A-Level Maths: Normal Distribution Key Exam Points | A-Level 数学：正态分布考点精讲

📚 A-Level Maths: Normal Distribution Key Exam Points | A-Level 数学：正态分布考点精讲

The normal distribution is one of the most important continuous probability distributions in A-Level Mathematics. It underpins a vast range of statistical methods and appears in almost every examination session. Understanding its shape, parameters, and how to calculate probabilities using standardisation is essential for achieving top marks. This revision guide covers every key point you need, from the bell curve to normal approximations, with clear bilingual explanations.

正态分布是 A-Level 数学中最重要的连续型概率分布之一。它支撑着大量统计方法，几乎在每次考试中都会出现。理解其形状、参数以及如何通过标准化计算概率，是获取高分的关键。这篇考点精讲从钟形曲线到正态近似，用清晰的中英双语解释为你覆盖每一个重要知识点。

1. The Bell-Shaped Curve | 钟形曲线

The normal distribution is often called the bell curve because its probability density function is symmetric and bell‑shaped. It is defined for all real numbers, and the total area under the curve equals 1. The curve is centred at the mean μ, and its spread is governed by the standard deviation σ. The maximum value occurs at x = μ, and the curve approaches, but never touches, the horizontal axis as x → ±∞.

正态分布常被称为钟形曲线，因为其概率密度函数是对称的钟形。它定义在所有实数上，曲线下的总面积为 1。曲线以均值 μ 为中心，其分散程度由标准差 σ 控制。最大值出现在 x = μ 处，并且当 x → ±∞ 时曲线无限趋近于横轴但永不触及。

Two key properties you must remember for the exam are symmetry about the mean, and the fact that the distribution is completely determined by μ and σ² (or σ). Changing μ shifts the curve left or right; increasing σ flattens and widens the curve.

考试中必须牢记的两个关键性质是关于均值的对称性，以及分布完全由 μ 和 σ²（或 σ）决定。改变 μ 会使曲线左右平移；增大 σ 则使曲线变得扁平和宽胖。

2. Probability Density Function (PDF) | 概率密度函数

If a continuous random variable X follows a normal distribution, we write X ~ N(μ, σ²). Its probability density function is given by

f(x) = 1/(σ√(2π)) · e^{–(x–μ)²/(2σ²)}, for –∞ < x < ∞.

如果连续型随机变量 X 服从正态分布，记为 X ~ N(μ, σ²)。其概率密度函数为

f(x) = 1/(σ√(2π)) · e^{–(x–μ)²/(2σ²)}，定义域为全体实数。

While you will not be asked to integrate this function in the exam (integration is done using numerical tables or calculators), you should recognise its components: the factor 1/(σ√(2π)) ensures the total area is 1, and the exponent contains the squared distance from the mean, scaled by 2σ². The function is never negative.

虽然考试中不会要求你对这个函数进行积分（积分通过数表或计算器完成），但你应当认识其组成部分：因子 1/(σ√(2π)) 确保总面积为 1，指数部分包含到均值的平方距离除以 2σ²。该函数始终非负。

3. Parameters: Mean and Variance | 参数：均值与方差

For X ~ N(μ, σ²), the mean is E(X) = μ and the variance is Var(X) = σ². The standard deviation is σ. These parameters have a direct geometric meaning: μ is the line of symmetry, and σ controls the points of inflection, which occur at μ ± σ. At these points the curve changes curvature.

对于 X ~ N(μ, σ²)，均值为 E(X) = μ，方差为 Var(X) = σ²。标准差是 σ。这些参数具有直接的几何意义：μ 是对称轴，σ 控制拐点位置，拐点出现在 μ ± σ 处。在拐点处曲线的弯曲方向发生改变。

Standard deviation σ: roughly 68% of data lies within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ (the empirical rule).
标准差 σ：大约 68% 的数据落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内（经验规则）。
Variance σ²: always non‑negative; a larger σ² indicates more spread.
方差 σ²：始终非负；σ² 越大表示数据越分散。

4. Standard Normal Distribution Z ~ N(0, 1) | 标准正态分布 Z ~ N(0, 1)

The standard normal distribution is the special case where μ = 0 and σ² = 1. We use Z to denote a standard normal random variable. Its PDF simplifies to

φ(z) = 1/√(2π) e^–½z².

