Normal Distribution for A-Level WJEC Mathematics: Key Points and Exam Focus | A-Level WJEC 数学：正态分布考点精讲

📚 Normal Distribution for A-Level WJEC Mathematics: Key Points and Exam Focus | A-Level WJEC 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 WJEC Mathematics. It models many natural phenomena and forms the foundation for statistical inference. In WJEC Unit S1 and the new specification, you are expected to calculate probabilities using standardisation and statistical tables, solve inverse problems, handle combinations of independent normal variables, and apply the normal approximation to the binomial distribution. This article provides a structured revision guide covering all key concepts, worked examples, common mistakes and exam-focused advice to help you secure top marks.

正态分布是 A-Level WJEC 数学中最重要的连续概率分布之一，能够模拟大量自然现象，也是统计推断的基础。在 WJEC S1 和新考纲中，考生需要掌握标准化查表求概率、逆正态问题、独立正态变量的组合以及正态近似二项分布。本文提供结构化复习指南，涵盖所有核心概念、例题解析、常见错误和考试技巧，帮助你拿下高分。

1. Introduction to the Normal Distribution | 正态分布导论

A continuous random variable X follows a normal distribution if its probability density function is bell-shaped, symmetric about the mean μ, and its spread is determined by the standard deviation σ. The total area under the curve equals 1. The notation is X ~ N(μ, σ²). Unlike discrete distributions, we cannot find P(X = a) — this is always zero; instead we calculate probabilities over intervals.

若连续随机变量 X 的概率密度函数呈钟形、关于均值 μ 对称且其分散程度由标准差 σ 决定，则称其服从正态分布，记作 X ~ N(μ, σ²)。曲线下总面积为 1。与离散分布不同，单点概率 P(X = a) 恒为零，我们总是计算区间概率。

The shape of the curve changes with the parameters. A larger μ shifts the curve to the right; a larger σ flattens and widens the curve. In WJEC questions, you often need to sketch the distribution to visualise probabilities, especially when using symmetry.

曲线的形状随参数变化：μ 增大会使曲线右移；σ 增大会使曲线变得扁平宽阔。WJEC 考题中，往往需要画出分布草图以直观展示概率，特别是在利用对称性时。

2. The Standard Normal Distribution and Z-scores | 标准正态分布与 Z 分数

The standard normal distribution has mean 0 and standard deviation 1, i.e. Z ~ N(0,1). Its cumulative probabilities are tabled as Φ(z) = P(Z < z). All normal probabilities can be converted to this scale using the Z-score formula: z = (x − μ) / σ. This process is called standardisation.

标准正态分布的均值为 0、标准差为 1，即 Z ~ N(0,1)。其累积概率已制成表格 Φ(z) = P(Z < z)。任何正态概率都可以通过 Z 分数公式 z = (x − μ)/σ 转换为标准正态，这个过程称为标准化。

Key properties to remember: the curve is symmetric, so P(Z < −z) = P(Z > z) = 1 − Φ(z). Also, P(Z < 0) = 0.5. The total area from −∞ to ∞ is 1. WJEC statistical tables usually give Φ(z) for positive z values only; negative z probabilities are found by symmetry.

需要牢记的关键性质：曲线对称，因此 P(Z < −z) = P(Z > z) = 1 − Φ(z)；同时 P(Z < 0) = 0.5。从 −∞ 到 ∞ 的总面积为 1。WJEC 给出的统计表通常只包含正 z 值，负 z 对应的概率需利用对称性求得。

3. Using the Standard Normal Table | 使用标准正态表

The WJEC statistical table provides values of Φ(z) for z from 0.00 to about 3.99, typically to 4 decimal places. To find Φ(z), locate the row for the first two digits (e.g. 1.2) and the column for the second decimal (e.g. 0.03) to read the probability for z = 1.23. Always draw a diagram to check whether you need Φ(z), 1 − Φ(z), or a difference.

WJEC 统计表提供 z 从 0.00 到约 3.99 的 Φ(z) 值，通常保留四位小数。查表时，先定位前两位数字所在行（如 1.2），再找到第二位小数所在列（如 0.03），即可读出 z = 1.23 的概率。务必画图确认你需要的是 Φ(z)、1 − Φ(z) 还是区间差值。

For negative z, e.g. z = −1.45, you use Φ(−1.45) = 1 − Φ(1.45). Remember that the table only gives cumulative probabilities for the lower tail, i.e. P(Z < z). If your question asks for P(Z > z), use the complement rule.

对于负 z，例如 z = −1.45，用 Φ(−1.45) = 1 − Φ(1.45)。切记表格给出的是下尾累积概率 P(Z < z)。如果问题要求的是 P(Z > z)，则利用互补规则。

4. Calculating Probabilities: Less Than, Greater Than, Between | 计算概率：小于、大于、介于

To find P(X < a) for X ~ N(μ, σ²), first obtain z = (a − μ)/σ. Then P(X < a) = Φ(z). Always check whether a is above or below the mean to use correct symmetry. For P(X > b), calculate z = (b − μ)/σ then P(X > b) = 1 − Φ(z).

