Year 11 CAIE Statistics: Formulas & Theorems Quick Reference | Year 11 CAIE 统计：公式定理速查手册

📚 Year 11 CAIE Statistics: Formulas & Theorems Quick Reference | Year 11 CAIE 统计：公式定理速查手册

This quick-reference handbook is designed for Year 11 CAIE Statistics students. It compiles the essential formulas and theorems needed across the syllabus, from descriptive statistics to hypothesis testing. Use it to revise key concepts efficiently and ensure accuracy in applying statistical methods.

本速查手册专为 Year 11 CAIE 统计课程学生设计，汇编了从描述性统计到假设检验等整个考纲必备的公式与定理。用它可以高效复习核心概念，确保在应用统计方法时准确无误。

1. Measures of Central Tendency and Dispersion | 集中趋势与离散程度的度量

Measures of central tendency indicate the centre of a data set, while dispersion measures describe the spread.

集中趋势的度量反映数据集的中心，离散程度的度量则描述数据的分散情况。

x̄ = Σx / n

样本均值（x̄）为所有观测值之和除以样本容量 n。

μ = ΣX / N

总体均值 μ 的计算方式相同，但用总体容量 N。

The median is the middle value when data are ordered. For odd n, it is the (n+1)/2 th value; for even n, it is the average of the n/2 th and (n/2 +1)th values.

中位数是排序后位于中间的值。n 为奇数时取第 (n+1)/2 个值；n 为偶数时取第 n/2 和第 n/2+1 个值的平均数。

The lower quartile Q₁ is the median of the lower half of data, and Q₃ is the median of the upper half. The interquartile range is IQR = Q₃ – Q₁.

下四分位数 Q₁ 是数据下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。四分位距 IQR = Q₃ – Q₁。

s² = Σ(x – x̄)² / (n – 1)

样本方差 s² 是偏差平方和除以 n−1。

σ² = Σ(x – μ)² / N

总体方差 σ² 是偏差平方和除以 N。

Standard deviation is the square root of variance: s = √s².

标准差是方差的算术平方根：s = √s²。

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2. Basic Probability Rules | 概率基本法则

The addition rule for any two events A and B is used to find the probability of A or B occurring.

对于任意两事件 A 和 B，加法法则用于计算 A 或 B 发生的概率。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

当 A 与 B 互斥时，P(A ∩ B)=0，此时 P(A ∪ B) = P(A) + P(B)。

Conditional probability gives the probability of A given that B has occurred.

条件概率表示在事件 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)

若 A 与 B 独立，则 P(A|B) = P(A) 且乘法法则变为 P(A ∩ B) = P(A) × P(B)。

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3. Discrete Random Variables | 离散随机变量

A discrete random variable X takes countable values with probabilities P(X = x). The expected value is the long-run average.

一个离散随机变量 X 取可数个值，每个值对应概率 P(X = x)。期望值就是长期平均。

E(X) = Σ x·P(X = x)

方差衡量 X 围绕期望值的离散程度。

Var(X) = E(X²) – [E(X)]²

Standard deviation of X is σ = √Var(X).

X 的标准差 σ = √Var(X)。

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4. The Binomial Distribution | 二项分布

If a trial has two outcomes (success/failure) with constant probability of success p, and n independent trials are performed, the number of successes X follows a binomial distribution.

若一次试验只有两种结果（成功/失败），成功的概率 p 保持不变，且进行 n 次独立试验，则成功次数 X 服从二项分布。

X ~ B(n, p)

The probability of exactly r successes is given by the binomial probability function.

恰好得到 r 次成功的概率由二项概率函数给出。

P(X = r) = ⁿC_r p^r (1 – p)^{n – r}

其中 ⁿC_r = n! / [r!(n – r)!]。

The mean and variance of a binomial random variable:

二项随机变量的均值与方差：

E(X) = np

Var(X) = np(1 – p)

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5. The Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution symmetric about the mean μ. Many natural phenomena follow it approximately.

正态分布是一种关于均值 μ 对称的连续概率分布，许多自然现象的分布近似于它。

X ~ N(μ, σ²)

To find probabilities, we convert X to the standard normal variable Z, which has mean 0 and variance 1.

为求概率，我们将 X 转换为均值为 0、方差为 1 的标准正态变量 Z。

Z = (X – μ) / σ

Probabilities are then obtained from standard normal tables, using symmetry if needed: P(Z ≤ -a) = 1 – P(Z ≤ a).

然后查标准正态表求概率，必要时利用对称性：P(Z ≤ -a) = 1 – P(Z ≤ a)。

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6. Sampling and Sampling Distributions | 抽样与抽样分布

When random samples of size n are drawn from a population with mean μ and variance σ², the sample mean X̄ is a random variable with its own distribution.

从均值为 μ、方差为 σ² 的总体中抽取容量为 n 的随机样本，样本均值 X̄ 是一个随机变量，有其自身的分布。

E(X̄) = μ

Var(X̄) = σ² / n

The standard error of the mean is SE = σ / √n. If the population is normal, X̄ is exactly normally distributed; for large n, the Central Limit Theorem ensures X̄ is approximately normal even if the population is not normal.

均值的标准误 SE = σ / √n。

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