📚 SQA Year 13 Statistics Formula & Theorem Quick Reference | SQA 高年级统计公式定理速查手册
This quick-reference handbook collates the essential formulae, theorems and statistical tables you will need for the SQA Advanced Higher Statistics course (Year 13). Use it to reinforce your understanding of probability, inference, regression and non‑parametric tests – every statement is paired with a Chinese translation to support bilingual learners.
这本速查手册整理了 SQA 高年级统计（Advanced Higher Statistics）所需的核心公式、定理和统计表格。每个知识点都配有中英双语解说，帮助你在概率、推断、回归和非参数检验等模块中快速定位关键内容，巩固备考。
1. Basic Probability Rules | 基本概率法则
The complement rule states that P(A’) = 1 – P(A), where A’ is the event ‘not A’.
补集法则：P(A’) = 1 – P(A)，其中 A’ 表示事件“非 A”。
For two events A and B, the addition rule is 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).
对于两个事件 A 和 B，加法法则为 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. Multiplication rule follows as P(A ∩ B) = P(A | B) P(B).
条件概率：P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。乘法法则即为 P(A ∩ B) = P(A | B) P(B)。
Events A and B are independent if and only if P(A ∩ B) = P(A) P(B) or equivalently P(A | B) = P(A).
事件 A 与 B 相互独立当且仅当 P(A ∩ B) = P(A) P(B)，或等价地 P(A | B) = P(A)。
2. Discrete Probability Distributions | 离散概率分布
Expected value (mean) of a discrete random variable X: E(X) = Σ x P(X = x). Variance: Var(X) = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – [E(X)]².
离散随机变量 X 的期望值（均值）：E(X) = Σ x P(X = x)。方差：Var(X) = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – [E(X)]²。
Binomial distribution: X ~ B(n, p). Then P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ, E(X) = np, Var(X) = np(1 – p).
二项分布：X ~ B(n, p)。则 P(X = k) = C(n, k) pᵏ (1 – p)ⁿ⁻ᵏ，E(X) = np，Var(X) = np(1 – p)。
Poisson distribution: X ~ Po(λ). P(X = k) = (λᵏ e⁻λ) / k!, E(X) = Var(X) = λ. Useful as an approximation to B(n, p) when n is large and p is small (≈ np).
泊松分布：X ~ Po(λ)。P(X = k) = (λᵏ e⁻λ) / k!，E(X) = Var(X) = λ。当 n 很大且 p 很小时（≈ np），可用作二项分布的近似。
3. Continuous Probability Distributions | 连续概率分布
For a continuous random variable X with probability density function f(x), the probability that X lies in [a, b] is ∫ₐᵇ f(x) dx. The total area under the curve is 1.
对于具有概率密度函数 f(x) 的连续随机变量 X，X 落在区间 [a, b] 内的概率为 ∫ₐᵇ f(x) dx。整个曲线下的面积为 1。
The uniform distribution U(a, b) has density f(x) = 1/(b – a), mean (a + b)/2, variance (b – a)²/12.
均匀分布 U(a, b) 的密度为 f(x) = 1/(b – a)，均值为 (a + b)/2，方差为 (b – a)²/12。
Normal distribution: X ~ N(μ, σ²). The standard normal Z = (X – μ)/σ ~ N(0, 1). Critical values such as z₀.₀₂₅ = 1.96 are used in inference.
正态分布：X ~ N(μ, σ²)。标准正态 Z = (X – μ)/σ ~ N(0, 1)。临界值如 z₀.₀₂₅ = 1.96 用于统计推断。
If X ~ N(μ, σ²) then a linear transformation aX + b is also normally distributed with mean aμ + b and variance a²σ².
若 X ~ N(μ, σ²)，则线性变换 aX + b 仍服从正态分布，均值为 aμ + b，方差为 a²σ²。
4. Sampling Distributions | 抽样分布
The sample mean X̄ from a population with mean μ and variance σ² has E(X̄) = μ and Var(X̄) = σ²/n. If the population is normal, X̄ ~ N(μ, σ²/n).
