📚 Percentage Points of the Normal Distribution | 正态分布的百分位点
The normal distribution is one of the most important continuous probability distributions in A-Level mathematics. Understanding how to find percentage points (also called quantiles or critical values) is essential for solving problems involving confidence intervals, hypothesis tests, and probability calculations.
正态分布是A-Level数学中最重要的连续概率分布之一。理解如何求百分位点（也称分位数或临界值）对于解决置信区间、假设检验和概率计算问题至关重要。
1. What Are Percentage Points? | 什么是百分位点？
For a continuous random variable X with probability density function f(x), the percentage point with probability p is the value xₚ such that P(X ≤ xₚ) = p. In other words, it is the value below which a specified percentage of the distribution lies.
对于具有概率密度函数 f(x) 的连续随机变量 X，概率为 p 的百分位点是一个值 xₚ，使得 P(X ≤ xₚ) = p。换句话说，它是分布中指定百分比的数据所落于其下的那个值。
For example, the 90th percentile is the value such that 90% of the data is below it. This is often written as x₀.₉₀ or P₉₀.
例如，第90百分位数就是使得90%的数据位于其下方的值，通常记为 x₀.₉₀ 或 P₉₀。
When the distribution is normal, percentage points are found by working backwards from a known probability using the standard normal distribution table or an inverse normal function on a calculator.
当分布为正态分布时，百分位点通过已知概率反向查找标准正态分布表或使用计算器上的反正态函数来求得。
2. The Standard Normal Distribution | 标准正态分布
The standard normal distribution has mean μ = 0 and standard deviation σ = 1. Its random variable is usually denoted Z. The cumulative distribution function Φ(z) = P(Z ≤ z) gives the area to the left of z.
标准正态分布的均值 μ = 0，标准差 σ = 1，其随机变量通常记为 Z。累积分布函数 Φ(z) = P(Z ≤ z) 表示 z 左侧的面积。
Any normal distribution X ~ N(μ, σ²) can be transformed to the standard normal distribution using z = (X − μ) / σ. Thus percentage points for any normal distribution can be found from those of Z.
任何正态分布 X ~ N(μ, σ²) 都可以通过 z = (X − μ) / σ 转换为标准正态分布。因此，任何正态分布的百分位点都可从 Z 的百分位点求出。
z = (x − μ) / σ
This transformation is crucial because it allows us to use one table for all normal distributions.
这一转换至关重要，因为它允许我们使用同一张表来处理所有正态分布。
3. Notation and Inverse Probability | 记号与逆概率
We define zₚ as the value such that P(Z ≤ zₚ) = p. For example, z₀.₉₅ is the 95th percentile of the standard normal distribution, meaning that 95% of the distribution lies below z₀.₉₅.
我们定义 zₚ 为满足 P(Z ≤ zₚ) = p 的值。例如，z₀.₉₅ 是标准正态分布的第95百分位数，表示分布的95%位于 z₀.₉₅ 之下。
In statistics, the notation z(α) is often used for the upper-tail percentage point: P(Z > z(α)) = α. Equivalently, z(α) = z₁₋ₐ.
在统计学中，记号 z(α) 常用于上尾百分位点：P(Z > z(α)) = α。等价地，z(α) = z₁₋ₐ。
Common values include z(0.05) = 1.6449, z(0.025) = 1.9600, and z(0.005) = 2.5758. These are widely used in confidence interval construction.
常用值包括 z(0.05) = 1.6449、z(0.025) = 1.9600 和 z(0.005) = 2.5758。这些值广泛用于构造置信区间。
4. Finding Percentage Points from the Table | 从正态分布表查百分位点
Most Edexcel formula booklets provide a table of the standard normal cumulative distribution function Φ(z) for z from 0 to a certain value. To find zₚ, you need to look up the probability p in the body of the table and find the corresponding z value.
大多数Edexcel公式册提供了标准正态累积分布函数 Φ(z) 的表格，z 通常从0开始到某个值。要找到 zₚ，你需要在表的主体中查找概率 p，并找到对应的 z 值。
However, the table usually only gives values of Φ(z) for z ≥ 0, i.e., probabilities greater than 0.5. For a left-tail probability less than 0.5, use the symmetry property: Φ(−z) = 1 − Φ(z).
但是，表通常只给出 z ≥ 0 时的 Φ(z) 值，即概率大于0.5。若左尾概率小于0.5，利用对称性质：Φ(−z) = 1 − Φ(z)。
For example, to find z₀.₀₂₅, note that Φ(z₀.₀₂₅) = 0.025. Since this is less than 0.5, we find the value such that Φ(z) = 0.975, which is z = 1.96. Therefore z₀.₀₂₅ = −1.96.
例如，求 z₀.₀₂₅，注意 Φ(z₀.₀₂₅) = 0.025。由于它小于0.5，我们找到满足 Φ(z) = 0.975 的 z 值，即 z = 1.96。因此 z₀.₀₂₅ = −1.96。
Φ(−z) = 1 − Φ(z)
Remember that the table gives the area to the left of z. If you need the upper-tail percentage point, you convert the upper tail probability α to the left-tail probability 1 − α and look that up.
