A-Level Mathematics: Solving Normal Distribution Probabilities | A-Level数学：正态分布概率求解方法

📚 A-Level Mathematics: Solving Normal Distribution Probabilities | A-Level数学：正态分布概率求解方法

The normal distribution is one of the most important probability models in A-Level Mathematics. It appears in both Statistics and Further Statistics modules, and students are expected to calculate probabilities, find unknown parameters, and apply continuity corrections when using the normal approximation to the binomial distribution.

正态分布是A-Level数学中最重要的概率模型之一。无论是统计学还是进阶统计学模块，你都可能遇到它。考试要求你计算概率、反求未知参数，并在用正态分布近似二项分布时进行连续性修正。

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

A normal distribution with mean μ and variance σ² is written as X ~ N(μ, σ²). The standard normal distribution has mean 0 and variance 1, written as Z ~ N(0, 1). To standardise any normal random variable, we subtract the mean and divide by the standard deviation:

均值为 μ、方差为 σ² 的正态分布记为 X ~ N(μ, σ²)。标准正态分布的均值为 0、方差为 1，记为 Z ~ N(0, 1)。要将任意正态随机变量标准化，我们减去均值并除以标准差：

Z = (X − μ) / σ

This transformation converts every normal distribution into the same standard scale, allowing us to use a single set of probability tables.

这个变换将所有正态分布统一到同一个标准尺度上，使我们可以使用同一套概率表。

2. Reading Probability Tables Correctly | 正确查表

There are two common types of standard normal tables. One gives P(Z < z) for positive z values; another gives P(0 < Z < z). You must check which table your exam board provides. In Edexcel, the table gives P(Z < z) for z ≥ 0.

常见的标准正态表有两种：一种给出 P(Z < z)（z 为正），另一种给出 P(0 < Z < z)。你必须确认你的考试局提供的是哪种表。以爱德思为例，其表格给出的是 z ≥ 0 时 P(Z < z) 的值。

For example, the table shows P(Z < 1.23) = 0.8907. But when dealing with negative z values, symmetry must be used:

例如，表中显示 P(Z < 1.23) = 0.8907。但处理负的 z 值时，必须利用对称性：

P(Z < −a) = P(Z > a) = 1 − P(Z < a)

P(Z > a) = 1 − P(Z < a)

A common mistake is using a table that gives P(0 < Z < z) and forgetting to add 0.5 for the lower half of the distribution. Always verify your table format before starting an exam.

一个常见错误是：你使用的表格给出的是 P(0 < Z < z)，却忘了加上下半个分布的 0.5。考试前务必确认表格格式。

3. Finding P(X < x) for a General Normal Distribution | 求一般正态分布的 P(X < x)

Suppose X ~ N(50, 4²), and we want P(X < 55). First standardise:

设 X ~ N(50, 4²)，要求 P(X < 55)。首先标准化：

z = (55 − 50) / 4 = 1.25

Then P(X < 55) = P(Z < 1.25) = 0.8944. The key is to perform the standardisation carefully, using the population standard deviation σ, not σ².

于是 P(X < 55) = P(Z < 1.25) = 0.8944。关键在于细心标准化，使用总体标准差 σ，而不是方差 σ²。

4. Finding P(X > x) Using the Complement Rule | 用对立事件求 P(X > x)

A direct table lookup gives left-tail probabilities only. For right-tail probabilities, use the complement:

查表直接得到的是左尾概率。对于右尾概率，要用补事件：

P(X > x) = 1 − P(X < x)

Example: X ~ N(30, 5²). Find P(X > 37).

例：X ~ N(30, 5²)，求 P(X > 37)。

z = (37 − 30) / 5 = 1.4

P(X > 37) = P(Z > 1.4) = 1 − P(Z < 1.4) = 1 − 0.9192 = 0.0808

Notice that for a continuous distribution, P(X = x) = 0, so strict and non-strict inequalities are equivalent. However, be careful when applying continuity corrections in discrete approximations.

注意：对连续分布而言，P(X = x) = 0，因此严格不等式与非严格不等式没有区别。但在离散近似中，使用连续性修正时就要小心。

5. Finding P(a < X < b) Between Two Values | 求区间概率 P(a < X < b)

To find the probability between two values, standardise both and subtract cumulative probabilities:

要求两个值之间的概率，先将两个端点标准化，再相减累积概率：

P(a < X < b) = P(Z < (b − μ)/σ) − P(Z < (a − μ)/σ)

Example: X ~ N(100, 10²). Find P(95 < X < 115).

