📚 IGCSE Cambridge Statistics: Quick Reference Formula & Theorem Handbook | IGCSE 剑桥统计：公式定理速查手册
This handbook provides a concise summary of the key formulas and theorems required for the IGCSE Cambridge Statistics syllabus. Each section covers essential concepts from descriptive statistics to hypothesis testing, all laid out for quick revision.
本手册简明总结了 IGCSE 剑桥统计课程所需的关键公式和定理。每个部分涵盖从描述性统计到假设检验的核心概念，适合快速复习。
1. Measures of Central Tendency | 集中趋势的度量
The arithmetic mean (average) for a sample of n observations is the sum of all values divided by n.
样本（n 个观测值）的算术平均数（平均值）是所有值之和除以 n。
&xbar; = Σx / n
For a population, the population mean is denoted by μ.
对于总体，总体均值用 μ 表示。
μ = Σx / N
The median is the middle value when the data are arranged in ascending order. Its position is found at (n + 1)/2.
中位数是数据按升序排列后位于中间位置的值。其位置通过 (n + 1)/2 确定。
Median position = (n + 1) / 2
The mode is the value that occurs most frequently in a data set. A set may have no mode, one mode (unimodal), or more than one mode (multimodal).
众数是数据集中出现频率最高的值。一组数据可以没有众数、有一个众数（单峰）或多个众数（多峰）。
2. Measures of Dispersion | 离散程度的度量
The range is the simplest measure of spread, calculated as the difference between the maximum and minimum values.
极差是最简单的离散程度度量，等于最大值与最小值之差。
Range = maximum – minimum
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁). It represents the spread of the middle 50% of data.
四分位距 (IQR) 是上四分位数 (Q₃) 与下四分位数 (Q₁) 之差，表示中间 50% 数据的分散程度。
IQR = Q₃ – Q₁
For ungrouped data, quartile positions are often determined using (n+1)/4 for Q₁ and 3(n+1)/4 for Q₃.
对于未分组数据，通常用 (n+1)/4 确定 Q₁ 的位置，3(n+1)/4 确定 Q₃ 的位置。
Variance and standard deviation measure the average squared deviation from the mean. For a population:
方差和标准差衡量数据偏离均值的平均平方距离。对于总体：
σ² = Σ(x – μ)² / N
σ = √[Σ(x – μ)² / N]
For a sample used to estimate the population variance, the denominator is (n – 1):
当用样本估计总体方差时，分母为 (n – 1)：
s² = Σ(x – &xbar;)² / (n – 1)
s = √[Σ(x – &xbar;)² / (n – 1)]
3. Frequency Distributions and Grouped Data | 频数分布与分组数据
For grouped data, the midpoint x of each class interval is used. The estimated mean is given by:
对于分组数据，使用每个组区间的组中值 x。估计均值由下式给出：
&xbar; = Σf x / Σf
The estimated variance for grouped data (population form) is:
分组数据的估计方差（总体形式）为：
σ² = Σf (x – &xbar;)² / Σf
To locate the median or quartiles in a grouped frequency table, linear interpolation is used:
在分组频数表中定位中位数或四分位数使用线性插值：
Median = L + [(n/2 – F) / f] × c
where L = lower boundary of the median class, F = cumulative frequency before the median class, f = frequency of the median class, c = class width.
其中 L 为中位数所在组的下限，F 为中位数所在组之前的累计频数，f 为中位数所在组的频数，c 为组距。
Similarly, for the lower quartile:
类似地，对于下四分位数：
Q₁ = L + [(n/4 – F) / f] × c
and for the upper quartile change n/4 to 3n/4.
上四分位数则将 n/4 替换为 3n/4。
4. Probability | 概率
Probability is a measure of the likelihood of an event, ranging from 0 to 1. The probability of event A is:
概率是对事件发生可能性的度量，取值范围从 0 到 1。事件 A 的概率为：
P(A) = n(A) / n(S)
where n(A) is the number of favourable outcomes and n(S) is the total number of possible outcomes.
其中 n(A) 是事件包含的有利结果数，n(S) 是样本空间中的总结果数。
The complement rule states: P(A') = 1 – P(A).
互补事件规则：P(A') = 1 – P(A)。
For mutually exclusive events, the probability that either A or B occurs is the sum of their probabilities:
对于互斥事件，A 或 B 发生的概率等于两者概率之和：
P(A ∪ B) = P(A) + P(B)
For independent events, the probability that both A and B occur is the product of their individual probabilities:
对于独立事件，A 和 B 同时发生的概率等于各自概率的乘积：
P(A ∩ B) = P(A) × P(B)
Conditional probability is the probability of event A given that B has occurred:
条件概率是在事件 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)
5. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values, each with a probability P(X = x). The sum of all probabilities is 1.
离散随机变量 X 取可数个值，每个值对应概率 P(X = x)。所有概率之和为 1。
The expected value (mean) of X is:
X 的期望值（均值）为：
E(X) = Σ x P(X = x)
The variance of X is:
X 的方差为：
Var(X) = Σ x² P(X = x) – [E(X)]²
or equivalently Var(X) = Σ (x – μ)² P(X = x).
或等价地 Var(X) = Σ (x – μ)² P(X = x)。
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. We write X ~ B(n, p).
二项分布描述了固定次数的独立试验中成功次数的概率分布，每次试验成功概率同为 p。记作 X ~ B(n, p)。
The probability of obtaining exactly r successes in n trials is given by the binomial probability mass function:
在 n 次试验中恰好获得 r 次成功的概率由二项概率质量函数给出：
P(X = r) = nCr pr (1 – p)n–r
where nCr = n! / [r! (n – r)!].
其中 nCr = n! / [r! (n – r)!]。
The mean and variance of a binomial random variable are:
二项随机变量的均值和方差为：
E(X) = n p
Var(X) = n p (1 – p)
7. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a bell-shaped curve. It is defined by the mean μ and variance σ², written as X ~ N(μ, σ²).
正态分布是一种具有钟形曲线的连续概率分布，由均值 μ 和方差 σ² 定义，记作 X ~ N(μ, σ²)。
To find probabilities for any normal variable, we standardise it to the standard normal distribution Z ~ N(0, 1) using:
为计算任意正态变量的概率，我们将其标准化为标准正态分布 Z ~ N(0, 1)，使用公式：
z = (x – μ) / σ
= P(Z ≤ z). For a given probability, inverse normal calculations find the corresponding x-value:
标准正态表提供 Φ(z) = P(Z ≤ z)。对于给定概率，逆正态计算可找出相应的 x 值：
x = μ + z σ
8. Sampling and Estimation | 抽样与估计
When samples of size n are drawn from a population with mean μ and standard deviation σ, the distribution of the sample mean &xbar; has:
从均值为 μ、标准差为 σ 的总体中抽取容量为 n 的样本时，样本均值 &xbar; 的分布具有：
E(&xbar;) = μ
Standard error = σ / √n
A confidence interval for the population mean μ when σ is known is:
当 σ 已知时，总体均值 μ 的置信区间为：
&xbar; ± z × (σ / √n)
Commonly used z-values for confidence intervals are:
常用的置信区间 z 值如下：
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