从均值为 μ、方差为 σ² 的总体中抽取的样本均值 X̄ 满足 E(X̄) = μ，Var(X̄) = σ²/n。若总体正态，则 X̄ ~ N(μ, σ²/n)。
Central Limit Theorem (CLT): For a large sample (n ≥ 30), X̄ is approximately normal regardless of the population shape.
中心极限定理 (CLT)：当样本量足够大（n ≥ 30）时，无论总体分布形状如何，样本均值 X̄ 近似服从正态分布。
The distribution of the sample proportion p̂ is approximately N( p, p(1 – p)/n ) provided np ≥ 5 and n(1 – p) ≥ 5.
样本比例 p̂ 的抽样分布近似为 N( p, p(1 – p)/n )，前提是 np ≥ 5 且 n(1 – p) ≥ 5。
5. Confidence Intervals | 置信区间
A 95% confidence interval for the population mean μ when σ is known: X̄ ± z₀.₀₂₅ × (σ / √n).
当总体标准差 σ 已知时，总体均值 μ 的 95% 置信区间为：X̄ ± z₀.₀₂₅ × (σ / √n)。
When σ is estimated by the sample standard deviation s, use the t‑distribution: X̄ ± tₐ/₂,ₙ₋₁ × (s / √n), where df = n – 1.
当用样本标准差 s 估计 σ 时，使用 t 分布：X̄ ± tₐ/₂,ₙ₋₁ × (s / √n)，自由度 df = n – 1。
Confidence interval for a population proportion p: p̂ ± zₐ/₂ × √[ p̂(1 – p̂) / n ].
总体比例 p 的置信区间：p̂ ± zₐ/₂ × √[ p̂(1 – p̂) / n ]。
For the difference of two means (independent samples), the interval is (X̄₁ – X̄₂) ± tₐ/₂,df × √(s₁²/n₁ + s₂²/n₂) assuming equal variances not required (Welch’s df).
两个独立样本均值之差的置信区间：（X̄₁ – X̄₂）± tₐ/₂,df × √(s₁²/n₁ + s₂²/n₂)，通常使用 Welch 自由度。
6. Hypothesis Testing – Core Steps | 假设检验 – 核心步骤
State the null hypothesis H₀ and the alternative hypothesis H₁. Decide the significance level α (commonly 0.05) and identify the test statistic.
陈述原假设 H₀ 与备择假设 H₁。确定显著性水平 α（通常为 0.05），并识别检验统计量。
Compare the p‑value with α: if p‑value ≤ α, reject H₀; otherwise, do not reject H₀. A p‑value is the probability of obtaining a result at least as extreme, assuming H₀ is true.
将 p 值与 α 比较：若 p 值 ≤ α，则拒绝 H₀；否则不拒绝 H₀。p 值是在原假设成立的条件下，获得当前结果或更极端结果的概率。
Type I error (α) is rejecting H₀ when true; Type II error (β) is failing to reject H₀ when false. Power = 1 – β.
第一类错误（α）为弃真错误，第二类错误（β）为存伪错误。检验功效 = 1 – β。
7. Tests for Means and Proportions | 均值与比例的检验
One‑sample z‑test for μ (σ known): test statistic Z = (X̄ – μ₀) / (σ / √n). One‑sample t‑test (σ unknown): t = (X̄ – μ₀) / (s / √n) with df = n – 1.
单样本 z 检验（σ 已知）：检验统计量 Z = (X̄ – μ₀) / (σ / √n)。单样本 t 检验（σ 未知）：t = (X̄ – μ₀) / (s / √n)，自由度 = n – 1。
Two‑sample t‑test for difference of means (unpaired) assuming equal variance: pooled variance s²ₚ = [(n₁ – 1)s₁² + (n₂ – 1)s₂²] / (n₁ + n₂ – 2); test statistic t = (X̄₁ – X̄₂) / (sₚ √(1/n₁ + 1/n₂)).
