(–z) = 1 – Φ(z). To find the probability between two values, compute P(a < X < b) = Φ((b–μ)/σ) – Φ((a–μ)/σ). For right‑tail probabilities, P(Z > z) = 1 – Φ(z).
标准正态表给出了 0 ≤ z ≤ 3.99 范围内的 Φ(z) = P(Z < z)。当 z 为负时，利用 Φ(–z) = 1 – Φ(z)。求两个值之间的概率时，使用 P(a < X < b) = Φ((b–μ)/σ) – Φ((a–μ)/σ)。右尾概率 P(Z > z) = 1 – Φ(z)。
asked for P(X > k) where z turns out negative, for instance, remember to convert carefully — sketch will show you that P(Z > –k) = 1 – Φ(–k) = Φ(k).
永远先画一张简图，标出对应区域。这能避免减错尾部的常见错误。若题目求 P(X > k) 而算出的 z 为负值，记住小心转换——示意图会告诉你 P(Z > –k) = 1 – Φ(–k) = Φ(k)。
| Query | Standardised form | Example result |
| P(X < 85) for X ~ N(100,12²) | z = (85–100)/12 = –1.25 | 1 – Φ(1.25) = 0.1056 |
| P(95 < X < 110) for same X | z₁ = –0.4167, z₂ = 0.8333 | Φ(0.8333) – (1–Φ(0.4167)) |
4. Symmetry and Tail Probabilities | 对称性与尾部概率
Symmetry is the key to unlocking many exam problems. For Z ~ N(0,1), P(Z < –a) = P(Z > a), and P(–a < Z < a) = 2Φ(a) – 1. Questions often give a symmetric interval probability and ask you to find the boundary.
对称性是解决许多考题的关键。对于 Z ~ N(0,1)，P(Z < –a) = P(Z > a)，而 P(–a < Z < a) = 2Φ(a) – 1。题目常给出对称区间的概率，要求你求出边界值。
If a problem states that the middle 95% of a normal population lies between two values, you know that the tail on each side is 2.5%. Hence the boundary corresponds to the z‑value such that Φ(z) = 0.975, giving z ≈ 1.96. This z‑value is then un‑standardised to the original scale.
若题目说明正态总体的中间 95% 落在两个值之间，你就知道每侧尾部面积为 2.5%。因此边界对应满足 Φ(z) = 0.975 的 z 值，即 z ≈ 1.96。然后将该 z 值转换回原始量纲。
Critical z‑values for common confidence levels: 90% → 1.645, 95% → 1.960, 99% → 2.576.
常用置信水平的临界 z 值：90% → 1.645，95% → 1.960，99% → 2.576。
5. Finding Unknown Mean or Standard Deviation | 求未知均值或标准差
eous equations using the standardisation formula and the table values. For example, P(X < 30) = 0.15 and P(X > 70) = 0.10.
Edexcel 经常给出 μ 或 σ 未知的题目，并提供两条概率信息。利用标准化公式和表格值建立两个联立方程。例如，已知 P(X < 30) = 0.15 和 P(X > 70) = 0.10。
Convert each probability to a z‑value: from the table, Φ(z₁) = 0.15 ⇒ z₁ ≈ –1.0364, and 1 – Φ(z₂) = 0.10 ⇒ Φ(z₂) = 0.90 ⇒ z₂ ≈ 1.2816. Then write –1.0364 = (30 – μ)/σ and 1.2816 = (70 – μ)/σ. Solve simultaneously for μ and σ.
将每个概率转换为 z 值：查表得 Φ(z₁) = 0.15 ⇒ z₁ ≈ –1.0364，而 1 – Φ(z₂) = 0.10 ⇒ Φ(z₂) = 0.90 ⇒ z₂ ≈ 1.2816。然后写出 –1.0364 = (30 – μ)/σ 和 1.2816 = (70 – μ)/σ，联立解出 μ 和 σ。
Always check that your solved parameters are sensible — variances must be positive, and the mean should lie between the given x‑values in most symmetric setups.
务必检查解出的参数是否合理——方差必须为正数，且在大多数对称情形下，均值应介于给定的 x 值之间。
6. Inverse Normal: From Probability to x‑Value | 逆正态：由概率求 x 值
The inverse normal problem asks: given P(X < x₀) = p (or a right‑tail probability), find x₀. First find the z‑score that corresponds to cumulative probability p by reading the table backwards, then un‑standardise: x₀ = μ + zσ.
逆正态问题要求：已知 P(X < x₀) = p（或右尾概率），求出 x₀。首先通过反向查表找出对应于累积概率 p 的 z 值，然后反标准化：x₀ = μ + zσ。
.6449σ. When the given probability is less than 0.5, the corresponding z will be negative.
如果概率涉及上尾，例如 P(X > x₀) = 0.05，注意 P(X < x₀) = 0.95，故找出使得 Φ(z) = 0.95 的 z（z ≈ 1.6449），于是 x₀ = μ + 1.6449σ。当给定概率小于 0.5 时，对应的 z 将为负值。
Exam reports consistently show that students forget to convert a “greater than” probability to “less than” before reading the table. Write down all intermediate steps clearly to avoid arithmetic errors.
考试报告一再显示，学生常忘记在查表前把“大于”概率转换为“小于”。务必清晰写出所有中间步骤，以避免算术错误。
7. Distribution of the Sample Mean | 样本均值的分布
When a random sample of size n is taken from a normal population X ~ N(μ, σ²), the sample mean X̄ also follows a normal distribution: X̄ ~ N(μ, σ²/n). The standard error of the mean is σ/√n, which decreases as the sample size grows.
