Year 12 CIE Statistics: Formula & Theorem Quick Reference Handbook

📚 Year 12 CIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This quick reference handbook covers all essential formulas and theorems for the CIE AS Mathematics Probability & Statistics 1 (9709) syllabus, designed for Year 12 students. It is organised by topic to facilitate rapid revision before exams.

本速查手册涵盖了CIE AS数学概率统计1（9709）课程中Year 12学生需要掌握的所有核心公式和定理，按主题编排，方便学生在考前快速复习。

1. Representing Data | 数据表示

Data can be displayed through stem-and-leaf diagrams, where each value is split into a stem and a leaf. A key must be provided to show the place value. Back-to-back stem-and-leaf diagrams compare two datasets.

数据可用茎叶图表示，每个数值分成茎和叶两部分，并附上图例说明数位。背靠背茎叶图可用于比较两组数据。

Box-and-whisker plots use the five‑number summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. Outliers are typically defined as values more than 1.5 × IQR below Q₁ or above Q₃.

箱线图利用五数概括：最小值、下四分位数(Q₁)、中位数(Q₂)、上四分位数(Q₃)和最大值。异常值通常定义为低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR 的数值。

Histograms are used for continuous grouped data. The area of each bar is proportional to frequency. The vertical axis is frequency density, calculated as frequency density = frequency ÷ class width. There are no gaps between bars, and class boundaries are used to ensure continuity.

直方图用于连续分组数据，每个柱形的面积与频数成正比。纵轴为 频率密度，计算公式：频率密度 = 频数 ÷ 组距。柱形之间无间隙，需使用组界以保证连续性。

Cumulative frequency graphs plot the running total of frequencies against the upper class boundaries. The curve is used to estimate the median, quartiles and percentiles. When connecting points, a smooth curve or straight lines joining points may be used as instructed.

累积频率图将累积频数对组上界描点。曲线用于估计中位数、四分位数和百分位数。连线时可按题目要求使用光滑曲线或用直线连接各点。

2. Measures of Central Tendency | 集中趋势度量

The mean of a set of n observations x₁, x₂, …, xₙ is x̄ = (Σx) / n. For grouped data with frequencies f, the mean is x̄ = (Σfx) / Σf, using class midpoints as x-values.

n 个观测值 x₁, x₂, …, xₙ 的均值为 x̄ = (Σx) / n。对于分组数据，使用组中值作为 x，均值为 x̄ = (Σfx) / Σf。

The median is the middle value when data are ordered. For n observations, the median is at position (n+1)/2. In a cumulative frequency diagram, the median is read at the 50th percentile.

中位数 是排序后位于中间的值。对于 n 个数据，中位数的位置为 (n+1)/2。在累积频率图中，可直接从 50% 处读取中位数。

The mode is the most frequent value. For grouped data, the modal class is the class with the highest frequency density.

众数是出现次数最多的值。对于分组数据，频率密度最高的组即为众数组。

When using coded data, if y = (x – a)/b, then x̄ = a + b·ȳ. Coding simplifies arithmetic but leaves the shape of the distribution unchanged.

若使用编码数据 y = (x – a)/b，则 x̄ = a + b·ȳ。编码可简化计算，且不改变分布的形状。

3. Measures of Variation | 离散程度度量

The range = maximum – minimum. The interquartile range (IQR) = Q₃ – Q₁. Both are measures of spread; the IQR is resistant to outliers.

极差 = 最大值 – 最小值。四分位距(IQR) = Q₃ – Q₁。两者均衡量离散程度，其中 IQR 不受异常值影响。

The variance of a population of N values is σ² = (Σ(x – μ)²) / N. For a sample of n observations, the standard notation in CIE S1 is s² = (Σ(x – x̄)²) / n (treating the data as the population). The standard deviation s is the positive square root of the variance.

N 个值的总体方差为 σ² = (Σ(x – μ)²) / N。对于 n 个观测值的样本，CIE S1 中通常将数据视为总体，使用 s² = (Σ(x – x̄)²) / n。标准差 s 是方差的正平方根。

A computationally efficient formula for variance is variance = (Σx²)/n – x̄². For grouped data, variance = (Σfx²)/Σf – x̄², where x denotes the class midpoint.

