📚 The Normal Distribution | 正态分布
The normal distribution is one of the most important continuous probability distributions in A-Level Mathematics. It models variables such as heights, weights, measurement errors and exam marks, where most observations cluster around a central mean and extreme values become less likely. Understanding its shape, parameters and standardisation process is essential for Edexcel Statistics questions.
正态分布是 A-Level 数学中最重要的连续概率分布之一。它用于描述身高、体重、测量误差和考试成绩等变量，这些数据大多集中在均值附近，极端值出现的概率较低。理解正态曲线的形状、参数以及标准化过程是解答 Edexcel 统计题的关键。
1. Definition and Continuous Nature | 定义与连续性
A continuous random variable X follows a normal distribution with mean μ and variance σ² if its probability density function is symmetric, bell-shaped and defined for all real x. We write X ~ N(μ, σ²). Note that the second parameter is variance, not standard deviation.
如果连续随机变量 X 服从均值为 μ、方差为 σ² 的正态分布，则其概率密度函数呈对称钟形，且对所有实数 x 有定义。我们记作 X ~ N(μ, σ²)。请注意，第二个参数是方差，而不是标准差。
f(x) = 1/(σ√(2π)) × e−½((x−μ)/σ)²
The total area under the probability density curve is exactly 1, because the normal distribution is a continuous probability distribution. This means the curve never gives a probability at a single point; instead, probabilities are found as areas under the curve over an interval.
概率密度曲线下的总面积恰好为 1，因为正态分布是一种连续概率分布。这意味着曲线在单点处并不给出概率；相反，概率表示为一个区间内曲线下的面积。
2. Shape and Symmetry | 形状与对称性
The normal curve is perfectly symmetrical about the vertical line x = μ. This implies P(X < μ) = P(X > μ) = 0.5. The curve has a single peak at the mean, and its tails extend indefinitely in both directions but never touch the horizontal axis.
正态曲线关于直线 x = μ 完全对称。因此 P(X < μ) = P(X > μ) = 0.5。曲线在均值处只有一个峰值，两端尾部无限延伸但永远不会接触横轴。
- Mean = median = mode for a normal distribution. | 正态分布的均值、中位数和众数相等。
- The total area under the curve equals 1. | 曲线下总面积为 1。
- The x-axis is a horizontal asymptote. | x 轴是水平渐近线。
Because of symmetry, probabilities on either side of the mean mirror each other. For example, if P(X < a) = p, then P(X > 2μ − a) = p as well. This property often allows exam questions to be simplified.
由于对称性，均值两侧的概率互为镜像。例如，如果 P(X < a) = p，那么 P(X > 2μ − a) 也等于 p。这一性质常常可以简化考题。
3. Parameters μ and σ | 参数 μ 与 σ
The mean μ determines the centre of the distribution. Changing μ shifts the entire curve left or right without changing its shape. The standard deviation σ controls the spread; a larger σ gives a flatter and wider curve, while a smaller σ gives a taller and narrower curve.
均值 μ 决定了分布的中心位置。改变 μ 只会使整条曲线左右平移，而不会改变形状。标准差 σ 控制离散程度；σ 越大，曲线越平坦宽阔；σ 越小，曲线越高窄。
In Edexcel notation, X ~ N(μ, σ²) always shows the variance as the second parameter. If a question says X ~ N(100, 225), then μ = 100 and σ² = 225, so σ = 15. You must identify this correctly before using a calculator.
在 Edexcel 记号中，X ~ N(μ, σ²) 始终用方差作为第二个参数。如果题目给出 X ~ N(100, 225)，则 μ = 100，σ² = 225，因此 σ = 15。在使用计算器之前必须正确识别这一点。
4. The 68-95-99.7 Rule | 68-95-99.7 规则
For any normal distribution, approximately 68% of observations lie within one standard deviation of the mean, 95% lie within two standard deviations, and 99.7% lie within three standard deviations. These percentages are useful for quick estimates but exact probabilities should be found using a calculator or tables.
对于任何正态分布，大约 68% 的观测值落在均值的一个标准差范围内，95% 落在两个标准差范围内，99.7% 落在三个标准差范围内。这些百分比可用于快速估计，但精确概率应使用计算器或表格求出。
| Interval | Approximate probability |
| μ ± 1σ | 0.68 |
| μ ± 2σ | 0.95 |
| μ ± 3σ | 0.997 |
This rule is often tested in multiple-choice or data-interpretation questions where a quick judgement about outliers or usual values is needed.
