AS Mathematics: Normal Distribution – Key Points | AS 数学：正态分布考点精讲

📚 AS Mathematics: Normal Distribution – Key Points | AS 数学：正态分布考点精讲

The normal distribution is the cornerstone of continuous probability in AS Mathematics. It models countless natural phenomena – from heights and weights to measurement errors. Mastering its properties and probability calculations is essential for exam success, as questions often demand precise use of the standard normal table and the standardising formula.

正态分布是 AS 数学中连续概率的核心内容。它模拟了从身高、体重到测量误差等无数自然现象。掌握其性质与概率计算对考试成功至关重要，因为考题通常要求精准运用标准正态分布表以及标准化公式。

1. What is a Normal Distribution? | 什么是正态分布？

A normal distribution is a continuous probability distribution defined by its bell-shaped, symmetric curve. The random variable X can take any real value, and the total area under the curve equals 1. The probability of X lying in any interval is given by the area under the curve over that interval.

正态分布是一种连续型概率分布，以其钟形、对称的曲线为特征。随机变量 X 可取任意实数值，且曲线下方的总面积等于 1。X 落在任一区间内的概率等于该区间上曲线下方的面积。

The probability density function is f(x) = (1/(σ√(2π))) e^{–(x–μ)²/(2σ²)}, but in AS exams you will rarely need this explicit form. Instead, you rely on the shape determined by the mean μ and standard deviation σ.

其概率密度函数为 f(x) = (1/(σ√(2π))) e^{–(x–μ)²/(2σ²)}，但在 AS 考试中你几乎不需要使用这一显式形式，而是依赖于由均值 μ 和标准差 σ 所决定的曲线形状。

2. Parameters: μ and σ | 参数：μ 和 σ

The mean μ fixes the centre of the curve. Changing μ shifts the whole curve left or right without altering its spread. The standard deviation σ controls the width of the bell. A small σ gives a tall, narrow spike; a large σ gives a flat, wide mound.

均值 μ 确定了曲线的中心位置。改变 μ 会使整条曲线左右平移而不改变其分散程度。标准差 σ 控制钟形曲线的宽度。较小的 σ 产生高而窄的尖峰；较大的 σ 则形成扁平而宽阔的山丘状。

Notation: We write X ~ N(μ, σ²). The second parameter is the variance σ², not σ. A common mistake is to use σ instead of σ² when specifying the distribution or entering data into a calculator.

符号：记作 X ~ N(μ, σ²)。第二个参数是方差 σ²，而非 σ。一个常见错误是在定义分布或在计算器中输入数据时使用了 σ 而不是 σ²。

3. Key Properties of the Normal Curve | 正态曲线的关键性质

The normal curve is symmetric about the mean, μ. This implies that mean = median = mode. The total area under the curve is 1. The tails extend to infinity in both directions and never touch the horizontal axis.

正态曲线关于均值 μ 对称。这意味着均值 = 中位数 = 众数。曲线下方总面积为 1。尾部向正负无穷延伸且永不触及水平轴。

Empirical Rule (68–95–99.7): Approximately 68% of observations lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ. This helps to make quick probability estimates and check the reasonableness of calculations.

经验法则（68–95–99.7）：约 68% 的观测值落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。这有助于快速估计概率并核查计算结果的合理性。

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

The standard normal distribution has mean 0 and variance 1. Its probability density is denoted by φ(z) and the cumulative distribution function by Φ(z). The Φ(z) table gives P(Z ≤ z), the area to the left of z under the standard normal curve.

标准正态分布的均值为 0，方差为 1。其概率密度函数记为 φ(z)，累积分布函数记为 Φ(z)。Φ(z) 表给出了 P(Z ≤ z)，即标准正态曲线下位于 z 左侧的面积。

Most exam tables provide left-tail probabilities. You must check whether your table gives P(Z < z) or P(0 < Z < z). For AS work we assume the cumulative left-tail table, so P(Z < z) = Φ(z).

大多数考试用表提供左侧尾部概率。你必须查清自己的表格给出的是 P(Z < z) 还是 P(0 < Z < z)。对于 AS 内容，我们假定使用左侧累积表，即 P(Z < z) = Φ(z)。

5. Standardising: Z = (X – μ) / σ | 标准化：Z = (X – μ) / σ

Any normal variable X ~ N(μ, σ²) can be transformed to a standard normal Z ~ N(0,1) by subtracting μ and dividing by σ. The formula is Z = (X – μ)/σ. Then P(X < a) = P(Z < (a–μ)/σ).

任何正态变量 X ~ N(μ, σ²) 均可通过减去 μ 再除以 σ 转化为标准正态变量 Z ~ N(0,1)。公式为 Z = (X – μ)/σ。于是 P(X < a) = P(Z < (a–μ)/σ)。

This transformation is the gateway to all normal probability calculations. Be meticulous: divide by σ, not σ², and always subtract μ before dividing.

这一变换是所有正态概率计算的关键。务必细致：除以的是 σ 而非 σ²，且一定要先减去 μ 再除以 σ。

6. Using the Z-table for Probabilities | 使用 Z 表求概率

To find P(X < a) for X ~ N(μ, σ²): calculate z = (a – μ)/σ, then look up Φ(z) in the table. If z is negative, use symmetry: Φ(–z) = 1 – Φ(z). Always sketch a bell curve and shade the region to avoid errors.