例：X ~ N(100, 10²)，求 P(95 < X < 115)。

z₁ = (95 − 100) / 10 = −0.5, z₂ = (115 − 100) / 10 = 1.5

P(95 < X < 115) = P(Z < 1.5) − P(Z < −0.5)

= 0.9332 − (1 − 0.6915) = 0.9332 − 0.3085 = 0.6247

Take care when both z-values are negative; draw a diagram to help visualise the required region.

当两个 z 值都为负时，画图能帮助你直观理解所求区域。

6. Finding Percentiles and Inverse Normal Problems | 反求百分位数与逆正态问题

Instead of finding probability from a value, we may be asked to find the value corresponding to a given probability. For example, find the 90th percentile of X ~ N(50, 8²).

有时题目不是由值求概率，而是由概率反求值。例如，求 X ~ N(50, 8²) 的第 90 百分位数。

Let a be the 90th percentile, so P(X < a) = 0.90. The inverse table gives z = 1.2816. Then:

设 a 为第 90 百分位数，即 P(X < a) = 0.90。反查表得 z = 1.2816，于是：

a = μ + zσ = 50 + 1.2816 × 8 = 60.2528

Similarly, we can find lower percentiles using negative z-values. When the given probability is on the right tail, convert it to a left-tail probability first. For instance, P(X > a) = 0.05 means P(X < a) = 0.95.

类似地，可以用负 z 值求低百分位数。当给定概率在右尾时，先转换为左尾概率。例如 P(X > a) = 0.05 意味着 P(X < a) = 0.95。

7. Finding Unknown μ and σ | 求解未知的 μ 与 σ

Exam questions often provide two probability statements and require solving for both μ and σ simultaneously. For example, X ~ N(μ, σ²), P(X < 60) = 0.8413 and P(X < 90) = 0.9772.

考试题常给出两个概率条件，要求联立求出 μ 和 σ。例如 X ~ N(μ, σ²)，P(X < 60) = 0.8413 且 P(X < 90) = 0.9772。

From the first: (60 − μ)/σ = 1. From the second: (90 − μ)/σ = 2. Solving simultaneously:

由第一个条件得 (60 − μ)/σ = 1；由第二个条件得 (90 − μ)/σ = 2。联立求解：

60 − μ = σ, 90 − μ = 2σ

σ = 30, μ = 30

After finding the parameters, you may be asked to calculate a new probability, such as P(X < 75). A good strategy is to convert each condition into a linear equation in μ and σ before solving.

求出参数后，题目可能要求继续计算新概率，如 P(X < 75)。好的策略是先把每个条件转化为关于 μ 和 σ 的线性方程，再联立求解。

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

When n is large, a binomial distribution X ~ B(n, p) can be approximated by a normal distribution with the same mean and variance:

当 n 很大时，二项分布 X ~ B(n, p) 可以用具有相同均值和方差的正态分布来近似：

μ = np, σ² = np(1 − p)

The common rule is to use this approximation when np and n(1−p) are both greater than 5. Because the binomial is discrete and the normal is continuous, a continuity correction is essential.

一般规则是当 np 和 n(1−p) 都大于 5 时使用该近似。由于二项分布是离散的而正态分布是连续的，连续性修正必不可少。

9. Applying Continuity Corrections Correctly | 正确应用连续性修正

Let X ~ B(n, p). Suppose we approximate X using Y ~ N(np, np(1−p)). To convert a discrete inequality into a continuous one, adjust the boundary by 0.5:

设 X ~ B(n, p)，用 Y ~ N(np, np(1−p)) 近似。要把离散不等式转化为连续不等式，需要用 0.5 调整边界：

P(X ≤ k) becomes P(Y < k + 0.5).
P(X < k) becomes P(Y < k − 0.5). Since X is integer-valued, P(X < k) is the same as P(X ≤ k−1).
P(X ≥ k) becomes P(Y > k − 0.5).
P(X > k) becomes P(Y > k + 0.5).

Example: X ~ B(200, 0.3). Find P(X ≤ 70).

