IB Mathematics: Standard Normal Distribution Z-Table and Calculation | IB数学：标准正态分布Z值查表与计算

📚 IB Mathematics: Standard Normal Distribution Z-Table and Calculation | IB数学：标准正态分布Z值查表与计算

The standard normal distribution is one of the most important tools in IB Mathematics. It allows us to calculate probabilities and compare data from different normal distributions using the Z-table. Mastering this skill is essential for Paper 2 and the internal assessment.

标准正态分布是IB数学中最重要的工具之一。它让我们能够通过Z值表计算概率，并比较来自不同正态分布的数据。掌握这项技能对于Paper 2和内部评估至关重要。

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1. What is the Standard Normal Distribution? | 什么是标准正态分布？

A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is often denoted as N(0, 1). The total area under the curve is exactly 1, representing the total probability.

标准正态分布是均值为0、标准差为1的正态分布，通常记作N(0, 1)。曲线下的总面积为1，代表总概率。

The curve is symmetric about the mean, and its shape is determined entirely by the Z-score. Any normal distribution can be transformed into a standard normal distribution using the Z-score formula.

曲线关于均值对称，其形状完全由Z值决定。任何正态分布都可以通过Z值公式转换为标准正态分布。

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2. Definition of Z-Score | Z分数的定义

The Z-score measures how many standard deviations a data point is from the mean. The formula is:

Z分数衡量一个数据点距离均值多少个标准差。公式为：

Z = (X – μ) / σ

Here, X is the raw value, μ is the population mean, and σ is the population standard deviation. A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it is below the mean.

其中X是原始值，μ是总体均值，σ是总体标准差。正Z分数表示该值高于均值，负Z分数表示低于均值。

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3. Structure of the Z-Table | 标准正态分布表的结构

The Z-table typically gives the cumulative probability P(Z ≤ z) for z values between -3.49 and 3.49. The table has two dimensions: the row gives the integer part and the first decimal of z, while the column gives the second decimal.

Z值表通常给出z在-3.49到3.49之间的累积概率P(Z ≤ z)。表格有两个维度：行给出z的整数部分和第一位小数，列给出第二位小数。

For example, to find P(Z ≤ 1.23), locate the row labeled 1.2 and the column labeled 0.03. The value at the intersection is the cumulative probability. Most IB tables provide probabilities for positive z only; for negative z, use symmetry.

例如，要查找P(Z ≤ 1.23)，找到行标签1.2和列标签0.03。交点处的值就是累积概率。大多数IB表格只提供正z的概率；对于负z，需要利用对称性。

z	0.00	0.01	0.02	0.03
1.2	0.8849	0.8869	0.8888	0.8907

A sample Z-table extract is shown above. The value 0.8907 means that about 89.07% of the data lies below z = 1.23.

上表是Z值表的示例片段。0.8907表示约89.07%的数据位于z = 1.23以下。

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4. How to Use the Z-Table to Find Cumulative Probability | 如何使用Z表查找累积概率

To find P(Z ≤ z) for a positive z, simply read the table directly. For a negative z, use the symmetry property: P(Z ≤ -z) = 1 – P(Z ≤ z). This is because the total area is 1 and the curve is symmetric.

要查找正z的P(Z ≤ z)，直接查表即可。对于负z，利用对称性：P(Z ≤ -z) = 1 – P(Z ≤ z)。因为总面积为1且曲线对称。

For example, P(Z ≤ -1.23) = 1 – P(Z ≤ 1.23) = 1 – 0.8907 = 0.1093. This means 10.93% of the data lies below -1.23.

例如，P(Z ≤ -1.23) = 1 – P(Z ≤ 1.23) = 1 – 0.8907 = 0.1093。这意味着10.93%的数据位于-1.23以下。

• Always check whether the table gives P(Z ≤ z) or P(0 ≤ Z ≤ z). Some tables are cumulative, others are tail-probability.
• 务必确认表格给出的是P(Z ≤ z)还是P(0 ≤ Z ≤ z)。有些是累积表，有些是尾概率表。

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5. Finding Right-Tail Probabilities | 查找右侧尾概率

The right-tail probability is P(Z ≥ z). Since the total area is 1, P(Z ≥ z) = 1 – P(Z ≤ z). For z = 1.23, the right-tail probability is 1 – 0.8907 = 0.1093.