请记住，表给出的是 z 左侧的面积。如果你需要上尾百分位点，将上尾概率 α 转换为左尾概率 1 − α，然后查找该值。
5. Common Percentage Points | 常用百分位点
The following table lists frequently used percentage points of the standard normal distribution. These values should be memorised or quickly retrieved from your formula book.
下表列出了标准正态分布中经常使用的百分位点。这些值应该牢记，或能快速从公式册中查到。
| β = P(Z ≤ zβ) | α = P(Z > zβ) | zβ |
| 0.900 | 0.100 | 1.2816 |
| 0.950 | 0.050 | 1.6449 |
| 0.975 | 0.025 | 1.9600 |
| 0.990 | 0.010 | 2.3263 |
| 0.995 | 0.005 | 2.5758 |
Note that z₀.₉₇₅ = 1.9600 is the celebrated critical value for a two-tailed 5% significance test. A handy approximation is 1.96 for 97.5%.
注意 z₀.₉₇₅ = 1.9600 是双尾5%显著性检验中著名的临界值。常用近似值为 1.96。
6. Using a Calculator: Inverse Normal | 使用计算器：反正态函数
Modern calculators (such as the Casio fx-991EX) have an inverse normal function. In the distribution menu, choose “Inverse Normal” and enter the tail probability and the parameters of the normal distribution.
现代计算器（如 Casio fx-991EX）具有反正态函数。在分布菜单中选择“Inverse Normal”（反正态），然后输入尾部概率和正态分布的参数。
For a left-tail probability p, with mean μ and standard deviation σ, the calculator returns x such that P(X ≤ x) = p. For a right-tail probability, you may need to select “right tail” or enter 1 − p in the left-tail mode.
对于左尾概率 p，均值 μ 和标准差 σ，计算器返回满足 P(X ≤ x) = p 的 x。对于右尾概率，你可能需要选择“右尾”或在左尾模式下输入 1 − p。
Always check that your answer makes sense. If μ = 0 and σ = 1, then the inverse normal for p = 0.975 should give approximately 1.96.
务必检查答案是否合理。如果 μ = 0，σ = 1，则查 p = 0.975 的反正态应约为 1.96。
When using a calculator, set the tail setting correctly: “left” means P(X ≤ x), “right” means P(X ≥ x), and “central” gives a symmetric interval centred on μ.
使用计算器时，请正确设置尾部：“左尾”表示 P(X ≤ x)，“右尾”表示 P(X ≥ x)，“中央”给出以 μ 为中心的对称区间。
7. Converting to a General Normal Distribution | 转换到一般正态分布
For a random variable X ~ N(μ, σ²), the percentage point xₚ corresponding to a given cumulative probability p is found using:
对于随机变量 X ~ N(μ, σ²)，对应于给定累积概率 p 的百分位点 xₚ 可通过下式求得：
xₚ = μ + zₚ × σ
where zₚ is the corresponding standard normal percentage point. This formula is derived by solving zₚ = (xₚ − μ) / σ for xₚ.
其中 zₚ 是相应的标准正态百分位点。该公式由 zₚ = (xₚ − μ) / σ 对 xₚ 求解得出。
Example: Suppose X ~ N(100, 15²). Find the 90th percentile. Here z₀.₉₀ = 1.2816, so x₀.₉₀ = 100 + 1.2816 × 15 = 119.224 ≈ 119.2.
例：设 X ~ N(100, 15²)。求第90百分位数。这里 z₀.₉₀ = 1.2816，因此 x₀.₉₀ = 100 + 1.2816 × 15 = 119.224 ≈ 119.2。
This procedure is essential when constructing tolerance intervals or deciding cut-off values in quality control.
此过程在构造容差区间或在质量控制中决定截止值时至关重要。
8. Upper-Tail and Lower-Tail Percentage Points | 上尾与下尾百分位点
In hypothesis testing, we often need the critical value that separates the rejection region from the non-rejection region. For a one-tailed test at significance level α, the critical value is the upper-tail percentage point z(α) such that P(Z > z(α)) = α.
在假设检验中，我们常常需要将拒绝域与接受域分开的临界值。对于显著性水平为 α 的单尾检验，临界值是上尾百分位点 z(α)，满足 P(Z > z(α)) = α。
For a two-tailed test at significance level α, each tail has area α/2. The critical values are therefore −z(α/2) and z(α/2). The total rejection area is α, with α/2 in each tail.
对于显著性水平为 α 的双尾检验，每个尾部的面积为 α/2。因此临界值为 −z(α/2) 和 z(α/2)。拒绝域总面积为 α，每个尾部各 α/2。
For example, a two-tailed test at α = 0.05 uses z(0.025) = 1.96, so the rejection regions are Z < −1.96 or Z > 1.96.
例如，α = 0.05 的雙尾檢驗使用 z(0.025) = 1.96，因此拒絕域為 Z < −1.96 或 Z > 1.96。
P(Z ≤ −1.96) + P(Z ≥ 1.96) = 0.025 + 0.025 = 0.05
Always identify whether the test is one-tailed or two-tailed before selecting the critical value.