例：X ~ B(200, 0.3)，求 P(X ≤ 70)。

Here np = 60 and np(1−p) = 42, so both conditions are satisfied. Using continuity correction:

此处 np = 60，np(1−p) = 42，条件均满足。使用连续性修正：

P(X ≤ 70) ≈ P(Y < 70.5)

= P(Z < (70.5 − 60) / √42) = P(Z < 1.62) = 0.9474

The most common error is forgetting the correction when p is close to 0.5; always write the corrected inequality before standardising.

最常见的错误是当 p 接近 0.5 时忘了修正；标准化之前先把修正后的不等式写出来。

10. Common Traps in Exam Questions | 考试中的常见陷阱

Students frequently lose marks on normal distribution questions due to a few recurring mistakes. The following table summarises the most important pitfalls and how to avoid them.

学生在正态分布题目中常因几个重复出现的错误而失分。下表总结了最重要的陷阱及规避方法。

Mistake 陷阱	Why it happens 错误原因	How to avoid 规避方法
Using σ² instead of σ in the z-score formula	Confusing variance with standard deviation	Always take the square root of variance before dividing
Not applying continuity correction	Treating discrete approximation as exact	Adjust boundaries by ±0.5 before standardising
Looking up the wrong tail	Misreading P(X > x) as P(X < x)	Use 1 − P(X < x) for right-tail probabilities
Using the wrong table format	Not checking whether table gives P(Z < z) or P(0 < Z < z)	Check table heading and add 0.5 if needed

By actively identifying these traps during practice, you can build the exam awareness needed to avoid them under time pressure.

在练习中主动识别这些陷阱，能帮助你在时间压力下建立考场意识，从容避开它们。

11. Using a Calculator Efficiently | 高效使用计算器

Most A-Level approved calculators can compute normal probabilities directly using the Normal CDF function. For example, the Casio ClassWiz and Texas Instruments calculators accept inputs such as lower bound, upper bound, mean, and standard deviation.

大多数A-Level允许的计算器可以直接用正态累积分布函数计算概率。例如卡西欧ClassWiz和德州仪器计算器可以输入下限、上限、均值和标准差。

For P(X < 55) with X ~ N(50, 4²), set lower = −10⁹⁹, upper = 55, μ = 50, σ = 4. The result should be 0.8944. For inverse normal, input the left-tail area and the calculator returns the corresponding value.

对于 X ~ N(50, 4²) 求 P(X < 55)，设下限为 −10⁹⁹，上限为 55，μ = 50，σ = 4，结果应为 0.8944。使用逆正态功能时，输入左尾面积，计算器即返回对应值。

However, do not become too reliant on the calculator. In multi-stage questions, you must still demonstrate clear working to earn method marks. Moreover, in some questions the table provides an exact z-value that a calculator rounding might slightly alter, so learn both approaches.

但不要过度依赖计算器。在多步骤大题中，你仍然需要写出清晰的步骤来获得方法分。此外，某些题目中查表得到的 z 值与计算器的近似值可能略有不同，因此两种方法都要掌握。

12. Practice Summary and Final Tips | 练习总结与最终建议

To master normal distribution probabilities, follow this systematic routine:

要掌握正态分布概率求解，请遵循以下系统化步骤：

Identify the distribution type: normal, binomial approximated by normal, or inverse normal.
Write down μ, σ² and take σ correctly.
Draw a small sketch to visualise the required area.
Apply continuity correction when approximating a discrete distribution.
Standardise using Z = (X − μ)/σ.
Convert the probability into a form the table handles, using 1 − P(Z < z) for right tails.
For inverse problems, convert the probability to a left-tail area before looking up z.

Regular practice with past paper questions is the most effective way to become fluent with these steps. Pay close attention to the wording of each question: whether it involves a sample mean, a continuity correction, or the use of a permissive calculator function can alter your entire approach.

定期练习历年真题是熟练掌握这些步骤的最有效方法。仔细阅读每道题的措辞：题目是否涉及样本均值、是否需要连续性修正、能否使用计算器函数，这些都会影响你的解题策略。

With consistent practice and a clear step-by-step method, normal distribution questions become routine rather than intimidating.

通过持续练习和清晰的逐步方法，正态分布题目将不再令人生畏，而会变得驾轻就熟。

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A-Level Mathematics: Solving Normal Distribution Probabilities | A-Level数学：正态分布概率求解方法