GCSE Cambridge Statistics: Formula and Theorem Quick Reference Handbook | GCSE Cambridge 统计：公式定理速查手册

📚 GCSE Cambridge Statistics: Formula and Theorem Quick Reference Handbook | GCSE Cambridge 统计：公式定理速查手册

This handbook provides a rapid revision reference for the key formulas and theorems required in GCSE Cambridge Statistics. It covers descriptive statistics, probability, probability distributions, sampling, correlation, regression, and index numbers. All symbols are presented in standard notation, and every section is carefully paired with both English and Chinese explanations to support bilingual learners preparing for the Cambridge IGCSE or O Level Statistics examinations.

本手册为 GCSE 剑桥统计学科提供关键公式和定理的快速复习参考。涵盖描述统计、概率、概率分布、抽样、相关、回归和指数等内容。所有符号均使用标准记法，每个部分均提供英文和中文的双语解释，以帮助正在准备剑桥 IGCSE 或 O Level 统计考试的学习者。

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

Measures of central tendency describe the centre of a data set. The three most common measures are the mean, median and mode. Each gives a single value that represents a typical observation.

集中趋势度量描述数据集的中心位置。最常用的三个度量是均值、中位数和众数。每个都给出一个能代表典型观测值的单一数值。

Arithmetic mean for ungrouped data: add all values and divide by the number of observations.

未分组数据的算术平均数：将所有数值相加后除以观测值的个数。

x̄ = (Σxᵢ) / n

For grouped data, the estimated mean is calculated using midpoints.

对于分组数据，使用组中值来计算估计均值。

x̄ = Σ(fᵢ mᵢ) / Σfᵢ

where fᵢ is the frequency of each class and mᵢ is the midpoint of that class.

其中 fᵢ 为每组的频数，mᵢ 为对应组的组中值。

The median is the middle value when data are ordered. For ungrouped data with n observations, the median is the (n+1)/2 th value. For grouped data, interpolation is used.

中位数是将数据排序后位于中间的值。未分组数据有 n 个观测值时，中位数是第 (n+1)/2 个值。对于分组数据，需要使用线性插值。

Median = L + ( (n/2 − F) / f ) × w

L: lower boundary of the median class; F: cumulative frequency before the median class; f: frequency of the median class; w: class width.

L 为中位数所在组的下限；F 为低于该组的累积频数；f 为该组的频数；w 为组距。

The mode is the most frequent value. For grouped data, the modal class is the class interval with the highest frequency, sometimes estimated using a formula based on neighbouring frequencies.

众数是出现次数最多的值。对于分组数据，众数组是频数最高的那个组，有时会基于相邻组的频数利用公式进行估计。

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2. Measures of Dispersion | 离散度量

Dispersion measures describe how spread out the data are. Common measures include range, interquartile range (IQR), variance and standard deviation.

离散度量描述数据的分散程度。常用的度量包括全距、四分位距 (IQR)、方差和标准差。

The range is simply the difference between the maximum and minimum values.

全距即最大值与最小值之差。

The interquartile range is IQR = Q₃ − Q₁, where Q₁ is the lower quartile (25th percentile) and Q₃ is the upper quartile (75th percentile).

四分位距 IQR = Q₃ − Q₁，其中 Q₁ 为下四分位数（第25百分位数），Q₃ 为上四分位数（第75百分位数）。

Variance and standard deviation measure the average squared deviation from the mean. For a sample, the formulas are:

方差和标准差度量各个数据值偏离均值的平方平均水平。对于样本，公式为：

s² = Σ(xᵢ − x̄)² / (n − 1)

s = √[ Σ(xᵢ − x̄)² / (n − 1) ]

where n − 1 is used to give an unbiased estimate of the population variance.

其中分母使用 n − 1 以给出总体方差的无偏估计。

For grouped data, xᵢ in the formulas is replaced by the class midpoint mᵢ, and (xᵢ − x̄)² is multiplied by the frequency fᵢ.

对于分组数据，公式中的 xᵢ 用组中值 mᵢ 替代，并将 (xᵢ − x̄)² 乘以该组的频数 fᵢ。

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3. Probability Fundamentals | 概率基础

Probability measures the chance that an event will occur. It is always a number between 0 and 1.