方差的一个简便计算公式为 方差 = (Σx²)/n – x̄²。对于分组数据，方差 = (Σfx²)/Σf – x̄²，其中 x 为组中值。

If coded data y = (x – a)/b are used, then sₓ² = b²·sᵧ² and sₓ = |b|·sᵧ. The mean and standard deviation can thus be recovered from the coded values.

若使用编码 y = (x – a)/b，则 sₓ² = b²·sᵧ² 且 sₓ = |b|·sᵧ。可以从编码值还原均值与标准差。

The standard deviation is always non‑negative and has the same units as the original data. A larger standard deviation indicates greater spread around the mean.

标准差始终非负，且与原数据单位相同。标准差越大，表明数据在均值周围的离散程度越高。

4. Basic Probability Rules | 概率基本法则

For any event A, the probability P(A) satisfies 0 ≤ P(A) ≤ 1. The complement rule: P(A′) = 1 – P(A).

对于任意事件 A，其概率满足 0 ≤ P(A) ≤ 1。补集法则：P(A′) = 1 – P(A)。

The addition rule for any two events A and B: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive (cannot occur together), then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

任意两事件 A 和 B 的加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥（不能同时发生），则 P(A ∩ B) = 0，从而 P(A ∪ B) = P(A) + P(B)。

Two events are independent if the occurrence of one does not affect the probability of the other. That is, P(A ∩ B) = P(A) × P(B). Equivalently, P(A|B) = P(A) provided P(B) > 0.

若一事件的发生不影响另一事件的概率，则两事件独立。即 P(A ∩ B) = P(A) × P(B)，等价地 P(A|B) = P(A)（P(B) > 0）。

Venn diagrams and tree diagrams are powerful tools for organising probabilities. On a tree diagram, branch probabilities multiply along a path, and probabilities of different outcomes on the same branch add.

文氏图和树形图是整理概率的有力工具。在树形图中，沿路径将分支概率相乘，同一分支下不同结果的概率相加。

5. Conditional Probability & Independence | 条件概率与独立

The conditional probability of A given B is P(A|B) = P(A ∩ B) / P(B), where P(B) > 0. This formula is fundamental for revising probabilities when partial information is known.

已知事件 B 发生的条件下，事件 A 的条件概率为 P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。此公式是在部分信息已知时修正概率的基石。

To test for independence, verify whether P(A ∩ B) = P(A)·P(B). If the equality holds, the events are independent; otherwise they are not.

检验独立性时，验证 P(A ∩ B) = P(A)·P(B) 是否成立。若等式成立则事件独立，否则不独立。

When working with a table of frequencies or probabilities, conditional probabilities can be found by restricting attention to the row or column of the given condition. The denominator changes to the total of that row or column.

使用频率表或概率表时，可通过将注意力限制在给定条件所在的行或列来求条件概率，此时分母变为该行或列的总和。

6. Permutations & Combinations | 排列与组合

The number of ways to arrange n distinct objects in a line is n! = n × (n–1) × … × 1, read as “n factorial”.

将 n 个不同物体排成一列的方法数为 n! = n × (n–1) × … × 1，读作“n 的阶乘”。

The number of arrangements of r objects chosen from n distinct objects is the permutation ⁿP_r = n! / (n – r)!. Order matters in permutations.

从 n 个不同物体中选取 r 个进行排列的方法数为排列 ⁿP_r = n! / (n – r)!。排列重视顺序。

The number of ways to choose r objects from n distinct objects, where order does not matter, is the combination ⁿC_r = n! / [r! (n – r)!]. This is also written as (ⁿᵣ).

从 n 个不同物体中选出 r 个而不考虑顺序的方法数为组合 ⁿC_r = n! / [r! (n – r)!]，也记作 (ⁿᵣ)。

When some objects are identical, the number of distinct arrangements of n items, where there are n₁ identical of type 1, n₂ of type 2, etc., is n! / (n₁! n₂! …).

当存在相同物体时，n 个物品中若有 n₁ 个类型1相同、n₂ 个类型2相同……则不同的排列数为 n! / (n₁! n₂! …)。

Combinations appear in the binomial theorem and binomial probability. The binomial coefficient ⁿC_r is the number of ways to obtain r successes in n independent trials.

