📚 IB & Edexcel Mathematics: Statistics Key Points | IB 及 Edexcel 数学：统计考点精讲
Statistics is a core component of both IB and Edexcel A-Level Mathematics, covering data analysis, probability, distributions, and inferential methods. This revision guide distils the essential concepts, formulas, and problem-solving strategies you need for exam success.
统计学是 IB 和 Edexcel A-Level 数学的核心组成部分，涵盖数据分析、概率、分布和推断方法。本考点精讲提炼了考试成功所必需的核心概念、公式和解题策略。
1. Descriptive Statistics | 描述统计
Descriptive statistics summarise data using measures of central tendency and dispersion. Understanding these measures is the first step in any statistical analysis.
描述统计通过集中趋势和离散程度的度量来总结数据。理解这些度量是任何统计分析的第一步。
The three common measures of centre are the mean, median, and mode. The mean (x̄) is the arithmetic average: x̄ values. The median is the middle value when data are ordered; it is resistant to outliers. The mode is the most frequent value and can be used for categorical data.
三种常见的中心度量是均值、中位数和众数。均值（x̄）是算术平均：x̄ = (Σxᵢ)/n，它对极端值敏感。中位数是数据排序后的中间值，对异常值具有抗扰性。众数是最频繁出现的值，可用于分类数据。
Measures of spread include the range, interquartile range (IQR), variance, and standard deviation. The range and IQR describe the spread of the middle 50% of data. Variance and standard deviation quantify variation around the mean.
离散度量包括极差、四分位距（IQR）、方差和标准差。极差和 IQR 描述了数据中间 50% 的散布程度。方差和标准差则量化围绕均值的变异。
Sample variance: s² = Σ(xᵢ - x̄)² / (n - 1)
样本方差：s² = Σ(xᵢ - x̄)² / (n - 1)
Sample standard deviation: s = √[ Σ(xᵢ - x̄)² / (n - 1) ]
样本标准差：s = √[ Σ(xᵢ - x̄)² / (n - 1) ]
| Measure | 度量 | Formula / Note |
|---|---|---|
| Mean | 均值 | x̄ = (Σxᵢ)/n |
| Median | 中位数 | Middle ordered value; for even n, average two middle values |
| Mode | 众数 | Most frequent value(s) |
| Variance | 方差 | s² = Σ(xᵢ - x̄)²/(n-1) |
| Standard deviation | 标准差 | s = √s² |
| IQR | 四分位距 | Q₃ - Q₁ |
2. Probability Basics | 概率基础
a random experiment. The probability of an event A is written as P(A), with 0 ≤ P(A) ≤ 1.
概率论是推断统计学的基础，处理随机实验中事件发生的可能性。事件 A 的概率记为 P(A)，满足 0 ≤ P(A) ≤ 1。
nditional probability describes the chance of event A given that event B has occurred: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
条件概率描述在事件 B 发生的条件下事件 A 发生的概率：P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。
Two events A and B are independent if P(A ∩ B) = P(A)P(B), or equivalently if P(A|B) = P(A). This concept is critical for product rule applications.
若 P(A ∩ B) = P(A)P(B)，或等价地 P(A|B) = P(A)，则事件 A 与 B 相互独立。这一概念在乘法法则应用中至关重要。
Bayes' theorem links conditional and marginal probabilities:
贝叶斯定理将条件概率与边际概率联系起来：
P(A|B) = [P(B|A) P(A)] / P(B)
P(A|B) = [P(B|A) P(A)] / P(B)
A common application is updating the probability of a hypothesis given new evidence. Tree diagrams and two-way tables are powerful tools for organising probability calculations.
常见应用是根据新证据更新假设的概率。树状图和双向表是组织概率计算的强大工具。
3. Discrete Random Variables | 离散随机变量
ies. The probability distribution P(X = x) must satisfy Σ P(X = x) = 1 for all possible x.
离散随机变量 X 取一列可数值，每个值有特定的概率。概率分布 P(X = x) 必须满足所有可能 x 的 Σ P(X = x) = 1。
The expected value (mean) of X is E[X] = μ = Σ x P(X = x). It represents the long‑run average if the experiment is repeated many times.
X 的期望值（均值）为 E[X] = μ = Σ x P(X = x)，表示若试验重复多次的长期平均值。
variance is Var(X) = E[(X - μ)²] = E[X²] - (E[X])². The standard deviation is σ = √Var(X).
方差为 Var(X) = E[(X - μ)²] = E[X²] - (E[X])²。标准差为 σ = √Var(X)。
Var(aX + b) = a² Var(X). These properties are frequently used in distribution derivations.
对于线性函数，有 E[aX + b] = a E[X] + b 且 Var(aX + b) = a² Var(X)。这些性质在分布推导中经常使用。
4. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in n independent trials, each with the same probability of success p. It is denoted X ~ B(n, p).
二项分布模型化 n 次独立试验中成功的次数，每次试验成功概率均为 p。记为 X ~ B(n, p)。
The probability mass function is:
概率质量函数为：
P(X = k) = ⁿCₖ pᵏ (1 - p)ⁿ⁻ᵏ
P(X = k) = ⁿCₖ pᵏ (1 - p)ⁿ⁻ᵏ
p and Var(X) = np(1 - p). The distribution is symmetric when p = 0.5 and skewed otherwise.
