📚 AS OCR Statistics: Quick Reference Handbook of Key Formulas and Theorems | AS OCR 统计：公式定理速查手册
Welcome to the AS OCR Statistics quick reference handbook. This article compiles essential formulas, theorems, and key concepts you need to master for the AS level examination. Use it to revise definitions, check your understanding of probability rules, discrete random variables, binomial distribution, hypothesis testing, and correlation/regression. Each section presents concise English explanations followed by Chinese translations to reinforce bilingual learning.
欢迎使用 AS OCR 统计速查手册。本文汇编了 AS 阶段考试必须掌握的核心公式、定理与关键概念。使用本文复习定义，检查对概率规则、离散随机变量、二项分布、假设检验及相关与回归的理解。每节先提供简洁英文解释，紧接中文翻译，强化双语学习。
1. Summary Statistics: Mean, Median, Mode, Range, IQR, Variance, Standard Deviation | 摘要统计量：均值、中位数、众数、极差、四分位距、方差、标准差
Mean: for a sample of size n with data values x₁, x₂, …, xₙ, the mean x̄ = (∑xᵢ)/n.
均值: 样本容量为 n，数据值为 x₁, x₂, …, xₙ，均值 x̄ = (∑xᵢ)/n。
x̄ = (∑xᵢ) / n
Median: the middle value when data are ordered. If n is odd, median is the (n+1)/2-th value; if n even, it is the average of the n/2-th and (n/2 +1)-th values.
中位数: 有序数据中间的值。n 为奇数时，中位数为第 (n+1)/2 个值；n 为偶数时，为第 n/2 与第 (n/2+1) 个值的平均数。
Mode: the most frequently occurring value(s).
众数: 出现频率最高的值。
Range: maximum value – minimum value.
极差: 最大值减最小值。
Interquartile Range (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，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 (sample): s² = [∑(xᵢ – x̄)²] / (n – 1). For a population the denominator is N.
方差（样本）: s² = [∑(xᵢ – x̄)²] / (n – 1)；对于总体，分母为 N。
s² = ∑(xᵢ – x̄)² / (n – 1)
Standard Deviation: s = √[∑(xᵢ – x̄)² / (n – 1)].
标准差: s = √[∑(xᵢ – x̄)² / (n – 1)]。
2. Probability Rules | 概率规则
Basic Rule: For any event A, 0 ≤ P(A) ≤ 1. P(A) = 0 means A is impossible; P(A) = 1 means A is certain.
基本规则: 对于任意事件 A，0 ≤ P(A) ≤ 1。P(A) = 0 表示 A 不可能发生；P(A) = 1 表示 A 必然发生。
Complement Rule: P(A’) = 1 – P(A), where A’ is the event “not A”.
补集规则: P(A’) = 1 – P(A)，其中 A’ 表示“非 A”事件。
Addition Rule: 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).
加法法则: 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)。
Conditional Probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
条件概率: P(A|B) = P(A ∩ B) / P(B)，前提为 P(B) > 0。
Multiplication Rule: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B). If A and B are independent, P(A ∩ B) = P(A) × P(B).
乘法法则: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)。若 A 与 B 独立，则 P(A ∩ B) = P(A) × P(B)。
Venn Diagrams and Tree Diagrams: useful tools for organising probabilities and solving multi‑stage problems.
文氏图与树状图: 用于整理概率和求解多阶段问题的实用工具。
3. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布
A discrete random variable X takes a countable number of values. Its probability distribution lists each possible value x together with P(X = x). The sum of all probabilities equals 1: ∑P(X = x) = 1.
离散随机变量 X 可取可数个值。其概率分布列出每个可能取值 x 及对应的 P(X = x)。所有概率之和等于 1：∑P(X = x) = 1。
Probability Mass Function (p.m.f.): P(X = x) = p(x). For all x outside the support, p(x) = 0.
概率质量函数 (p.m.f.): P(X = x) = p(x)。对于所有不在支撑集中的 x，p(x) = 0。
Cumulative Distribution Function: F(x) = P(X ≤ x) = ∑_{t ≤ x} p(t).
累积分布函数: F(x) = P(X ≤ x) = ∑_{t ≤ x} p(t)。
Example: For a fair six‑sided die, X ~ Uniform {1,2,3,4,5,6}, P(X = x) = 1/6 for each x.
例如：对于公平的六面骰子，X ~ 均匀分布 {1,2,3,4,5,6}，每个 x 的 P(X = x) = 1/6。
4. Expectation, Variance and Standard Deviation of Discrete Random Variables | 离散随机变量的期望、方差与标准差
Expected Value (Mean): μ = E(X) = ∑[x · P(X = x)].
期望值（均值）: μ = E(X) = ∑[x · P(X = x)]。
E(X) = ∑ x p(x)
Variance: Var(X) = E[(X – μ)²] = ∑[(x – μ)² · P(X = x)]. An equivalent computational formula is Var(X) = E(X²) – [E(X)]².
方差: Var(X) = E[(X – μ)²] = ∑[(x – μ)² · P(X = x)]。等价的简便计算公式为 Var(X) = E(X²) – [E(X)]²。
Var(X) = E(X²) – (E(X))²
Standard Deviation: σ = √Var(X).
标准差: σ = √Var(X)。
Linear Transformations: For a random variable X and constants a, b:
E(aX + b) = a E(X) + b.
Var(aX + b) = a² Var(X).
线性变换: 对随机变量 X 及常数 a、b：
E(aX + b) = a E(X) + b。
Var(aX + b) = a² Var(X)。
5. The Binomial Distribution: Formula and Conditions | 二项分布：公式与条件
A binomial distribution arises from a fixed number n of independent trials, each having two outcomes (success/failure) with constant probability of success p. The random variable X = number of successes follows B(n, p).