对于 X ~ N(μ, σ²)，求 P(X < a) 先得到 z = (a − μ)/σ，然后 P(X < a) = Φ(z)。要检查 a 在均值的上方还是下方，以便正确使用对称性。对于 P(X > b)，计算 z = (b − μ)/σ，然后 P(X > b) = 1 − Φ(z)。

For an interval P(a < X < b), compute z₁ = (a − μ)/σ and z₂ = (b − μ)/σ. Then P(a < X < b) = Φ(z₂) − Φ(z₁). This is one of the most common WJEC exam requirements. Sketching a normal curve and shading the region helps avoid sign errors.

对于区间 P(a < X < b)，计算 z₁ = (a − μ)/σ 和 z₂ = (b − μ)/σ，则 P(a < X < b) = Φ(z₂) − Φ(z₁)。这是 WJEC 最常见的考点之一。画出正态曲线并涂黑对应区域有助于避免符号错误。

When a question provides the variance σ², remember to take the square root to obtain σ before standardising. Many candidates mistakenly use the variance instead of the standard deviation, which will destroy the Z-score.

若题目给出的是方差 σ²，记得在标准化前先开平方得到 σ。很多考生误将方差当作标准差代入 Z 分数，会导致整题错误。

5. The Inverse Normal Distribution | 逆正态分布

Inverse normal problems give a probability and ask you to find the corresponding x-value. The typical phrasing: “Find the value of k such that P(X < k) = 0.95.” For X ~ N(μ, σ²), first find z where Φ(z) = 0.95. Using the table, you locate the z that gives this cumulative probability, often requiring interpolation if not directly tabled. Then x = μ + zσ.

逆正态问题给出概率，要求求出对应的 x 值。典型问法：“求 k 使得 P(X < k) = 0.95”。对于 X ~ N(μ, σ²)，先找到满足 Φ(z) = 0.95 的 z，通常需要查表并可能使用插值。然后通过 x = μ + zσ 还原 x 值。

If the probability refers to an upper tail, e.g. P(X > k) = 0.05, rewrite it as P(X < k) = 0.95 and proceed. The inverse normal procedure is heavily tested in WJEC, especially in contexts like quality control, weights, volumes and examination marks.

若概率为上尾概率，如 P(X > k) = 0.05，需转化为 P(X < k) = 0.95 再求解。逆正态过程在 WJEC 考试中频繁出现，常见于质量控制、重量、体积和考试成绩等场景。

6. Finding the Mean or Standard Deviation | 求均值或标准差

WJEC often asks you to determine unknown μ or σ (or both) when given two probability statements. You set up two equations involving Φ(z₁) and Φ(z₂), convert to Z-scores, and solve simultaneously. For example, given P(X < 150) = 0.8 and P(X > 170) = 0.1, you first find z₁ such that Φ(z₁) = 0.8 and z₂ such that P(Z > z₂) = 0.1, i.e. Φ(z₂) = 0.9. Then write (150 − μ)/σ = z₁ and (170 − μ)/σ = z₂, and solve.

WJEC 常给出两个概率条件，要求确定未知的 μ 或 σ（或两者）。此时需建立两个含有 Φ(z₁) 和 Φ(z₂) 的方程，转化为 Z 分数并联立求解。例如，已知 P(X < 150) = 0.8 和 P(X > 170) = 0.1，先找出使 Φ(z₁)=0.8 的 z₁ 以及使 P(Z > z₂)=0.1 即 Φ(z₂)=0.9 的 z₂，然后写出 (150 − μ)/σ = z₁ 与 (170 − μ)/σ = z₂ 并求解。

Linearity is essential here: always work with σ, not σ², in the equations. After finding σ, you may be required to state the variance. Be careful with signs — drawing a diagram of each tail and the mean position prevents algebraic errors.

这里要注意线性关系：方程中必须使用 σ 而非 σ²。求出 σ 后，题目可能要求给出方差。务必注意符号——画出每个尾部及均值位置的图示，可以避免代数错误。

7. Sums and Differences of Independent Normal Variables | 独立正态变量的和与差

When two or more independent normal variables are combined linearly, the result is also normally distributed. If X ~ N(μₓ, σₓ²) and Y ~ N(μ_y, σ_y²) and they are independent, then:

若将两个或多个独立正态变量线性组合，结果仍服从正态分布。设 X ~ N(μₓ, σₓ²) 与 Y ~ N(μ_y, σ_y²) 独立，则：

X + Y ~ N(μₓ + μ_y, σₓ² + σ_y²)
X − Y ~ N(μₓ − μ_y, σₓ² + σ_y²) ← note: variances always add

The variance of both sum and difference is the sum of the individual variances. This surprises many students. For a scaled variable aX + bY, the mean is aμₓ + bμ_y and the variance is a²σₓ² + b²σ_y².