该规则常用于选择题或数据解释题，需要快速判断异常值或正常取值范围。
5. Standardising to Z | 标准化为 Z
To find probabilities for any normal variable X ~ N(μ, σ²), we convert X to the standard normal variable Z using Z = (X − μ)/σ. The standard normal distribution has mean 0 and variance 1, written Z ~ N(0, 1). Standardisation allows all normal probabilities to be compared on the same scale.
为了求任意正态变量 X ~ N(μ, σ²) 的概率，我们使用 Z = (X − μ)/σ 将 X 转换为标准正态变量 Z。标准正态分布的均值为 0，方差为 1，记作 Z ~ N(0, 1)。标准化使得所有正态概率可以在同一尺度上比较。
Z = (X − μ) / σ
For example, if X ~ N(70, 25) and you want P(X < 74), first find σ = 5. Then Z = (74 − 70)/5 = 0.8, so P(X < 74) = P(Z < 0.8). This is the value you look up or obtain from the calculator.
例如，若 X ~ N(70, 25)，要求 P(X < 74)，首先得到 σ = 5。然后 Z = (74 − 70)/5 = 0.8，因此 P(X < 74) = P(Z < 0.8)。这就是你要查表或从计算器获得的值。
6. Calculating Probabilities | 计算概率
Edexcel questions often ask for P(X < a), P(X > a) or P(a < X < b). Because the normal distribution is continuous, P(X < a) = P(X ≤ a) and P(X > a) = P(X ≥ a). Use the symmetry property and the complementary rule P(Z > z) = 1 − P(Z < z) when necessary.
Edexcel 题目常要求计算 P(X < a)、P(X > a) 或 P(a < X < b)。由于正态分布是连续分布，P(X < a) = P(X ≤ a)，并且 P(X > a) = P(X ≥ a)。必要时使用对称性和互补规则 P(Z > z) = 1 − P(Z < z)。
On most approved calculators, you can enter the lower bound, upper bound, μ and σ directly to find P(a < X < b). For one-tailed probabilities, use a very small lower bound such as −10⁹ or a very large upper bound such as 10⁹ to approximate the infinite tail.
在大多数考试允许的计算器上，你可以直接输入下界、上界、μ 和 σ 来求 P(a < X < b)。对于单尾概率，可使用一个很小的下界（如 −10⁹）或很大的上界（如 10⁹）来近似无限尾部。
- P(X < a): lower = −10⁹, upper = a. | P(X < a)：下界 −10⁹，上界 a。
- P(X > a): lower = a, upper = 10⁹. | P(X > a)：下界 a，上界 10⁹。
- P(a < X < b): lower = a, upper = b. | P(a < X < b)：下界 a，上界 b。
7. Inverse Normal Calculations | 正态反查
If a probability is given and you need to find the corresponding boundary value, use the inverse normal function on your calculator. For example, to find k such that P(X < k) = 0.95, enter area 0.95, mean μ and standard deviation σ into the inverse normal function.
如果已知概率，需要求对应的临界值，可使用计算器上的正态反查功能。例如，要求满足 P(X < k) = 0.95 的 k，在反查功能中输入面积 0.95、均值 μ 和标准差 σ 即可。
For right-tail problems, convert the condition first. If P(X > k) = 0.05, then P(X < k) = 0.95. If X ~ N(100, 15²), then k ≈ 124.7. This means only the top 5% of values are above 124.7.
对于右尾问题，要先转化条件。如果 P(X > k) = 0.05，则 P(X < k) = 0.95。若 X ~ N(100, 15²)，则 k ≈ 124.7。这意味着只有最高的 5% 数值大于 124.7。
The inverse normal function always works with left-tail cumulative probabilities. Writing the correct left-tail probability is the most common skill required in exam questions.
正态反查功能始终使用左尾累积概率。写出正确的左尾概率是考试题中最常考查的技能。
8. Finding Unknown μ and σ | 求解未知 μ 和 σ
Some exam questions provide two probability statements and ask you to find unknown μ and σ. Write each statement in standardised form, for example P(X < a) = p becomes P(Z < (a−μ)/σ) = p. Then use the inverse normal Z-value and solve the simultaneous equations.
