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

📚 A-Level OCR Maths: Normal Distribution Key Points | A-Level OCR 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 OCR Mathematics. It underpins statistical inference, hypothesis testing and modelling of natural phenomena. This revision guide covers all essential concepts, formulas and techniques you need to master for the exam.

正态分布是 A-Level OCR 数学中最重要的连续概率分布之一，支撑着统计推断、假设检验和自然现象建模。本考点精讲涵盖所有你需要掌握的核心理念、公式和技巧。

1. The Normal Probability Density Function | 正态概率密度函数

A continuous random variable X follows a normal distribution with mean μ and variance σ², written X ~ N(μ, σ²). Its probability density function (pdf) is symmetric and bell‑shaped.

若连续随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，记为 X ~ N(μ, σ²)。其概率密度函数对称且呈钟形。

f(x) = 1/(σ√(2π)) · exp(–(x – μ)² / (2σ²))

The total area under the curve equals 1. Probabilities correspond to areas under the curve between given x‑values. There is no simple closed‑form integral, so tables or calculators are used.

曲线下的总面积为 1。概率是给定 x 值之间曲线下的面积。没有简单的积分表达式，因此使用表格或计算器。

2. Parameters μ and σ | 参数 μ 和 σ

The mean μ determines the centre of the distribution; the curve is symmetric about x = μ. The standard deviation σ controls the spread: larger σ gives a flatter, wider curve. Approximately 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

均值 μ 决定分布的中心；曲线对称于 x = μ。标准差 σ 控制离散程度：σ 越大，曲线越扁平、越宽。约 68% 的数据落在 μ ± σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。

You must be able to interpret μ and σ in context, and use these empirical percentages for quick estimation.

你必须能在实际情境中解释 μ 和 σ，并利用这些经验百分比进行快速估算。

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

To use statistical tables, we standardise any normal variable X into a z‑score. The standard normal variable Z has mean 0 and variance 1.

为使用统计表，我们将任意正态变量 X 标准化为 z 分数。标准正态变量 Z 的均值为 0，方差为 1。

z = (x – μ) / σ

This transformation shifts and rescales the distribution so that we can compare different normals using a single set of tables.

该变换通过平移和重新缩放，使得我们可以用同一套表来比较不同的正态分布。

4. Using the Z‑table | 使用 Z 表

OCR provides tables of the cumulative distribution function Φ(z) = P(Z ≤ z) for positive z. The table gives the area to the left of z. For negative z, use symmetry: Φ(–z) = 1 – Φ(z).

OCR 提供正 z 值的累积分布函数表 Φ(z) = P(Z ≤ z)。表格给出 z 左侧的面积。对于负 z 值，利用对称性：Φ(–z) = 1 – Φ(z)。

Always sketch a bell curve and shade the required probability region. This prevents sign errors and helps decide whether to add or subtract table values.

始终画一个钟形曲线并给所求概率区域涂上阴影。这可以防止符号错误，并帮助判断表值该相加还是相减。

5. Finding Probabilities for X ~ N(μ, σ²) | 计算 X ~ N(μ, σ²) 的概率

To find P(X < a), standardise to P(Z < (a – μ)/σ) and use Φ. For P(X > a) use 1 – Φ((a – μ)/σ). For between values P(a < X < b), compute Φ((b – μ)/σ) – Φ((a – μ)/σ).

求 P(X < a) 时，标准化为 P(Z < (a – μ)/σ) 并查 Φ；P(X > a) 用 1 – Φ((a – μ)/σ)；区间概率 P(a < X < b) 计算 Φ((b – μ)/σ) – Φ((a – μ)/σ)。

Remember: the probability of exactly a single point is zero for continuous distributions, so P(X ≤ a) = P(X < a) and inequalities can be treated interchangeably.

注意：连续分布中单点概率为零，因此 P(X ≤ a) = P(X < a)，不等号可以互换。

6. Reverse Lookup: Finding z Given Φ(z) | 反向查表：已知 Φ(z) 求 z

When given a probability and asked to find the corresponding value, first identify the area to the left of z. Look inside the table body for the closest probability, then read off the z‑value by combining the row and column. If the probability refers to a right‑tail, convert to left‑tail via 1 – p.

当给出概率求对应取值时，先确定 z 左侧的面积。在表内找到最接近的概率，然后通过行列读出 z 值。若概率是右尾，用 1 – p 转换为左尾。

Then un‑standardise using x = μ + σz to recover the original variable value. Interpolation may be required if the exact probability is not listed.

随后用 x = μ + σz 去标准化，得到原变量的值。若表中无精确概率值，可能需要进行线性插值。

7. Linear Combinations of Normal Variables | 正态变量的线性组合

If X₁ ~ N(μ₁, σ₁²) and X₂ ~ N(μ₂, σ₂²) are independent, then any linear combination aX₁ + bX₂ + c is also normal. Its mean is aμ₁ + bμ₂ + c, and variance is a²σ₁² + b²σ₂².