期望和方差为 E[X] = np 和 Var(X) = np(1 - p)。当 p = 0.5 时分布对称，否则呈现偏态。
n be found using tables, technology, or the normal approximation for large np and n(1-p). Requirements: n is fixed, trials are independent, only two outcomes, and p remains constant.
累积概率 P(X ≤ k) 可使用表格、计算器，或在 np 和 n(1-p) 较大时用正态近似求得。应用条件：n 固定、试验独立、只有两种结果、且 p 保持不变。
5. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution defined by its mean μ and variance σ²: X ~ N(μ, σ²). Its bell-shaped curve is symmetric around μ.
正态分布是由均值 μ 和方差 σ² 定义的连续概率分布：X ~ N(μ, σ²)。钟形曲线关于 μ 对称。
For standardisation, we use the z‑score:
标准化时使用 z 分数：
Z = (X - μ) / σ ~ N(0, 1)
Z = (X - μ) / σ ~ N(0, 1)
any normal variable, P(X ≤ x) = Φ((x - μ)/σ). Key interval probabilities: approximately 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
标准正态表给出 Φ(z) = P(Z ≤ z)。对任意正态变量，P(X ≤ x) = Φ((x - μ)/σ)。关键区间概率：约 68% 的数据落在 μ ± σ，95% 在 μ ± 2σ，99.7% 在 μ ± 3σ。
Inverse normal calculations solve for x given a left‑tail probability. The normal distribution is also the limiting distribution for sample means through the Central Limit Theorem.
逆向正态计算是在给定左尾概率下求解 x。正态分布还是样本均值在大样本下的极限分布（中心极限定理）。
6. Sampling Distributions & Confidence Intervals | 抽样分布与置信区间
size n are drawn from a population with mean μ and standard deviation σ, the sample mean x̄ follows a sampling distribution with E[x̄] = μ and standard error SE = σ/√n.
当从均值为 μ、标准差为 σ 的总体中抽取容量为 n 的随机样本时，样本均值 x̄ 的抽样分布满足 E[x̄] = μ，标准误 SE = σ/√n。
If the population is normal, x̄ is exactly N(μ, σ²/n). For large n (typically n ≥ 30), the Central Limit Theorem ensures x̄ is approximately normal regardless of the population shape.
若总体正态，则 x̄ 精确服从 N(μ, σ²/n)。当 n 较大（通常 n ≥ 30）时，根据中心极限定理，无论总体分布如何，x̄ 近似正态。
A confidence interval provides a range of plausible values for the population mean. For known σ:
置信区间为总体均值提供一个可信范围。当 σ 已知时：
x̄ ± z* × (σ/√n)
x̄ ± z* × (σ/√n)
When σ is unknown and n is small, use the t‑distribution with n - 1 degrees of freedom: x̄ ± t* × (s/√n).
当 σ 未知且 n 较小时，使用自由度为 n - 1 的 t 分布：x̄ ± t* × (s/√n)。
The width of the interval depends on the confidence level (e.g., 90%, 95%, 99%), sample size n, and variability. Larger n reduces the margin of error.
区间的宽度取决于置信水平（如 90%、95%、99%）、样本量 n 和变异程度。n 越大，误差界限越小。
7. Hypothesis Testing | 假设检验
Hypothesis testing is a formal procedure for deciding between two competing claims about a population parameter. The null hypothesis H₀ represents the status quo, while the alternative H₁ represents the research claim.
假设检验是在关于总体参数的两个对立论断之间作出决策的正式程序。零假设 H₀ 代表现状，备择假设 H₁ 代表研究主张。
H₀ and H₁; (2) Choose significance level α; (3) Compute the test statistic; (4) Find the p‑value or critical region; (5) Make a conclusion: if p‑value ≤ α, reject H₀; otherwise, do not reject H₀.
步骤：(1) 陈述 H₀ 和 H₁；(2) 选择显著性水平 α；(3) 计算检验统计量；(4) 求 p 值或确定临界域；(5) 做出结论：若 p 值 ≤ α，拒绝 H₀；否则不拒绝 H₀。
For a one-sample z‑test of the mean (σ known), test statistic:
对于已知 σ 的单样本均值 z 检验，检验统计量为：
z = (x̄ - μ₀) / (σ/√n)
z = (x̄ - μ₀) / (σ/√n)
A p‑value is the probability of observing a test statistic at least as extreme as the one obtained, assuming H₀ is true. Rejection regions can be one‑tailed or two‑tailed depending on H₁. Never accept H₀; only fail to reject it.
p 值是在 H₀ 为真的条件下，观察到至少与所得统计量同样极端的检验统计量的概率。否定域根据 H₁ 可为单侧或双侧。永远不要接受 H₀，只能不拒绝它。
8. Correlation & Linear Regression | 相关与线性回归
easures the strength and direction of a linear relationship between two variables. Pearson's correlation coefficient r satisfies -1 ≤ r ≤ 1.
相关度量两个变量之间线性关系的强度和方向。皮尔逊相关系数 r 满足 -1 ≤ r ≤ 1。
The formula for r is:
r 的计算公式为：
r = Σ(x - x̄)(y - ȳ) / √[ Σ(x - x̄)² Σ(y - ȳ)² ]
r = Σ(x - x̄)(y
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