有些考题会给出两个概率条件，要求求解未知的 μ 和 σ。将每个条件写成标准化形式，例如 P(X < a) = p 变为 P(Z < (a−μ)/σ) = p。然后使用反查得到的 Z 值，建立并解联立方程。
Suppose X ~ N(μ, σ²), P(X < 50) = 0.2 and P(X < 60) = 0.7. The corresponding Z-values are approximately −0.8416 and 0.5244. This gives the simultaneous equations 50 = μ − 0.8416σ and 60 = μ + 0.5244σ. Solving gives σ ≈ 7.32 and μ ≈ 56.16.
假设 X ~ N(μ, σ²)，P(X < 50) = 0.2 且 P(X < 60) = 0.7。对应的 Z 值约为 −0.8416 和 0.5244。由此得到联立方程 50 = μ − 0.8416σ 和 60 = μ + 0.5244σ。解得 σ ≈ 7.32，μ ≈ 56.16。
Always keep the Z-values to at least four significant figures during calculation. Rounding too early can produce inaccurate final answers, especially when the equations are close together.
计算过程中 Z 值至少保留四位有效数字。过早舍入可能导致最终答案不准确，尤其是当两个方程比较接近时。
9. Normal Approximation to 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). The usual Edexcel condition is that both np and nq are greater than 5 (some texts use 10).
当 n 较大且 p 不太接近 0 或 1 时，二项分布 B(n, p) 可用均值为 np、方差为 np(1−p) 的正态分布近似。Edexcel 通常要求 np 和 nq 都大于 5（有些教材使用 10）。
If X ~ B(n, p) and np > 5, nq > 5, then X ~ N(np, npq) approximately.
Here q = 1 − p. This approximation is useful when n is too large for exact binomial calculations. In exam questions you should state the approximation, justify the conditions, and then apply a continuity correction.
这里 q = 1 − p。当 n 太大而难以进行精确的二项计算时，这种近似非常有用。在考试题中，你应写出近似、验证条件，然后进行连续性校正。
10. Continuity Correction | 连续性校正
Since binomial is discrete and normal is continuous, we apply a continuity correction. Replace P(X = a) with P(a−0.5 < X < a+0.5), P(X ≤ a) with P(X < a+0.5), and P(X ≥ a) with P(X > a−0.5) in the normal approximation.
由于二项分布是离散的，而正态分布是连续的，我们需要进行连续性校正。在正态近似中，用 P(a−0.5 < X < a+0.5) 代替 P(X = a)，用 P(X < a+0.5) 代替 P(X ≤ a)，用 P(X > a−0.5) 代替 P(X ≥ a)。
| Discrete event | Normal approximation |
| P(X = a) | P(a−0.5 < X < a+0.5) |
| P(X ≤ a) | P(X < a+0.5) |
| P(X ≥ a) | P(X > a−0.5) |
| P(X < a) | P(X < a−0.5) |
| P(X > a) | P(X > a+0.5) |
Forgetting the continuity correction is a common error. Even if the question does not explicitly say ‘use a normal approximation’, you must include the correction whenever a discrete binomial is approximated by a continuous normal distribution.
忘记连续性校正是常见错误。即使题目没有明确说“使用正态近似”，只要用连续的正态分布近似离散的二项分布，就必须进行校正。
11. Exam Tips and Common Errors | 考试技巧与常见错误
Always check whether the given parameter is variance or standard deviation. If X ~ N(100, 225), then σ² = 225 and σ = 15. Many students incorrectly use 225 as σ. Also remember that the calculator needs σ, not σ², for normal and inverse normal functions.
务必检查题目给出的参数是方差还是标准差。如果 X ~ N(100, 225)，则 σ² = 225，σ = 15。许多学生错误地将 225 当作 σ。此外，计算器中的正态分布和反查功能需要输入 σ，而不是 σ²。
- Do not confuse P(X < a) with P(X ≤ a); they are equal for continuous variables, but the notation still matters in binomial approximations. | 不要混淆 P(X < a) 和 P(X ≤ a)；对于连续变量它们相等，但在二项近似中记号仍然重要。
- State the standardisation formula clearly before using the calculator. | 在使用计算器前要清晰写出标准化公式。
- Sketch the normal curve to see whether you need a left tail, right tail or symmetrical area. | 画出正态曲线草图，判断需要左尾、右尾还是对称面积。
- For unknown μ and σ problems, keep at least four significant figures in Z-values. | 求解未知 μ 和 σ
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