右侧尾概率是P(Z ≥ z)。由于总面积为1，P(Z ≥ z) = 1 – P(Z ≤ z)。对于z = 1.23，右侧尾概率为1 – 0.8907 = 0.1093。

This is often used in one-tailed hypothesis tests. In IB, you may be asked to find the area in the upper tail directly from the table, especially when dealing with significance levels like 5% or 1%.

这常用于单尾假设检验。在IB考试中，可能会要求你直接从表中找到上尾面积，特别是在处理5%或1%的显著性水平时。

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6. Calculating Interval Probabilities | 计算区间概率

For an interval probability P(a ≤ Z ≤ b), find P(Z ≤ b) and subtract P(Z ≤ a). For example, P(-0.5 ≤ Z ≤ 1.0) = P(Z ≤ 1.0) – P(Z ≤ -0.5).

对于区间概率P(a ≤ Z ≤ b)，先找到P(Z ≤ b)再减去P(Z ≤ a)。例如，P(-0.5 ≤ Z ≤ 1.0) = P(Z ≤ 1.0) – P(Z ≤ -0.5)。

Using the table, P(Z ≤ 1.0) = 0.8413 and P(Z ≤ -0.5) = 1 – P(Z ≤ 0.5) = 1 – 0.6915 = 0.3085. Therefore, the interval probability is 0.8413 – 0.3085 = 0.5328.

查表得P(Z ≤ 1.0) = 0.8413，P(Z ≤ -0.5) = 1 – P(Z ≤ 0.5) = 1 – 0.6915 = 0.3085。因此区间概率为0.8413 – 0.3085 = 0.5328。

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7. Inverse Z: Finding Z from Probability | 逆Z值：从概率找Z值

Sometimes you know the probability and need to find the corresponding Z-score. For example, if P(Z ≤ z) = 0.95, what is z? Look for 0.9500 inside the Z-table and read the corresponding row and column.

有时已知概率要求对应的Z分数。例如，若P(Z ≤ z) = 0.95，求z。在Z表内部查找0.9500，对应行和列即可。

The closest value to 0.9500 is 0.9505 at z = 1.65, or 0.9495 at z = 1.64. IB typically uses an inverse normal table or a calculator. On the calculator, use invNorm(0.95, 0, 1) to get z ≈ 1.6449.

最接近0.9500的值是0.9505（z = 1.65）或0.9495（z = 1.64）。IB通常使用逆正态表或计算器。在计算器上，使用invNorm(0.95, 0, 1)可得z ≈ 1.6449。

For a right-tail probability, such as P(Z ≥ z) = 0.05, note that P(Z ≤ z) = 0.95, so the same z is used. For two-tailed at α = 0.05, the critical z is 1.96.

对于右侧尾概率，如P(Z ≥ z) = 0.05，注意P(Z ≤ z) = 0.95，所以使用相同的z。对于α = 0.05的双尾检验，临界z为1.96。

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8. Common Pitfalls and Tips | 常见陷阱与技巧

One common mistake is forgetting to standardize before using the Z-table. Always convert X to Z first. Another is confusing P(Z ≤ z) with P(Z ≥ z). Draw a small sketch to clarify which tail you need.

一个常见错误是忘记查表前必须先标准化。总是先将X转换为Z。另一个是把P(Z ≤ z)和P(Z ≥ z)混淆。画个简图可以帮助确认需要哪一侧的尾部。

• Check if the table is one-sided or two-sided. IB might provide a cumulative table for positive z only.
• 检查表格是单侧还是双侧。IB可能只提供正z的累积表。
• For negative z, always use symmetry, never try to read a negative value directly unless the table includes it.
• 对于负z，务必使用对称性，除非表格包含负值，否则不要直接读取。
• When using a calculator, know the order of parameters: invNorm(area, mean, standard deviation).
• 使用计算器时，要了解参数顺序：invNorm(面积, 均值, 标准差)。

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9. Calculator vs Z-Table | 计算器与Z表对比

In the IB exam, you may use a GDC (Graphing Display Calculator) for normal distribution calculations. The GDC gives more precise results than the Z-table because the table is rounded to four decimal places.