二项分布源于固定次数 n 的独立试验，每次试验仅两种结果（成功/失败），成功概率 p 恒定。随机变量 X = 成功次数，服从 B(n, p)。
Conditions: fixed number of trials, independence, two possible outcomes per trial, constant probability p.
条件: 试验次数固定、独立、每次试验两种可能结果、概率 p 恒定。
Probability Formula: P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, for r = 0, 1, 2, …, n, where ⁿCᵣ = C(n, r) = n! / [r!(n – r)!].
概率公式: P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ，其中 r = 0, 1, 2, …, n，ⁿCᵣ = C(n, r) = n! / [r!(n – r)!]。
P(X = r) = C(n, r) × pʳ × (1 – p)⁽ⁿ⁻ʳ⁾
Mean and Variance: E(X) = np, Var(X) = np(1 – p) = npq, where q = 1 – p.
均值与方差: E(X) = np, Var(X) = np(1 – p) = npq，其中 q = 1 – p。
Using tables or calculators: For cumulative probabilities P(X ≤ r), use binomial tables or a statistical calculator.
使用表格或计算器: 求累积概率 P(X ≤ r) 时，可使用二项分布表或统计计算器。
6. Hypothesis Testing for the Binomial Proportion p | 二项分布比例 p 的假设检验
A hypothesis test assesses whether evidence from a sample supports a claim about the population proportion p. The test uses an observed value from a binomial distribution B(n, p).
假设检验评价样本证据是否支持关于总体比例 p 的断言。检验基于来自二项分布 B(n, p) 的观测值。
Null Hypothesis H₀: p = p₀.
原假设 H₀: p = p₀。
Alternative Hypothesis H₁: p ≠ p₀ (two‑tailed), p < p₀ (left‑tailed), or p > p₀ (right‑tailed).
备择假设 H₁: p ≠ p₀（双尾），p < p₀（左尾），或 p > p₀（右尾）。
Significance Level α: usually 0.05 or 0.01, the maximum probability of wrongly rejecting H₀ when it is true.
显著性水平 α: 通常取 0.05 或 0.01，是当 H₀ 为真时错误拒绝 H₀ 的最大概率。
Test Statistic: the observed number of successes, r.
检验统计量: 观测到的成功次数 r。
p‑value: the probability of obtaining a result at least as extreme as the observed value, assuming H₀ is true. For a left‑tailed test, p‑value = P(X ≤ r | p = p₀); for a right‑tailed test, p‑value = P(X ≥ r | p = p₀); for a two‑tailed test, consider both tails.
p 值: 假设 H₀ 为真时，得到至少与观测值一样极端结果的概率。左尾检验：p‑value = P(X ≤ r | p = p₀)；右尾检验：p‑value = P(X ≥ r | p = p₀)；双尾检验需考虑两侧。
Conclusion: if p‑value ≤ α, reject H₀ and there is sufficient evidence for H₁; otherwise, do not reject H₀.
结论: 若 p‑值 ≤ α，拒绝 H₀，有充分证据支持 H₁；否则，不拒绝 H₀。
Critical Region: the set of values of the test statistic that lead to rejection of H₀. Determined from the binomial distribution with p = p₀.
临界域: 导致拒绝 H₀ 的检验统计量取值集合。由 p = p₀ 的二项分布确定。
7. Correlation: Product Moment Correlation Coefficient | 相关：积矩相关系数
The product moment correlation coefficient (PMCC), r, measures the strength and direction of a linear relationship between two variables x and y. –1 ≤ r ≤ 1.
积矩相关系数 (PMCC) r 衡量两个变量 x 与 y 之间线性关系的强度和方向。 –1 ≤ r ≤ 1。
Formulae:
Sxx = ∑(xᵢ – x̄)² = ∑xᵢ² – (∑xᵢ)²/n
Syy = ∑(yᵢ – ȳ)² = ∑yᵢ² – (∑yᵢ)²/n
Sxy = ∑(xᵢ – x̄)(yᵢ – ȳ) = ∑xᵢyᵢ – (∑xᵢ)(∑yᵢ)/n
公式:
Sxx = ∑(xᵢ – x̄)² = ∑xᵢ² – (∑xᵢ)²/n
Syy = ∑(yᵢ – ȳ)² = ∑yᵢ² – (∑yᵢ)²/n
Sxy = ∑(xᵢ – x̄)(yᵢ – ȳ) = ∑xᵢyᵢ – (∑xᵢ)(∑yᵢ)/n
r = Sxy / √(Sxx × Syy)
Interpretation: r close to +1 indicates strong positive correlation; r close to –1 indicates strong negative correlation; r near 0 suggests no linear correlation.
解释: r 接近 +1 表示强正相关；r 接近 –1 表示强负相关；r 接近 0 表示没有线性相关性。
Remember that correlation does not imply causation.
记住相关不意味着因果。
8. Linear Regression: Least Squares Regression Line | 线性回归：最小二乘回归线
The least squares regression line of y on x has equation y = a + bx, where the slope b and intercept a minimise the sum of squared vertical distances from the points to the line.
y 对 x 的最小二乘回归线方程为 y = a + bx，其中斜率 b 和截距 a 使数据点到直线的垂直距离平方和最小。
Slope: b = Sxy / Sxx.
斜率: b = Sxy / Sxx。
Intercept: a = ȳ – b x̄.
截距: a = ȳ – b x̄。
y = ȳ + b(x – x̄)
Prediction: use the regression line to estimate y for a given x. Care should be taken when extrapolating beyond the data range.
预测: 使用回归线对给定的 x 估计 y。外推超出数据范围
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