标准正态分布是 μ = 0 且 σ² = 1 的特殊情形。我们用 Z 表示标准正态随机变量。其概率密度函数简化成

φ(z) = 1/√(2π) e^–½z²。

The cumulative distribution function (CDF) of Z is denoted by Φ(z) = P(Z ≤ z). The values of Φ(z) are tabulated in the standard normal table provided in exams (or can be found using a calculator). Because the distribution is symmetric, Φ(–z) = 1 – Φ(z) always holds – a hugely helpful trick in questions.

Z 的累积分布函数记为 Φ(z) = P(Z ≤ z)。Φ(z) 的值在考试提供的标准正态分布表中列出（或可用计算器求得）。由于分布是对称的，恒有 Φ(–z) = 1 – Φ(z)，这是解题中极为有用的技巧。

5. Standardisation and the Z‑Score | 标准化与 z 分数

Any normally distributed variable X ~ N(μ, σ²) can be transformed into Z by subtracting the mean and dividing by the standard deviation:

Z = (X – μ) / σ.

The resulting Z follows N(0, 1). This process is called standardisation, and the value z = (x – μ)/σ is called the z‑score of x. The z‑score tells you how many standard deviations x is from the mean.

任何服从正态分布的变量 X ~ N(μ, σ²) 都可以通过减去均值再除以标准差转换成 Z：

Z = (X – μ) / σ。

得到的 Z 服从 N(0, 1)。这个过程叫作标准化，数值 z = (x – μ)/σ 称为 x 的 z 分数。z 分数告诉你 x 距离均值有多少个标准差。

In exam problems, you must always standardise before using the normal table unless your calculator handles non‑standard normal directly. Even then, showing the standardisation step earns method marks.

在考试题目中，除非你的计算器能直接处理非标准正态，否则你总是需要先标准化再查表。即便如此，写出标准化步骤仍然可以获得方法分。

6. Using the Standard Normal Table | 使用标准正态分布表

The standard normal table gives the cumulative probability Φ(z) = P(Z ≤ z) for positive z values. Typically the table lists z to one decimal place down the side and the second decimal across the top. For example, to find Φ(1.23), locate 1.2 in the left column and 0.03 in the top row; the intersection gives the probability. A small extract might look like:

标准正态分布表给出了正 z 值的累积概率 Φ(z) = P(Z ≤ z)。表格通常左侧列出一位小数，顶行列出第二位小数。例如，要查找 Φ(1.23)，在左列找到 1.2，顶行找到 0.03，交点处即为概率。下为一个简表示例：

z	0.00	0.01	0.02	0.03
1.0	0.8413	0.8438	0.8461	0.8485
1.1	0.8643	0.8665	0.8686	0.8708
1.2	0.8849	0.8869	0.8888	0.8907

To handle negative z, use Φ(–z) = 1 – Φ(z). For instance, Φ(–1.23) = 1 – 0.8907 = 0.1093. Some tables provide negative z directly; always check the table format in your exam.

对于负的 z，使用 Φ(–z) = 1 – Φ(z)。例如 Φ(–1.23) = 1 – 0.8907 = 0.1093。有些表直接给出负 z 的值；考试时请先确认表格格式。

7. Calculating Probabilities | 计算概率

Once you have z‑scores, most questions ask for one of three probability types:

一旦获得 z 分数，大多数题目会要求计算以下三类概率之一：

P(Z < a) – directly read Φ(a) from the table.
P(Z < a) – 直接从表中读取 Φ(a)。
P(Z > a) = 1 – Φ(a). Because the total area is 1, the right‑tail probability is the complement.
P(Z > a) = 1 – Φ(a)。由于总面积为 1，右尾概率为其补集。
P(a < Z < b) = Φ(b) – Φ(a). Subtract the left cumulative area from the right cumulative area.
P(a < Z < b) = Φ(b) – Φ(a)。用右侧累积面积减去左侧累积面积。

Always sketch a bell curve and shade the required region. This visual check prevents sign mistakes and helps you decide whether to subtract from 1 or to use symmetry.