在IB考试中，可以使用图形计算器（GDC）进行正态分布计算。GDC比Z表更精确，因为表格四舍五入到四位小数。

However, the Z-table is still tested in Paper 1 (non-calculator) for some schools. You should be comfortable with both methods. The table helps you understand the meaning of probabilities, while the calculator is faster for complex problems.

然而，一些学校在Paper 1（不允许计算器）中仍然会考Z表。你应该熟悉两种方法。表格帮助你理解概率的含义，而计算器在复杂问题中更快。

When using a GDC, you can find P(a ≤ X ≤ b) directly with normalcdf(a, b, μ, σ) on many models. But always define your variables clearly in the working.

使用GDC时，许多型号可以直接用normalcdf(a, b, μ, σ)计算P(a ≤ X ≤ b)。但答题时要清楚定义变量。

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10. Worked Examples | 例题演练

Example 1: Let X ~ N(50, 10²). Find P(X < 65). First, calculate Z = (65 - 50) / 10 = 1.5. Then P(Z < 1.5) = 0.9332 (from the table). So P(X < 65) = 0.9332.

例1：设X ~ N(50, 10²)。求P(X < 65)。首先计算Z = (65 - 50) / 10 = 1.5。然后查表P(Z < 1.5) = 0.9332。因此P(X < 65) = 0.9332。

Example 2: Let X ~ N(100, 15²). Find P(85 < X < 120). Compute Z₁ = (85 - 100)/15 = -1, Z₂ = (120 - 100)/15 = 1.333. Then P(-1 < Z < 1.333) = P(Z < 1.333) - P(Z < -1). From table, P(Z < 1.33) ≈ 0.9082 and P(Z < -1) = 0.1587. The answer is 0.9082 - 0.1587 = 0.7495.

例2：设X ~ N(100, 15²)。求P(85 < X < 120)。计算Z₁ = (85 - 100)/15 = -1，Z₂ = (120 - 100)/15 = 1.333。然后P(-1 < Z < 1.333) = P(Z < 1.333) - P(Z < -1)。查表P(Z < 1.33) ≈ 0.9082，P(Z < -1) = 0.1587。答案为0.9082 - 0.1587 = 0.7495。

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11. Exam Tips | 考试要点

Always write the Z-score formula before plugging in numbers. Show the standardization step clearly to earn method marks. When using the table, write the probabilities you read, not just the final answer.

务必先写出Z分数公式再代入数值。清楚展示标准化过程以获得方法分。使用表格时，写下你查到的概率，而不只写最终答案。

For inverse problems, set up the equation P(Z ≤ z) = p, then use the table or calculator. Draw a normal curve and shade the required region for every probability question.

对于逆问题，建立方程P(Z ≤ z) = p，然后查表或使用计算器。每个概率问题都要画出正态曲线并标出所需区域。

Remember that the normal distribution is continuous, so P(X < a) = P(X ≤ a). The inequalities do not change the probability.

记住正态分布是连续分布，所以P(X < a) = P(X ≤ a)。不等号不改变概率。

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12. Summary | 总结

The standard normal distribution and Z-table are foundational for IB probability and statistics. You must be able to standardize, read the table, find tail probabilities, and inverse values accurately.

标准正态分布和Z表是IB概率与统计的基础。你必须能够准确标准化、查表、找尾概率和求逆值。

Practice with both the table and your GDC to ensure flexibility in exams. Always sketch the curve, write the Z formula, and interpret your result in context.

同时用表格和GDC练习，确保考试时灵活应变。总是画出曲线，写出Z公式，并结合情境解释结果。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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