和与差的方差均为各自方差之和，这一点常让学生感到意外。对于缩放组合 aX + bY，均值为 aμₓ + bμ_y，方差为 a²σₓ² + b²σ_y²。

WJEC applications typically involve total weight of several items, differences in length or combined errors. You will need to find probabilities for the total or difference, standardising with the new mean and new standard deviation (square root of the new variance).

WJEC 的应用题常涉及多个物体的总重量、长度之差或组合误差。考生需使用新均值和新标准差（新方差开平方）进行标准化，求解总总量或差值落在某一区间的概率。

8. Normal Approximation to the Binomial | 正态近似二项分布

The binomial distribution X ~ B(n, p) can be approximated by a normal distribution with μ = np and σ² = np(1 − p) when n is large. The WJEC guideline expects both np > 5 and n(1 − p) > 5 for the approximation to be valid.

当 n 较大时，二项分布 X ~ B(n, p) 可用均值为 μ = np、方差为 σ² = np(1 − p) 的正态分布近似。WJEC 要求同时满足 np > 5 与 n(1 − p) > 5 方可近似。

Because the binomial is discrete and the normal is continuous, a continuity correction must be applied. For example, P(X ≤ 10) is approximated by P(Y < 10.5) where Y ~ N(np, np(1−p)). Similarly, P(X ≥ 12) is approximated by P(Y > 11.5), and P(X = 8) by P(7.5 < Y < 8.5). Missing this correction is a major cause of lost marks.

由于二项分布为离散型而正态连续，必须进行连续性校正。例如，P(X ≤ 10) 近似为 P(Y < 10.5)；P(X ≥ 12) 近似为 P(Y > 11.5)；P(X = 8) 近似为 P(7.5 < Y < 8.5)，其中 Y ~ N(np, np(1−p))。漏掉连续性校正是严重失分点。

After setting up the corrected boundary, standardise and use the normal table. Always state the approximate distribution and justify that np and nq exceed 5, as this earns method marks in WJEC mark schemes.

在校正边界后，标准化并查表求概率。务必写出近似分布并说明 np 与 nq 均大于 5，WJEC 的评分方案会因此给予步骤分。

9. Checking Assumptions and Conditions | 检查假设与条件

Using the normal distribution in real-world contexts requires that the data be continuous, roughly symmetric and unimodal, with no extreme outliers. WJEC questions may present a histogram, box plot or summary statistics and ask if the normal model is appropriate. Look for a bell-shaped histogram and check that the mean ≈ median. You might also be asked to plot on normal probability paper, where a straight line indicates normality.

在实际情境下使用正态分布，要求数据连续、大致对称单峰且无极端离群值。WJEC 题目可能给出直方图、箱线图或汇总统计量，询问正态模型是否合适。应观察直方图形状和均值与中位数的接近程度。有时会要求用到正态概率纸，直线表明数据符合正态性。

For the binomial approximation, explicitly state “np = … > 5 and n(1−p) = … > 5, hence the normal approximation is suitable.” This statement alone often carries one mark.

对于二项近似，需明确写出“np = … > 5 且 n(1−p) = … > 5，因此正态近似适用”，此陈述本身通常占一分。

10. Common Pitfalls and Exam Advice | 常见陷阱与考试建议

Many students confuse variance with standard deviation, especially when standardising. Always convert variance σ² to σ before calculating the Z-score. Another common error is misreading the table: WJEC tables give P(Z < z), so adapt accordingly for P(Z > z) or negative z. Forgetting to square a coefficient when combining variables (e.g. using aσₓ² instead of a²σₓ²) is also frequently penalised. With the binomial approximation, omitting the continuity correction can halve your accuracy.

许多学生混淆方差与标准差，尤其在标准化环节。务必先将方差 σ² 化为 σ 再求 Z 分数。另一个常见错误是误读表格：WJEC 表给出的是 P(Z < z)，对于 P(Z > z) 或负 z 需要相应调整。在组合变量时忘记将系数平方（如使用 aσₓ² 而非 a²σₓ²）也是高频扣分点。对于二项近似，省略连续性校正会使精度减半。

Always draw a diagram. A quick sketch of the normal curve with the mean, the shaded region and the z boundaries vastly reduces sign mistakes. Practise using the statistical table accurately and quickly by covering a variety of past WJEC papers. When solving inverse problems, check that your final x-value makes sense relative to the mean — if P(X < k) = 0.95, then k should be greater than μ.