等方差假设下的两独立样本 t 检验：合并方差 s²ₚ = [(n₁ – 1)s₁² + (n₂ – 1)s₂²] / (n₁ + n₂ – 2)；检验统计量 t = (X̄₁ – X̄₂) / (sₚ √(1/n₁ + 1/n₂))。
Paired t‑test: compute the differences dᵢ = X₁ᵢ – X₂ᵢ, then perform a one‑sample t‑test on d̄ using t = d̄ / (s_d / √n).
配对 t 检验：计算差值 dᵢ = X₁ᵢ – X₂ᵢ，然后对 d̄ 进行单样本 t 检验，t = d̄ / (s_d / √n)。
One‑sample proportion test: Z = (p̂ – p₀) / √[p₀(1 – p₀)/n]. Two‑sample proportion test uses a pooled estimate under H₀.
单样本比例检验：Z = (p̂ – p₀) / √[p₀(1 – p₀)/n]。两样本比例检验在原假设下使用合并估计值。
8. Chi‑Squared Tests | 卡方检验
Chi‑squared goodness‑of‑fit test: χ² = Σ (Oᵢ – Eᵢ)² / Eᵢ, where Oᵢ is observed frequency and Eᵢ is expected frequency. Degrees of freedom df = k – 1 – m (k categories, m estimated parameters).
卡方拟合优度检验：χ² = Σ (Oᵢ – Eᵢ)² / Eᵢ，Oᵢ 为观测频数，Eᵢ 为期望频数。自由度 df = k – 1 – m（k 个类别，m 个估计参数）。
Chi‑squared test for independence (contingency table): expected frequency E = (row total × column total) / grand total. df = (r – 1)(c – 1).
独立性卡方检验（列联表）：期望频数 E = (行合计 × 列合计) / 总计。自由度 df = (r – 1)(c – 1)。
Conditions: all expected frequencies should be at least 5. If not, consider combining categories or using Fisher’s exact test for 2 × 2 tables.
条件：所有期望频数应至少为 5。若不满足，可考虑合并类别或对 2 × 2 表使用 Fisher 精确检验。
9. Correlation and Regression | 相关与回归
Pearson’s product‑moment correlation coefficient r = Σ (xᵢ – x̄)(yᵢ – ȳ) / √[ Σ (xᵢ – x̄)² Σ (yᵢ – ȳ)² ]. It measures linear association; –1 ≤ r ≤ 1.
皮尔逊积矩相关系数 r = Σ (xᵢ – x̄)(yᵢ – ȳ) / √[ Σ (xᵢ – x̄)² Σ (yᵢ – ȳ)² ]。度量线性相关程度；–1 ≤ r ≤ 1。
Test for zero correlation: t = r √(n – 2) / √(1 – r²), with df = n – 2.
检验相关性是否为零：t = r √(n – 2) / √(1 – r²)，自由度 df = n – 2。
Simple linear regression model: y = β₀ + β₁ x + ε. Least‑squares estimates: β₁ = Σ (xᵢ – x̄)(yᵢ – ȳ) / Σ (xᵢ – x̄)², and β₀ = ȳ – β₁ x̄.
简单线性回归模型：y = β₀ + β₁ x + ε。最小二乘估计：β₁ = Σ (xᵢ – x̄)(yᵢ – ȳ) / Σ (xᵢ – x̄)²，β₀ = ȳ – β₁ x̄。
Confidence interval for slope β₁: β₁ ± tₐ/₂,ₙ₋₂ × SE(β₁), where SE(β₁) = s / √[ Σ (xᵢ – x̄)² ] and s² = Σ (yᵢ – ŷᵢ)² / (n – 2).
斜率 β₁ 的置信区间：β₁ ± tₐ/₂,ₙ₋₂ × SE(β₁)，其中 SE(β₁) = s / √[ Σ (xᵢ – x̄)² ]，s² = Σ (yᵢ – ŷᵢ)² / (n – 2)。
Prediction interval for a new response is wider than the confidence interval for the mean response, because it incorporates individual variation.