一定要画一张钟形曲线并标出所求区域。这一可视化检查能避免符号错误，并帮助你判断该用 1 减去还是利用对称性。

8. Inverse Normal – Finding z Given a Probability | 正态分布的反向查找 – 已知概率求 z

If a question gives a probability and asks for the corresponding z (or x) value, you need the inverse normal procedure. For example, find z such that P(Z < z) = 0.975. Look inside the main body of the normal table for 0.975; the corresponding row and column give z ≈ 1.96. If the probability is a left‑tail, you can directly read z. For a right‑tail of 0.025, use P(Z > z) = 0.025 ⇒ Φ(z) = 0.975, giving the same 1.96.

如果题目给出一个概率，要求对应的 z（或 x）值，你需要使用反向正态查找。例如，求 z 使得 P(Z < z) = 0.975。在正态分布表的主体中搜索 0.975，对应的行与列给出 z ≈ 1.96。若概率为左尾，可以直接读取 z。对于右尾概率 0.025，用 P(Z > z) = 0.025 ⇒ Φ(z) = 0.975，结果同为 1.96。

Many calculators have an inverse normal function, typically denoted invNorm(area, μ, σ). When using this, be very precise about whether the area is left‑tail, right‑tail, or central. Drawing a sketch remains the best way to avoid confusion.

许多计算器带有逆正态函数，通常表示为 invNorm(面积, μ, σ)。使用时务必明确面积属于左尾、右尾还是中心区域。画草图依然是避免混淆的最佳方法。

9. Working with Non‑Standard Normal X ~ N(μ, σ²) | 处理非标准正态分布

Exam questions usually give μ and σ² (or σ) and ask for probabilities concerning X. The flow is always: standardise → use Z‑table → interpret. For instance, if X ~ N(100, 15²), find P(X < 115). Standardise: z = (115 – 100)/15 = 1.00, so P(X < 115) = Φ(1.00) ≈ 0.8413. If the question asks for P(X > 85), standardise: z = (85 – 100)/15 = –1.00, so P(Z > –1.00) = P(Z < 1.00) = 0.8413 by symmetry.

考试题目通常给出 μ 和 σ²（或 σ），要求计算关于 X 的概率。流程始终是：标准化 → 查 Z 表 → 解读。例如，若 X ~ N(100, 15²)，求 P(X < 115)。标准化：z = (115 – 100)/15 = 1.00，因此 P(X < 115) = Φ(1.00) ≈ 0.8413。若问题要求 P(X > 85)，标准化：z = (85 – 100)/15 = –1.00，则 P(Z > –1.00) = P(Z < 1.00) = 0.8413（根据对称性）。

When the given probability is known and you need a boundary x, first find the appropriate z from the table, then back‑transform: x = μ + zσ. For example, find the 95th percentile of X ~ N(200, 30²). The 95th percentile corresponds to P(Z < z) = 0.95 ⇒ z ≈ 1.645. Thus x = 200 + 1.645 × 30 ≈ 249.35.

当已知概率而你需要求界限 x 时，先从表中找到对应的 z，再反代回：x = μ + zσ。例如，求 X ~ N(200, 30²) 的第 95 百分位数。第 95 百分位数对应 P(Z < z) = 0.95 ⇒ z ≈ 1.645。因此 x = 200 + 1.645 × 30 ≈ 249.35。

10. Normal Approximation to the Binomial | 二项分布的正态近似

When the number of trials n is large and p is not too close to 0 or 1, a binomial distribution B(n, p) can be approximated by a normal distribution with μ = np and σ² = np(1 – p). This is extremely useful because exact binomial probabilities become cumbersome for large n. The usual condition for the approximation to be valid is that both np > 5 and n(1 – p) > 5 (some syllabuses use np > 10).

当试验次数 n 很大且 p 不太接近 0 或 1 时，二项分布 B(n, p) 可以用均值为 μ = np，方差为 σ² = np(1 – p) 的正态分布来近似。这在 n 很大时极其有用，因为精确二项概率计算会变得繁琐。近似有效的常见条件是 np > 5 且 n(1 – p) > 5（部分课程要求 np > 10）。

However, because a binomial distribution is discrete and a normal distribution is continuous, a continuity correction must be applied to improve accuracy. You will lose marks if you forget this correction. The correction involves expanding the integer x by 0.5 in the direction that makes the interval contain the discrete bar.