求取 X ~ N(μ, σ²) 的 P(X < a) 时：计算 z = (a – μ)/σ，然后在表中查找 Φ(z)。若 z 为负数，利用对称性：Φ(–z) = 1 – Φ(z)。务必画出钟形曲线并标出区域以避免错误。

Example: X ~ N(100, 15²). Find P(X < 115). z = (115–100)/15 = 1.00. Φ(1.00) = 0.8413. So the probability is 0.8413.

示例：X ~ N(100, 15²)。求 P(X < 115)。z = (115–100)/15 = 1.00。Φ(1.00) = 0.8413。因此概率为 0.8413。

For P(X > a), use P(X > a) = 1 – P(X < a). For P(a < X < b), calculate P(X < b) – P(X < a). The continuity correction is not needed for continuous data.

对于 P(X > a)，使用 P(X > a) = 1 – P(X < a)。对于 P(a < X < b)，计算 P(X < b) – P(X < a)。连续数据无需进行连续性校正。

7. Finding Probabilities for Symmetric Intervals | 求对称区间的概率

Many exam questions ask for P(μ – kσ < X < μ + kσ). By standardising, this becomes P(–k < Z < k) = Φ(k) – Φ(–k) = 2Φ(k) – 1. This trick saves time and sidesteps arithmetic mistakes.

许多考题会要求计算 P(μ – kσ < X < μ + kσ)。通过标准化，这等价于 P(–k < Z < k) = Φ(k) – Φ(–k) = 2Φ(k) – 1。这一技巧能节省时间并避免算术错误。

For instance, within one standard deviation of the mean: k=1, probability = 2Φ(1)–1 ≈ 2×0.8413–1 = 0.6826, matching the 68% rule.

例如，在均值的一个标准差之内：k=1，概率 = 2Φ(1)–1 ≈ 2×0.8413–1 = 0.6826，与 68% 规则相符。

8. Inverse Normal: Finding X Given a Probability | 逆正态：已知概率求 X

When you know P(X < x₀) = p, first find z such that Φ(z) = p from the table. Then unstandardise: x₀ = μ + zσ. This gives the required percentile.

当已知 P(X < x₀) = p 时，先从表中找出使得 Φ(z) = p 的 z 值，然后反标准化：x₀ = μ + zσ。这就得到了所需的分位数。

Important z-values to memorise for typical significance levels are summarised below. They appear repeatedly in hypothesis testing and confidence intervals later in the course.

务必记住常见显著性水平对应的 z 值，汇列如下。它们将在后续课程的假设检验和置信区间中反复出现。

Key z-values:

Probability in left tail	z-value
0.90	1.2816
0.95	1.6449
0.975	1.9600
0.99	2.3263
0.995	2.5758

If the probability refers to a right-tail (e.g., the top 5%), restate the problem in terms of the left tail: the point x with 95% below it is the same as the point with 5% above.

若概率涉及右侧尾部（例如最高的 5%），则将问题用左侧尾部重新表述：下方有 95% 的点等同于上方有 5% 的点。

9. Calculator Use and Checking | 计算器的使用与检查

Many modern exam boards allow the use of calculators with built-in normal distribution functions. You can directly compute P(X < x) or inverse normal without standardising. However, you must still show sufficient working, such as stating the distribution and parameters, and often the standardised value z, to earn method marks.

许多现代考试局允许使用带有内置正态分布功能的计算器。你可以直接计算 P(X < x) 或逆正态而无需标准化。然而，你仍必须展示足够的解题步骤，例如说明分布与参数，并且通常还要写出标准化值 z，以获得方法分。

Always cross-check calculator results with a rough sketch and the 68–95–99.7 rule. If the probability looks impossible (e.g., negative or >1), you have likely confused σ and σ² or misapplied a table.

务必通过粗略草图和 68–95–99.7 法则交叉核对计算器结果。若概率看上去不可能（例如为负数或 >1），你很可能混淆了 σ 与 σ²，或错误地使用了表格。

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

Pitfall 1: Forgetting that P(X = a) = 0 for any continuous distribution. In normal distribution questions, always work with inequalities, not equalities.

陷阱 1：忘记对于任何连续分布均有 P(X = a) = 0。在正态分布题目中，始终要用不等式来处理，而不是等号。

Pitfall 2: Using σ² in the denominator of the Z formula. The correct standardised value is (x – μ)/σ. Double-check the variance and standard deviation given in the problem.

陷阱 2：在 Z 公式的分母中误用 σ²。正确的标准化值是 (x – μ)/σ。务必再次检查题目给出的方差和标准差。

Pitfall 3: Misreading the table. If your table gives P(0 < Z < z), you must add 0.5 to get the left-tail probability. In AS exams, cumulative left-tail tables are more common, but confirm with your exam board.

陷阱 3：误读表格。若你的表格提供的是 P(0 < Z < z)，则必须加上 0.5 才能得到左侧尾部概率。在 AS 考试中，左尾累积表更为常见，但仍需根据你所属考试局确认。

Exam tip: Always draw a clearly labelled bell curve with the mean, relevant x-values, and shaded area. Even if you use a calculator, the sketch demonstrates understanding and can earn marks if the final number is slightly off.

应试技巧：始终画出一个标注清晰的钟形曲线，标明均值、相关 x 值以及阴影面积。即使你使用计算器，草图也能体现你的理解，并在最终数字稍有偏差时赢得分数。

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AS Mathematics: Normal Distribution – Key Points | AS 数学：正态分布 考点精讲