务必画图。快速画出正态曲线、标注均值、阴影区域和 z 边界，可以大幅减少符号错误。通过练习历年 WJEC 真题提高查表速度和准确性。在解逆正态问题时，检查 x 值是否与均值合理匹配——若 P(X < k) = 0.95，则 k 应大于 μ。

Finally, show all steps clearly: state the distribution, specify the standardisation formula, write the probability in terms of Z, look up Φ(z), and give a concluding statement in context. WJEC mark schemes reward clear method, even if minor arithmetic slips occur.

最后，要清晰地写出所有步骤：声明分布类型、写出标准化公式、将概率转换为 Z 形式、查表得到 Φ(z)，并结合上下文给出结论。即使出现小的计算错误，WJEC 评分标准仍会奖励清晰的方法。

11. Worked Examples | 例题解析

Example 1 – Simple Probability:
X ~ N(100, 15²). Find P(X < 115).
z = (115 − 100)/15 = 1.00. Φ(1.00) = 0.8413.
Thus P(X < 115) = 0.8413.

例1 – 简单概率：
X ~ N(100, 15²)。求 P(X < 115)。
z = (115 − 100)/15 = 1.00。Φ(1.00) = 0.8413。
因此 P(X < 115) = 0.8413。

Example 2 – Between:
Heights of men ~ N(175, 12²). Proportion between 170 cm and 180 cm?
z₁ = (170−175)/12 = −0.4167, z₂ = (180−175)/12 = 0.4167.
Φ(0.4167) ≈ 0.6616 (by interpolation). Φ(−0.4167) = 1 − 0.6616 = 0.3384.
P = 0.6616 − 0.3384 = 0.3232 (about 32.3%).

例2 – 区间概率：
男性身高 ~ N(175, 12²)。介于 170 cm 与 180 cm 的比例？
z₁ = (170−175)/12 = −0.4167，z₂ = (180−175)/12 = 0.4167。
Φ(0.4167) ≈ 0.6616（插值），Φ(−0.4167) = 1 − 0.6616 = 0.3384。
概率 = 0.6616 − 0.3384 = 0.3232（约 32.3%）。

Example 3 – Inverse:
Machine fills jars, contents ~ N(μ, 2.5²). Find the value exceeded by 1% of jars.
P(X > k) = 0.01 ⇒ P(X < k) = 0.99. z for 0.99 ≈ 2.326. k = μ + 2.326×2.5.

例3 – 逆正态：
机器填装罐子，内装物 ~ N(μ, 2.5²)。求超过 1% 罐子的填入量。
P(X > k) = 0.01 ⇒ P(X < k) = 0.99。0.99 对应的 z ≈ 2.326。k = μ + 2.326×2.5。

Example 4 – Sum of variables:
Weight of an apple ~ N(150, 30²), weight of an orange ~ N(120, 25²) independent. Find probability that total of 1 apple and 1 orange exceeds 300 g.
T = X + Y ~ N(150+120, 30²+25²) = N(270, 1525). σ_T = √1525 ≈ 39.05.
P(T > 300): z = (300 − 270)/39.05 ≈ 0.768. 1 − Φ(0.768) ≈ 1 − 0.7789 = 0.2211.

例4 – 变量和：
苹果重量 ~ N(150, 30²)，橙子重量 ~ N(120, 25²)，二者独立。求一个苹果加一个橙子总重量超过 300 g 的概率。
T = X + Y ~ N(150+120, 30²+25²) = N(270, 1525)。σ_T = √1525 ≈ 39.05。
P(T > 300): z = (300 − 270)/39.05 ≈ 0.768。1 − Φ(0.768) ≈ 1 − 0.7789 = 0.2211。

Example 5 – Binomial approximation:
A fair coin tossed 200 times. Probability of 90 to 110 heads?
X ~ B(200, 0.5). np = 100, nq = 100 > 5, so use normal approx.
Y ~ N(100, 50). Using continuity correction: P(89.5 < Y < 110.5).
z₁ = (89.5−100)/√50 ≈ −1.484, z₂ = (110.5−100)/√50 ≈ 1.484.
Φ(1.484) ≈ 0.931, Φ(−1.484) ≈ 0.069. Probability ≈ 0.931 − 0.069 = 0.862.

例5 – 二项近似：
公平硬币抛 200 次，正面次数介于 90 到 110 的概率？
X ~ B(200, 0.5)。np = 100, nq = 100 > 5，可用正态近似。
Y ~ N(100, 50)。连续性校正：P(89.5 < Y < 110.5)。
z₁ = (89.5−100)/√50 ≈ −1.484，z₂ = (110.5−100)/√50 ≈ 1.484。
Φ(1.484) ≈ 0.931，Φ(−1.484) ≈ 0.069。概率 ≈ 0.931 − 0.069 = 0.862。

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Normal Distribution for A-Level WJEC Mathematics: Key Points and Exam Focus | A-Level WJEC 数学：正态分布 考点精讲