Pre-U AQA Statistics: Formula & Theorem Quick Reference Handbook | Pre-U AQA 统计：公式定理速查手册

📚 Pre-U AQA Statistics: Formula & Theorem Quick Reference Handbook | Pre-U AQA 统计：公式定理速查手册

This quick reference handbook compiles the essential formulas, theorems, and statistical distributions required for the Pre-U AQA Statistics course. Use it to reinforce your understanding of probability, inference, and modelling. Each section pairs concise English explanations with Chinese translations to help you master the concepts bilingually.

本速查手册汇集了 Pre-U AQA 统计课程所需的核心公式、定理和统计分布。利用它巩固概率、推断和建模知识。每个小节将简洁的英文解释与中文译文配对，帮助您双语掌握概念。

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1. Basic Probability Rules | 基础概率法则

P(A') = 1 − P(A)

The complement rule: the probability of an event not occurring is one minus the probability that it does occur.

补集规则：事件不发生的概率等于一减去事件发生的概率。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

ddition rule works for any two events A and B. If A and B are mutually exclusive, then P(A ∩ B) = 0.

一般加法法则适用于任意两事件 A 和 B。若 A 与 B 互斥，则 P(A ∩ B) = 0。

P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)

ation rule links joint probability to conditional probability. For independent events, P(A ∩ B) = P(A) × P(B).

乘法法则将联合概率与条件概率联系起来。对于独立事件，P(A ∩ B) = P(A) × P(B)。

P(A|B) = P(A ∩ B) / P(B), P(B) > 0

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

条件概率给出在 B 已发生时 A 的概率。

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2. Bayes' Theorem | 贝叶斯定理

P(A|B) = [P(B|A) × P(A)] / P(B)

Bayes' theorem allows us to update the probability of event A after observing B. The denominator P(B) can be expanded as P(B|A)P(A) + P(B|A')P(A').

贝叶斯定理允许我们在观察到 B 后更新事件 A 的概率。分母 P(B) 可展开为 P(B|A)P(A) + P(B|A')P(A')。

It is especially useful in diagnostic testing and decision making when prior probabilities are known.

当已知先验概率时，它在诊断测试和决策制定中尤为有用。

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3. Discrete Random Variables | 离散随机变量

For a discrete random variable X taking values xᵢ with probabilities pᵢ = P(X = xᵢ):

对于离散随机变量 X，取值 xᵢ 的概率为 pᵢ = P(X = xᵢ)：

E(X) = μ = Σ xᵢ pᵢ

The expected value E(X) is the probability‑weighted average of all possible values.

期望值 E(X) 是所有可能值的概率加权平均。

Var(X) = σ² = E[(X − μ)²] = E(X²) − μ²

Variance measures the spread of a distribution. The standard deviation is σ = √Var(X).

方差衡量分布的离散程度。标准差为 σ = √Var(X)。

E(aX + b) = a E(X) + b

Var(aX + b) = a² Var(X)

Linear transformations of random variables shift the mean and scale the variance accordingly.

随机变量的线性变换相应地平移均值并缩放方差。

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4. Binomial Distribution | 二项分布

X ~ B(n, p) P(X = k) = ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ

A binomial model counts the number of successes in n independent Bernoulli trials, each with success probability p.

二项模型统计在 n 次独立伯努利试验中成功的次数，每次成功概率为 p。

E(X) = np, Var(X) = np(1 − p)

The conditions for a binomial distribution are: fixed number of trials, two outcomes per trial, independent trials, and constant probability p.

二项分布的条件：固定试验次数、每次试验两种结果、独立试验、概率 p 恒定。

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5. Poisson Distribution | 泊松分布

X ~ Po(λ) P(X = k) = (e⁻λ λᵏ) / k! , k = 0, 1, 2, …

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming events occur independently at a constant average rate λ.

泊松分布用于建模在固定时间或空间间隔内发生的事件数，假设事件以恒定平均速率 λ 独立发生。

E(X) = λ, Var(X) = λ

n distribution can also approximate a binomial when n is large and p is small with λ = np.

均值与方差相等。当 n 很大且 p 很小时，泊松分布可用 λ = np 近似二项分布。

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6. Geometric Distribution | 几何分布

X ~ Geo(p) P(X = k) = p (1 − p)ᵏ⁻¹ , k = 1, 2, 3, …

The geometric distribution counts the number of trials up to and including the first success in a sequence of independent Bernoulli trials.

几何分布统计在独立伯努利试验序列中直到首次成功（含成功）的试验次数。

E(X) = 1 / p, Var(X) = (1 − p) / p²

It possesses the memoryless property: P(X > s + t | X > s) = P(X > t).

它具有无记忆性：P(X > s + t | X > s) = P(X > t)。

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7. Normal Distribution | 正态分布

X ~ N(μ, σ²) Z = (X − μ) / σ ~ N(0, 1)

The normal distribution is a continuous, symmetric bell‑shaped curve defined completely by its mean μ and variance σ². Standardising converts any normal variable to the standard normal Z.