新观测值的预测区间宽于均值响应的置信区间，因为它包含了个体变异。
10. Analysis of Variance (ANOVA) | 方差分析
One‑way ANOVA partitions total variability SST = SSTreatment + SSE. The test statistic is F = (SSTreatment / (k – 1)) / (SSE / (N – k)), which under H₀ follows an F‑distribution with df₁ = k – 1, df₂ = N – k.
单因素方差分析将总变异分解为 SST = SSTreatment + SSE。检验统计量 F = (SSTreatment / (k – 1)) / (SSE / (N – k))，在原假设下服从 F 分布，自由度 df₁ = k – 1，df₂ = N – k。
ANOVA assumptions: independence, normality of residuals, and homogeneity of variances. Check with residual plots or Levene’s test.
方差分析假设：独立性、残差正态性以及方差齐性。可通过残差图或 Levene 检验进行诊断。
Post‑hoc comparisons (e.g. Tukey’s HSD) are performed only if the overall F‑test is significant, to identify which means differ.
若整体 F 检验显著，则进行事后多重比较（如 Tukey HSD），以确定哪些均值之间存在差异。
11. Non‑parametric Tests | 非参数检验
Wilcoxon signed‑rank test (paired or one‑sample): rank the absolute differences, then sum the ranks for positive differences (W⁺). The test statistic is compared to a critical value or converted to a Z‑score.
Wilcoxon 符号秩检验（配对或单样本）：对差值的绝对值进行排序，计算正差值的秩和 W⁺。检验统计量与临界值比较或转换为 Z 值。
Mann‑Whitney U test (two independent samples): ranks all observations together and computes U = n₁n₂ + n₁(n₁+1)/2 – R₁, where R₁ is the sum of ranks in group 1.
Mann‑Whitney U 检验（两独立样本）：将所有观测值统一排序，计算 U = n₁n₂ + n₁(n₁+1)/2 – R₁，其中 R₁ 为第 1 组秩和。
Spearman’s rank correlation: rₛ is Pearson’s r calculated on the ranks. Used when data are ordinal or not bivariate normal. Test of ρₛ = 0 uses the same t‑transformation.
Spearman 秩相关系数：rₛ 是基于排序值的皮尔逊 r。适用于顺序数据或非双变量正态数据。检验 ρₛ = 0 使用相同的 t 变换。
12. Probability Generating Functions & Bivariate Data | 概率生成函数与二元数据
The probability generating function (PGF) of a non‑negative integer‑valued random variable X is Gₓ(t) = E(tᵡ) = Σ tᵏ P(X = k). It yields moments: Gₓ'(1) = E(X), Gₓ”(1) = E[X(X – 1)].
非负整数值随机变量 X 的概率生成函数 (PGF) 定义为 Gₓ(t) = E(tᵡ) = Σ tᵏ P(X = k)。可用于求矩：Gₓ'(1) = E(X)，Gₓ”(1) = E[X(X – 1)]。
For the sum of independent random variables X and Y with PGFs Gₓ and Gᵧ, the PGF of X + Y is Gₓ(t) × Gᵧ(t).
若独立随机变量 X 和 Y 的 PGF 分别为 Gₓ 和 Gᵧ，则 X + Y 的 PGF 为 Gₓ(t) × Gᵧ(t)。
Covariance and correlation: Cov(X, Y) = E(XY) – E(X)E(Y). Correlation ρ = Cov(X, Y) / [σₓ σᵧ]. Indicate strength and direction of linear relationship.
协方差与相关系数：Cov(X, Y) = E(XY) – E(X)E(Y)。相关系数 ρ = Cov(X, Y) / [σₓ σᵧ]，衡量线性关系的强度与方向。
The bivariate normal distribution is characterised by five parameters: μₓ, μᵧ, σₓ², σᵧ², and ρ. Conditional means lie on straight regression lines.
二元正态分布由五个参数描述：μₓ, μᵧ, σₓ², σᵧ² 和 ρ。条件均值位于直线回归线上。
Published by TutorHao | SQA Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导