概率衡量事件发生的可能性大小，总是一个介于 0 和 1 之间的数。

P(A) = n(A) / n(S)

where n(A) is the number of outcomes in event A and n(S) is the total number of equally likely outcomes in the sample space.

其中 n(A) 是事件 A 包含的结果个数，n(S) 是样本空间中所有等可能结果的总数。

Addition rule: for any events A and B,

加法公式：对任意两个事件 A 和 B，

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive (cannot happen together), then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

如果 A 和 B 互斥（不能同时发生），则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

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

补集公式：P(A’) = 1 − P(A)，其中 A’ 表示 A 不发生的事件。

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

Conditional probability is the probability that event A occurs given that event B has already occurred.

条件概率是指在事件 B 已经发生的条件下，事件 A 发生的概率。

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

Two events A and B are independent if the occurrence of one does not affect the probability of the other. Then:

如果两个事件 A 和 B 相互独立，则一个事件的发生不影响另一个事件发生的概率。此时：

P(A ∩ B) = P(A) × P(B)

and consequently P(A | B) = P(A) and P(B | A) = P(B).

并且有 P(A | B) = P(A)，P(B | A) = P(B)。

Tree diagrams are useful for visualising conditional probabilities. Probabilities multiply along branches and add across final outcomes.

树形图有助于可视化条件概率。概率沿支路相乘，最终结果相加。

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5. Permutations and Combinations | 排列与组合

Counting principles help determine the number of ways events can occur. The factorial n! = n × (n−1) × … × 1.

计数原则用于确定事件可能发生的方式总数。阶乘 n! = n × (n−1) × … × 1。

Permutations: the number of ways to arrange r objects chosen from n distinct objects, where order matters.

排列：从 n 个不同对象中选出 r 个进行排列（有序）的方法数。

nPr = n! / (n − r)!

Combinations: the number of ways to choose r objects from n distinct objects, where order does not matter.

组合：从 n 个不同对象中选出 r 个（无序）的方法数。

nCr = n! / [r! (n − r)!]

This is also written as C(n, r) or (nr). Combinations are used in binomial probability.

也记为 C(n, r) 或 (nr)。组合数在二项概率中发挥关键作用。

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6. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p.

二项分布用于描述在固定次数的独立试验中成功的次数，每次试验的成功概率 p 相同。

If X ~ B(n, p), then the probability of exactly k successes is:

若 X ~ B(n, p)，则恰好有 k 次成功的概率为：

P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ

Conditions: the number of trials n is fixed; each trial has only two outcomes (success/failure); trials are independent; and p remains constant.

条件：试验次数 n 固定；每次试验只有两种结果（成功/失败）；试验互相独立；p 保持恒定。

Mean and variance of a binomial distribution:

二项分布的均值与方差：

μ = np

σ² = np(1 − p)

These formulas are used to describe the centre and spread of the distribution.

这些公式用于描述分布的中心位置和离散程度。

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7. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution symmetric about its mean, with a bell-shaped curve. It is fully described by its mean μ and standard deviation σ.

正态分布是一种连续概率分布，其图形关于均值对称，呈钟形曲线。它完全由其均值 μ 和标准差 σ 描述。

To find probabilities from standard normal tables, convert any normal variable X to the standard normal variable Z:

为了从标准正态分布表中查找概率，需要将任意正态变量 X 转化为标准正态变量 Z：

Z = (X − μ) / σ

Z has a mean of 0 and a standard deviation of 1. The tables give P(Z < z) for various z-values.

Z 的均值为 0，标准差为 1。正态表中列出了不同 z 值对应的 P(Z < z)。

The total area under the curve is 1. The probability P(X < a) equals the area to the left of a. Because the distribution is symmetric, P(Z < −z) = 1 − P(Z < z).

曲线下的总面积为 1。概率 P(X < a) 等于 a 左侧的面积。由于对称性，P(Z < −z) = 1 − P(Z < z)。

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8. Sampling Methods and Bias | 抽样方法与偏差

In statistics, conclusions are often based on samples. A sampling method must be chosen carefully to obtain a representative

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