组合数出现在二项式定理和二项概率中。二项系数 ⁿC_r 表示在 n 次独立试验中获得 r 次成功的方法数。

7. Discrete Random Variables | 离散随机变量

A discrete random variable X takes a countable number of values xᵢ, each with probability P(X = xᵢ) = pᵢ. The sum of all probabilities must equal 1: Σ pᵢ = 1.

离散随机变量 X 取可数个值 xᵢ，每个值的概率为 P(X = xᵢ) = pᵢ。所有概率之和必为 1：Σ pᵢ = 1。

The expected value (mean) of X is E(X) = Σ [xᵢ · P(X = xᵢ)]. It represents the long‑run average outcome.

X 的 期望值（均值）为 E(X) = Σ [xᵢ · P(X = xᵢ)]，表示长期平均结果。

The variance of X is Var(X) = Σ [(xᵢ – μ)² · P(X = xᵢ)], with μ = E(X). A convenient computational form is Var(X) = E(X²) – [E(X)]², where E(X²) = Σ [xᵢ² · P(X = xᵢ)].

X 的方差为 Var(X) = Σ [(xᵢ – μ)² · P(X = xᵢ)]，μ = E(X)。实用的计算公式为 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ [xᵢ² · P(X = xᵢ)]。

The standard deviation is σ = √Var(X). Both variance and standard deviation measure the spread of the distribution.

标准差为 σ = √Var(X)。方差和标准差都衡量分布的离散程度。

For a linear transformation Y = aX + b, the expectation and variance follow simple rules: E(aX + b) = a E(X) + b and Var(aX + b) = a² Var(X). Adding a constant shifts the mean but does not affect variance or standard deviation.

对于线性变换 Y = aX + b，期望和方差满足：E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。加上常数只移动均值，不影响方差或标准差。

8. Binomial Distribution | 二项分布

A binomial distribution arises from a fixed number n of independent Bernoulli trials, each with the same probability of success p. Then X ~ B(n, p). The number of successes X can take values 0, 1, …, n.

当固定次数 n 的独立伯努利试验，每次成功概率均为 p 时，成功次数服从二项分布，记作 X ~ B(n, p)。X 可取值 0, 1, …, n。

The probability of exactly r successes is P(X = r) = ⁿC_r pʳ (1 – p)^{n – r}. Here ⁿC_r is the binomial coefficient.

恰好获得 r 次成功的概率为 P(X = r) = ⁿC_r pʳ (1 – p)^{n – r}。其中 ⁿC_r 为二项系数。

The mean of a binomial distribution is E(X) = np. The variance is Var(X) = np(1 – p).

二项分布的均值为 E(X) = np，方差为 Var(X) = np(1 – p)。

To find cumulative probabilities such as P(X ≤ x), use statistical tables or sum individual probabilities. The symmetry rule P(X ≤ r) = P(Y ≥ n – r) where Y ~ B(n, 1 – p) can sometimes simplify calculations.

求累积概率 P(X ≤ x) 时，可查统计表或逐项相加。利用对称性 P(X ≤ r) = P(Y ≥ n – r)，其中 Y ~ B(n, 1 – p)，有时可简化计算。

9. Geometric Distribution | 几何分布

The geometric distribution models the number of trials up to and including the first success in a sequence of independent Bernoulli trials, each with success probability p. We write X ~ Geo(p). The random variable X takes values 1, 2, 3, ….

几何分布描述在独立伯努利试验序列中，直到首次成功为止所进行的试验次数。记作 X ~ Geo(p)，随机变量 X 取值为 1, 2, 3, …。

The probability that the first success occurs on the r‑th trial is P(X = r) = (1 – p)^{r – 1} p. The sequence of failures before the first success follows a geometric progression.

首次成功发生在第 r 次试验的概率为 P(X = r) = (1 – p)^{r – 1} p。首次成功前的失败次数序列呈几何级数递减。

The mean of a geometric distribution is E(X) = 1 / p. The variance is Var(X) = (1 – p) / p².

几何分布的均值（期望次数）为 E(X) = 1 / p，方差为 Var(X) = (1 – p) / p²。

For the geometric distribution, cumulative probabilities can be found using the formula P(X ≤ r) = 1 – (1 – p)ʳ. This is useful when tables are not available

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Year 12 CIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册