当从正态总体 X ~ N(μ, σ²) 中抽取一个容量为 n 的随机样本时，样本均值 X̄ 也服从正态分布：X̄ ~ N(μ, σ²/n)。均值的标准误差为 σ/√n，它会随着样本量的增大而减小。
the original population is not normally distributed, the Central Limit Theorem says that X̄ ely normal provided n is large enough (usually n ≥ 30). The standardisation becomes z = (x̄ – μ) / (σ/√n).
即便原始总体并非正态分布，中心极限定理指出，只要 n 足够大（通常 n ≥ 30），X̄ 就近似服从正态分布。此时标准化公式变为 z = (x̄ – μ) / (σ/√n)。
Questions may ask for the probability that the mean of a sample falls within a certain interval. Always identify that you are dealing with X̄, not X, and therefore divide the variance by n. Using the wrong standard deviation is the single most common mistake in this topic.
考题可能要求计算样本均值落在某个区间内的概率。务必识别出你处理的是 X̄ 而不是 X，因此要将方差除以 n。使用错误的标准差是该专题中最常见的单一错误。
8. Normal Approximation to the Binomial | 二项分布的正态近似
If X ~ B(n, p) and both np > 5 and n(1 – hen X can be approximated by a normal distribution with mean μ = np and variance σ² = np(1 – ). The conditions ensure the binomial distribution is sufficiently symmetric and unimodal.
如果 X ~ B(n, p) 且满足 np > 5 与 n(1 – p) > 5，则 X 可以用均值为 μ = np、方差为 σ² = np(1 – p) 的正态分布来近似。这两个条件确保了二项分布足够对称且单峰。
Write X ≈ N(np, np(1 – Without the correction, the approximation is less accurate and may lose marks in the exam.
记作 X ≈ N(np, np(1 – p))。你必须进行连续性校正，因为你是在用一个连续分布去近似离散分布。若不校正，近似精度会降低，并可能在考试中失分。
Common approximations: for X ~ B(100, 0.4), np = 40, n(1–p) = 60, so X ≈ N(40, 24). Then P(X < 45) is approximated by P(Y < 44.5) where Y ~ N(40, 24).
常见近似：对于 X ~ B(100, 0.4)，np = 40，n(1–p) = 60，故 X ≈ N(40, 24)。那么 P(X < 45) 被近似为 P(Y < 44.5)，其中 Y ~ N(40, 24)。
9. Continuity Correction | 连续性校正
The continuity correction adjusts the discrete boundaries by 0.5 to account for the width of the bars in the binomial distribution. The corrected inequalities are:
连续性校正将离散边界调整 ±0.5，以补偿二项分布中柱条的宽度。校正后的不等式如下：
- P(X ≤ a) ≈ P(Y < a + 0.5)
P(X ≤ a) ≈ P(Y < a + 0.5) - P(X < a) ≈ P(Y < a – 0.5)
P(X < a) ≈ P(Y < a – 0.5) - P(X ≥ a) ≈ P(Y > a – 0.5)
P(X ≥ a) ≈ P(Y > a – 0.5) - P(X > a) ≈ P(Y > a + 0.5)
P(X > a) ≈ P(Y > a + 0.5) - P(a ≤ X ≤ b) ≈ P(a – 0.5 < Y < b + 0.5)
P(a ≤ X ≤ b) ≈ P(a – 0.5 < Y < b + 0.5)
Note that whether the inequality in the normal probability is “<” or “≤” hange the probability for a continuous distribution, so both can be used after correction.
注意，对于连续分布，使用“<”还是“≤”并不影响概率，因此校正后两者可以互换。
A typical exam question gives a binomial probability and asks for the approximate normal probability. The mark scheme expects you to state the continuity correction explicitly. Missing it results in the loss of accuracy marks even if the subsequent arithmetic is correct.
典型考题会给出一个二项概率并求其正态近似。阅卷标准期望你明确写出连续性校正。忽略这一步即使后续计算正确也会丢失精度分。
10. Common Exam Pitfalls and Tips | 常见考试陷阱与技巧
The most frequent errors in Edexcel normal distribution questions include confusing variance (σ²) with standard deviation (σ), forgetting to square or square‑root when substituting, using the wrong tail probability, misapplying the continuity correction, and mixing up X̄ and X standard deviations.
Edexcel 正态分布考题中最常见的错误包括：混淆方差（σ²）与标准差（σ），代入时忘记平方或开方，使用错误的尾部概率，错误地应用连续性校正，以及混淆 X̄ 与 X 的标准差。
Always write down the distribution you are using: “X ~ N(…)” or “X̄ ~ N(…)” ps you choose the correct variance. When dealing with the sample total T = ΣX, note that T ~ N(nμ, nσ²).
任何计算前务必清晰写出你使用的分布：“X ~ N(…)” 或 “X̄ ~ N(…)”，这有助于你选择正确的方差。当处理样本总和 T = ΣX 时，注意 T ~ N(nμ, nσ²)。
Double‑check the “greater than/less than” direction. If the table gives Φ(z) for P(Z < z), but the question asks for P(Z > z), you must do 1 – Φ(z). A quick sketch alongside your working will save many lost marks.
反复检查“大于/小于”的方向。如果表格给出的是 P(Z < z) 的值，而题目要求 P(Z > z)，你必须做 1 – Φ(z)。在解题时随手画出简图可挽回大量失分。
Finally, practise reading the normal table accurately. Interpolate linearly only when instructed; most Edexcel markschemes accept the nearest tabulated value or require reading between rows only in specific contexts. Know when to use the percentage points table for inverse normal problems with standard tail probabilities.
最后，练习准确读取正态分布表。除非明确要求，否则不进行线性插值；大多数 Edexcel 评分标准接受最接近的表格值，或仅在特定语境下要求行间读取。懂得何时使用百分比点表处理标准尾部概率的逆正态问题。
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