若 X₁ ~ N(μ₁, σ₁²) 与 X₂ ~ N(μ₂, σ₂²) 独立，则任意线性组合 aX₁ + bX₂ + c 仍服从正态分布。其均值为 aμ₁ + bμ₂ + c，方差为 a²σ₁² + b²σ₂²。

This is extremely useful for totals, differences, or sample means. For a sample mean X̄ from n independent observations of X ~ N(μ, σ²), we have X̄ ~ N(μ, σ²/n).

这对总和、差值或样本均值极为有用。对于来自 X ~ N(μ, σ²) 的 n 个独立观测的样本均值 X̄，有 X̄ ~ N(μ, σ²/n)。

A common OCR question type: the difference D = X₁ – X₂ has mean μ₁ – μ₂ and variance σ₁² + σ₂² (variance adds, not subtracts).

OCR 常见题型：差值 D = X₁ – X₂ 的均值为 μ₁ – μ₂，方差为 σ₁² + σ₂²（方差是相加，而不是相减）。

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

When 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). A common rule of thumb is that both np and n(1 – p) should be greater than 5 (or 10).

当 n 较大且 p 不接近 0 或 1 时，二项分布 B(n, p) 可用正态分布近似，μ = np，σ² = np(1 – p)。通常的经验法则是 np 和 n(1 – p) 都应大于 5（或 10）。

The approximation allows us to compute binomial probabilities quickly using the normal distribution, especially for large numbers of trials where exact binomial calculation is impractical.

该近似使我们能用正态分布快速计算二项概率，尤其当试验次数多、精确二项计算不便时。

9. Continuity Correction | 连续性校正

Because the binomial is discrete and the normal is continuous, we apply a continuity correction. Replace a discrete x with an interval: for P(X = r) use P(r – 0.5 < Y < r + 0.5) where Y ~ N(μ, σ²).

由于二项分布是离散的而正态分布是连续的，我们需进行连续性校正。将离散的 x 替换为一个区间：P(X = r) 用 P(r – 0.5 < Y < r + 0.5)，其中 Y ~ N(μ, σ²)。

For P(X ≥ r) use P(Y > r – 0.5); for P(X ≤ r) use P(Y < r + 0.5). Always adjust the boundary by 0.5 in the direction that makes the continuous region match the discrete probability.

P(X ≥ r) 用 P(Y > r – 0.5)；P(X ≤ r) 用 P(Y < r + 0.5)。始终按使连续区域与离散概率相匹配的方向调整 0.5。

10. Finding μ or σ Given Probabilities | 已知概率求 μ 或 σ

OCR exams often present a problem where you know a probability statement about X, for example P(X < 10) = 0.2, and you need to find μ or σ. Convert to the standard normal: P(Z < (10 – μ)/σ) = 0.2, then use the inverse table to find the corresponding z‑value.

OCR 考试常出现已知 X 的概率陈述（如 P(X < 10) = 0.2）需要求 μ 或 σ 的问题。将其转化为标准正态：P(Z < (10 – μ)/σ) = 0.2，然后利用反向查表找到相应的 z 值。

Set up an equation: (10 – μ)/σ = z. If two probabilities are given, you can form simultaneous equations to solve for both μ and σ.

建立方程：(10 – μ)/σ = z。若给出两个概率，则可联立方程组同时解出 μ 和 σ。

11. Common Pitfalls and Exam Tips | 常见陷阱与应试技巧

• Always check whether the variance or standard deviation is given in the question. If σ is given, square it to get variance when needed.

  始终确认题目给出的是方差还是标准差。若给出 σ，需要时平方得方差。

• When using the standardisation formula, remember to subtract μ and then divide by σ. A misplaced bracket leads to wrong z‑scores.

  使用标准化公式时，记得先减 μ 再除以 σ。括号位置错误会导致 z 分数错误。

• For the binomial approximation, never forget the continuity correction unless the question explicitly says not to use it. Missing it can cost many marks.

  二项近似时，除非题目明确说明不使用，否则千万不要忘记连续性校正，漏掉会丢很多分。

• Sketch the normal curve for every problem. Mark the mean, boundaries and shade the required area. This visual check will catch sign and direction mistakes.

  每道题都画正态曲线示意图，标出均值、边界并涂上阴影区域。这种可视化检查能捕捉符号和方向错误。

• When reading the z‑table, ensure you are using the correct column (adding the second decimal place). For negative z, apply symmetry correctly.

  查 z 表时，确保使用正确的列（加上第二位小数）。对于负 z，正确应用对称性。

12. Summary of Key Formulas | 核心公式一览

Concept	Formula
Standardisation	z = (x – μ) / σ
Un‑standardising	x = μ + σz
Linear combination mean	E(aX + bY) = aμₓ + bμᵧ
Linear combination variance	Var(aX + bY) = a²σₓ² + b²σᵧ² (for independent X,Y)
Sample mean variance	Var(X̄) = σ² / n
Binomial normal approx.	B(n, p) ≈ N(np, np(1 – p))
Continuity correction (X = r)	P(r – 0.5 < Y < r + 0.5)

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