然而，由于二项分布是离散的而正态分布是连续的，必须应用连续性校正来提高精确度。如果你忘记这一校正，将会失分。校正的做法是将整数 x 向包含离散长条的方向扩展 0.5。

11. Continuity Correction in Detail | 连续性校正详解

For a binomial random variable X ~ B(n, p) approximated by Y ~ N(np, np(1–p)), the following corrections apply:

对于用 Y ~ N(np, np(1–p)) 近似的二项随机变量 X ~ B(n, p)，应用以下校正：

P(X = k) ≈ P(k – 0.5 < Y < k + 0.5). The exact probability of a single value is approximated by the area of an interval of length 1 centred at k.
P(X = k) ≈ P(k – 0.5 < Y < k + 0.5)。单个值的精确概率近似为以 k 为中心、长度为 1 的区间面积。
P(X ≤ k) ≈ P(Y < k + 0.5). Include the full bar up to k.
P(X ≤ k) ≈ P(Y < k + 0.5)。包含直到 k 的完整条形。
P(X ≥ k) ≈ P(Y > k – 0.5). Include the bar from k onward.
P(X ≥ k) ≈ P(Y > k – 0.5)。包含从 k 开始的条形。
P(X < k) ≈ P(Y < k – 0.5), and P(X > k) ≈ P(Y > k + 0.5).
P(X < k) ≈ P(Y < k – 0.5)，以及 P(X > k) ≈ P(Y > k + 0.5)。

After applying the correction, standardise the resulting Y‑boundaries as usual. Always state the approximation and show the correction step clearly. Some exam boards also accept the use of a normal approximation to a Poisson distribution when λ > 15, but the binomial case is the most common.

应用校正后，照常将得到的 Y 界限标准化。务必说明进行了近似，并清晰展示校正步骤。一些考试局也接受当 λ > 15 时用正态近似泊松分布，但二项情形最为常见。

12. Common Pitfalls and Exam Tips | 常见错误与考试技巧

One of the most frequent errors is using σ instead of σ² when writing X ~ N(μ, σ²). The second parameter is the variance, not the standard deviation. Always check which is given in the question. Another pitfall is forgetting to subtract from 1 for P(Z > z) or misreading the table. Many students incorrectly look up the z‑value itself when they should be finding the corresponding probability inside the table.

最常见的错误之一是在写出 X ~ N(μ, σ²) 时使用 σ 而非 σ²。第二个参数是方差，不是标准差。务必核对题目给出的是哪一个。另一个陷阱是忘记用 1 减去来求 P(Z > z)，或者读错表格。许多学生在应当查找表内概率时，却错误地去查找 z 值本身。

During revision, practise full working: define the variable, state the distribution, show standardisation, draw the curve, write the probability statement, and quote the final value. This methodical approach secures method marks even if a calculator slip occurs. Use symmetry hacks: P(Z < –a) = P(Z > a) = 1 – Φ(a). Memorise key z‑values: 1.645 (one‑tailed 5%), 1.96 (two‑tailed 5%), 2.576 (1%). These will save time in hypothesis testing questions.

复习时要练习完整的解题过程：定义变量，写出分布，展示标准化，绘制曲线，写出概率表达式，并引用最终结果。这种有条理的方法即使计算器按错也能获得方法分。善用对称技巧：P(Z < –a) = P(Z > a) = 1 – Φ(a)。熟记关键 z 值：1.645（单尾 5%），1.96（双尾 5%），2.576（1%）。这将在假设检验题目中为你节省时间。

Finally, when approximating, never omit the continuity correction unless the question explicitly states an uncorrected normal approximation is acceptable. Check your syllabus document for exact conditions on np and n(1–p).

最后，进行近似时，除非题目明确说明可以使用未经校正的正态近似，否则永远不要省略连续性校正。请查阅你的课程大纲关于 np 和 n(1–p) 的具体条件。

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A-Level Maths: Normal Distribution Key Exam Points | A-Level 数学：正态分布 考点精讲