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

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

This quick reference guide brings together all the essential formulas and theorems for the Cambridge AS Level Probability & Statistics 1 course (9709). It is designed as a rapid revision tool, covering measures of central tendency and spread, probability laws, counting techniques, discrete random variables, the binomial and normal distributions, data presentation, and coding. Use it to consolidate your knowledge and for last-minute exam preparation.

本速查指南汇总了剑桥 AS 级概率与统计1 (9709) 课程的所有核心公式和定理。它旨在作为快速复习工具，涵盖集中趋势与离散程度度量、概率法则、计数方法、离散随机变量、二项分布与正态分布、数据呈现以及编码。用它来巩固知识并进行考前冲刺。

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

The arithmetic mean of a set of n numbers x₁, x₂, …, xₙ is the sum divided by n. It is the most commonly used measure of centre and is denoted by x̄.

x̄ = Σx / n

对一组 n 个数据 x₁, x₂, …, xₙ，算术平均数是总和除以 n。它是最常用的中心度量，记作 x̄。

For grouped data, use class midpoints: x̄ ≈ Σ(f × m) / Σf, where f is the frequency and m is the class midpoint.

对于分组数据，使用组中点：x̄ ≈ Σ(f × m) / Σf，其中 f 是频数，m 是组中点。

The median is the middle value when data are arranged in ascending order. If n is odd, median is the (n+1)/2th value; if n is even, median is the average of the n/2th and (n/2+1)th values.

中位数是将数据升序排列后的中间值。若 n 为奇数，中位数是第 (n+1)/2 个值；若 n 为偶数，则是第 n/2 和第 n/2+1 个值的平均数。

For grouped data, the median can be estimated from the cumulative frequency graph or by linear interpolation within the median class.

对于分组数据，中位数可以从累积频率图中估算，或通过中位数所在组的线性插值求得。

The mode is the value with the highest frequency. For grouped data, the modal class is the class with the highest frequency density, not simply the highest frequency.

众数是出现频率最高的值。对于分组数据，众数所在的组是频数密度最高的组，而不仅仅是频数最高的组。

2. Measures of Dispersion | 离散程度度量

The range is the difference between the largest and smallest values in the data set. It is sensitive to outliers.

极差是数据集中最大值与最小值的差值。它对离群值敏感。

The interquartile range (IQR) is Q3 – Q₁, where Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile). The IQR measures the spread of the middle 50% of the data.

四分位距 (IQR) 是 Q₃ – Q₁，其中 Q₁ 是下四分位数（第25百分位数），Q₃ 是上四分位数（第75百分位数）。IQR 度量了中间50%数据的分散程度。

For a population data set of n values, the variance σ² and standard deviation σ are defined using divisor n:

σ² = Σ(x − x̄)² / n = (Σx² / n) − x̄²

σ = √[Σ(x − x̄)² / n]

对于包含 n 个值的总体数据集，方差 σ² 和标准差 σ 使用除数 n 定义。

If the data are a sample and you need an unbiased estimate of the population variance, use divisor n−1: s² = Σ(x − x̄)² / (n−1). In exam questions, the context will indicate which formula to use.

如果数据是一个样本且需要总体方差的无偏估计，则使用除数 n−1：s² = Σ(x − x̄)² / (n−1)。考试题目会说明该用哪个公式。

For grouped data, x represents the class midpoint and the formulas are applied with frequencies: σ² = Σf(x − x̄)² / Σf.

对于分组数据，x 代表组中点，公式中加入频数：σ² = Σf(x − x̄)² / Σf。

The standard deviation is the square root of the variance and gives a measure of dispersion in the same units as the original data.

标准差是方差的平方根，以与原始数据相同的单位衡量离散程度。

3. Probability Basics | 概率基础

The probability of an event A is given by P(A) = number of outcomes in A / total number of outcomes in the sample space S, provided all outcomes are equally likely.

事件 A 的概率为 P(A) = A 中的结果数 / 样本空间 S 中的总结果数，前提是所有结果等可能发生。

The complement rule: P(A′) = 1 − P(A), where A′ (or Aᶜ) is the event that A does not occur.

互补事件法则：P(A′) = 1 − P(A)，其中 A′ 表示 A 不发生的事件。

For any two events A and B, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

对于任意两个事件 A 和 B，加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

For independent events, the multiplication rule is P(A ∩ B) = P(A) × P(B). This can be used to check for independence.

对于独立事件，乘法法则为 P(A ∩ B) = P(A) × P(B)。此式可用于检验独立性。

4. Conditional Probability and Independence | 条件概率与独立性

The conditional probability of A given B is defined as:

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0

在事件 B 发生的条件下，A 发生的条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。

Rearranging gives the general multiplication rule: P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A).

移项得到一般乘法法则：P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A)。

Events A and B are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(A | B) = P(A) when P(B) > 0.

事件 A 与 B 相互独立，当且仅当 P(A ∩ B) = P(A) × P(B)，或等价的 P(A | B) = P(A)（当 P(B) > 0 时）。

Tree diagrams are extremely useful for visualising conditional probability problems. Multiply along branches for joint probabilities and add relevant terminal probabilities for total probabilities.

树状图对于可视化条件概率问题极有帮助。沿分支相乘得到联合概率，将相关终端概率相加得到全概率。

5. Permutations and Combinations | 排列与组合

The factorial of a positive integer n is n! = n × (n−1) × (n−2) × … × 2 × 1. By definition, 0! = 1.

正整数 n 的阶乘为 n! = n × (n−1) × (n−2) × … × 2 × 1。定义 0! = 1。

A permutation is an arrangement of objects in a specific order. The number of permutations of r objects chosen from n distinct objects is:

ⁿPᵣ = n! / (n − r)!

排列是指对象按特定顺序的排列方式。从 n 个不同对象中选取 r 个的排列数为 ⁿPᵣ = n! / (n − r)!。

A combination is a selection of objects without regard to order. The number of combinations of r objects from n distinct objects is:

ⁿCᵣ = n! / [r! (n − r)!]

组合是不考虑顺序的选择方式。从 n 个不同对象中选取 r 个的组合数为 ⁿCᵣ = n! / [r! (n − r)!]。

These coefficients ⁿCᵣ are also known as binomial coefficients and appear in the binomial expansion.

这些系数 ⁿCᵣ 也称为二项式系数，并出现在二项展开式中。

When items are repeated, divide by the factorial of each repeated group: arrangements of n items with an identical group of p, q, … is n! / (p! q! …).

当有相同物品时，除以每组重复物品的阶乘：n 个物品中有相同的 p 个、q 个 …… 的排列数为 n! / (p! q! …)。

6. Discrete Random Variables | 离散随机变量

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

离散随机变量 X 可取可数个可能值，每个值具有概率 P(X = x)。所有概率之和必须等于 1。

The expected value (mean) of X is:

E(X) = μ = Σ x · P(X = x)

X 的期望值（均值）为 E(X) = μ = Σ x · P(X = x)。

The variance of X is given by:

Var(X) = σ² = Σ (x − μ)² P(X = x) = Σ x² P(X = x) − μ²

X 的方差为 Var(X) = σ² = Σ (x − μ)² P(X = x) = Σ x² P(X = x) − μ²。

For a linear transformation Y = aX + b, where a and b are constants, the rules are:

E(Y) = a E(X) + b

Var(Y) = a² Var(X)

对于线性变换 Y = aX + b，其中 a 和 b 为常数，有 E(Y) = a E(X) + b，Var(Y) = a² Var(X)。

The standard deviation of X is σ

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