正态分布是由均值 μ 和方差 σ² 完全确定的连续、对称的钟形曲线。标准化可将任意正态变量转换为标准正态 Z。

f(x) = 1/(σ√2π) e^(−½((x−μ)/σ)²)

Empirical rule: about 68% of data lie within μ ± 1σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

经验法则：约 68% 的数据落在 μ ± 1σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。

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8. Central Limit Theorem | 中心极限定理

random sample of size n from any population with mean μ and variance σ², the sample mean X̄ is approximately normally distributed when n is large (usually n ≥ 30).

对于从均值为 μ、方差为 σ² 的任意总体中抽取的大小为 n 的随机样本，当 n 足够大（通常 n ≥ 30）时，样本均值 X̄ 近似服从正态分布。

X̄ ⁓ N(μ, σ²/n) approximately

The CLT justifies the use of normal‑based confidence intervals and hypothesis tests for means, even when the population is not normal.

中心极限定理为均值的正态置信区间和假设检验提供了依据，即使总体不服从正态分布。

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9. Confidence Intervals | 置信区间

A (1 − α) × 100% confidence interval for a population mean μ when σ is known:

当 σ 已知时，总体均值 μ 的 (1 − α) × 100% 置信区间：

x̄ ± z* × σ / √n

When σ is unknown, use the t‑distribution with n − 1 degrees of freedom:

当 σ 未知时，使用自由度为 n − 1 的 t 分布：

x̄ ± t*ₙ₋₁ × s / √n

For a population proportion p: p̂ ± z* × √[p̂(1 − p̂)/n]. The margin of error decreases as sample size increases.

对于总体比例 p：p̂ ± z* × √[p̂(1 − p̂)/n]。误差边际随样本量增大而减小。

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10. Hypothesis Testing – Z, t, χ² Tests | 假设检验—Z检验、t检验、卡方检验

The p‑value is the probability, under the null hypothesis H₀, of obtaining a test statistic at least as extreme as the observed one. Reject H₀ if p‑value < α.

p 值是在原假设 H₀ 下获得至少与观测值一样极端的检验统计量的概率。若 p 值 < α，则拒绝 H₀。

Z = (x̄ − μ₀) / (σ/√n)

One‑sample Z‑test for a mean (σ known). The critical z* values are based on the standard normal distribution.

单样本均值 Z 检验（σ 已知）。临界值 z* 基于标准正态分布。

t = (x̄ − μ₀) / (s/√n), df = n − 1

One‑sample t‑test for a mean when σ is unknown. It is robust to moderate departures from normality.

σ 未知时均值的单样本 t 检验。它对适度偏离正态性具有稳健性。

χ² = Σ [(Oᵢ − Eᵢ)² / Eᵢ]

Chi‑squared test for goodness‑of‑fit or independence. Oᵢ are observed frequencies, Eᵢ expected frequencies under H₀. Degrees of freedom depend on the number of categories and constraints.

卡方检验用于拟合优度或独立性检验。Oᵢ 为观测频数，Eᵢ 为 H₀ 下的期望频数。自由度取决于类别数和约束条件。

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11. Correlation and Regression | 相关与回归

r = Sₓᵧ / √Sₓₓ Sᵧᵧ

Pearson’s product‑ res the strength and direction of a linear relationship between two variables. −1 ≤ r ≤ 1.

皮尔逊积矩相关系数 r 衡量两变量间线性关系的强度和方向。−1 ≤ r ≤ 1。

The least‑squares regression line of y on x is given by:

y 对 x 的最小二乘回归直线为：

ŷ = a + b x, where b = Sₓᵧ / Sₓₓ, a = ȳ − b x̄

ient of determination R² = r² indicates the proportion of variability in y explained by x.

决定系数 R² = r² 表示由 x 解释的 y 变异的比例。

Residuals (eᵢ = yᵢ − ŷᵢ) should be randomly scattered. A pattern suggests a non‑linear relationship or non‑constant variance.

残差 (eᵢ = yᵢ − ŷᵢ) 应随机分布。出现模式则暗示非线性关系或方差不齐。

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12. ANOVA (Analysis of Variance) | 方差分析

One‑way ANOVA compares the means of three or more independent groups. It partitions total variability into between‑group (B) and within‑group (W) variation.

单因素方差分析比较三个或更多独立组的均值。它将总变异分解为组间 (B) 和组内 (W) 变异。

F = MSB / MSW = (SSB / df_B) / (SSW / df_W)

Under H₀: μ₁ = μ₂ = … = μₖ, the F‑statistic follows an F‑ (k−1, N−k) degrees of freedom. A large F value suggests the group means are not all equal.

在 H₀：μ₁ = μ₂ = … = μₖ 下，F 统计量服从自由度为 (k−1, N−k) 的 F 分布。较大的 F 值表明各组均值不全相等。

Source	SS	df	MS	F
Between	SSB	k−1	MSB	MSB/MSW
Within	SSW	N−k	MSW
Total	SST	N−1

SSB = Σ nᵢ (x̄ᵢ − x̄)², SSW = Σ (nᵢ−1)sᵢ². Assumptions: normality, homogeneity of variances, and independence.

SSB = Σ nᵢ (x̄ᵢ − x̄)², SSW = Σ (nᵢ−1)sᵢ²。假设：正态性、方差齐性和独立性。

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