在选择临界值之前，务必判断检验是单尾还是双尾。
9. Worked Example | 例题详解
A machine fills packets of crisps with masses normally distributed with mean μ = 50 g and standard deviation σ = 2 g. The manufacturer wants to guarantee that at most 2% of packets are underweight. What mass should be printed as the minimum?
一台机器分装薯片，质量服从正态分布，均值 μ = 50 g，标准差 σ = 2 g。生产商想要保证最多有2%的包装重量不足。应印刷的最低质量是多少？
We need the 2nd percentile x₀.₀₂, since P(X < x₀.₀₂) = 0.02. From the standard normal table, Φ(z) = 0.02 implies z = −2.0537 (since Φ(−2.0537) = 0.02). Then:
我们需要第2百分位数 x₀.₀₂，因为 P(X < x₀.₀₂) = 0.02。查标准正态表，Φ(z) = 0.02 意味着 z = −2.0537（因为 Φ(−2.0537) = 0.02）。于是：
x₀.₀₂ = μ + z × σ = 50 + (−2.0537)(2) = 50 − 4.1074 = 45.8926 g
Therefore the minimum mass should be printed as 45.9 g to ensure no more than 2% of packets fall below this value.
因此，应将最低质量印刷为 45.9 g，以确保不超过2%的包装低于此值。
This example demonstrates how percentage points are used directly in real-world decision making.
这个例子展示了百分位点如何直接用于现实世界的决策。
10. Common Mistakes and Tips | 常见错误与提示
Mistake 1: Using the wrong tail. Always check whether the question asks for the area to the left, right, or between two values. Drawing a graph of the normal curve helps.
错误一：用错尾部。始终检查题目要求的是左侧、右侧还是两个值之间的面积。画出正态曲线图会有所帮助。
Mistake 2: Forgetting to convert a right-tail probability to a left-tail one when using a left-tail table. Remember: if P(Z > z) = α, then P(Z < z) = 1 − α.
错误二：使用左尾表时忘记将右尾概率转换为左尾概率。请记住：若 P(Z > z) = α，则 P(Z < z) = 1 − α。
Mistake 3: Confusing σ and σ². In most questions the variance is given; take the square root to obtain the standard deviation before using the formula.
错误三：混淆 σ 与 σ²。在多数题目中给出的是方差；使用公式前需开平方得到标准差。
Tip: Always write down the probability statement (e.g., P(Z ≤ z) = 0.95) before using a table or calculator. This avoids sign errors.
提示：在使用表或计算器之前，先写出概率语句（例如 P(Z ≤ z) = 0.95）。这样可以避免符号错误。
Tip: For the Edexcel exam, you are expected to know the standard normal table and how to interpolate when a probability lies between two tabulated values. Linear interpolation is usually sufficient.
提示：在Edexcel考试中，你需要会查标准正态表，并且当概率介于表中两个值之间时能够进行插值。线性插值通常就足够了。
11. Relationship with Confidence Intervals | 与置信区间的关系
Confidence intervals for a population mean use the percentage points of the normal distribution. For a 95% confidence interval, the critical value is z₀.₉₇₅ = 1.96. The interval is given by:
总体均值的置信区间使用正态分布的百分位点。对于95%置信区间，临界值为 z₀.₉₇₅ = 1.96。区间由下式给出：
x̄ ± z₀.₉₇₅ × (σ / √n)
Here x̄ is the sample mean, σ is the population standard deviation (if known), and n is the sample size.
其中 x̄ 是样本均值，σ 是总体标准差（若已知），n 是样本容量。
For other confidence levels, replace 1.96 with the appropriate percentage point (e.g., 1.645 for 90%, 2.576 for 99%).
对于其他置信水平，将1.96替换为相应的百分位点（例如90%用1.645，99%用2.576）。
Understanding that a 95% interval captures 95% of sample means in repeated sampling, not that it has a 95% chance of containing the parameter, is a common interpretation pitfall.
理解95%区间在重复抽样中能捕获95%的样本均值，而不是有95%的概率包含参数，是一个常见的解释陷阱。
12. Summary | 总结
Percentage points of the normal distribution are essential tools in statistical inference. For the standard normal distribution, zₚ satisfies P(Z ≤ zₚ) = p. For a general normal distribution, xₚ = μ + zₚ × σ.
正态分布的百分位点是统计推断中的基本工具。对于标准正态分布，zₚ 满足 P(Z ≤ zₚ) = p。对于一般正态分布，xₚ = μ + zₚ × σ。
You should be able to find percentage points from both statistical tables and using calculator functions. Remember the symmetry property, distinguish between one-tailed and two-tailed critical values, and practise with real examples.
你应该能够从统计表和计算器函数中求出百分位点。记住对称性质，区分单尾和双尾临界值，并通过实际例子进行练习。
Mastering these concepts will give you a strong foundation for hypothesis testing, confidence intervals, and more advanced statistical methods in your further studies.
掌握这些概念将为你在后续学习中的假设检验、置信区间以及更高级的统计方法打下坚实基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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