Year 13 WJEC Statistics: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 统计：公式定理速查手册

📚 Year 13 WJEC Statistics: Formula & Theorem Quick Reference Handbook | Year 13 WJEC 统计：公式定理速查手册

For Year 13 WJEC Statistics students, having a reliable quick reference of essential formulas and theorems is crucial for exam success. This handbook consolidates the key statistical tools you will encounter, from probability distributions and hypothesis testing to correlation and non-parametric methods. Use it as your go-to revision companion to master the statistical reasoning required at A2 level.

对于 Year 13 WJEC 统计学学生来说，拥有一份可靠的公式和定理速查手册是考试成功的关键。本手册整合了你将遇到的关键统计工具，涵盖概率分布、假设检验、相关性分析以及非参数方法。请把它作为你复习的首选伴侣，以掌握 A2 阶段所需的统计推理能力。

1. Probability & Random Variables | 概率与随机变量

Understanding the behaviour of random variables underpins all inference. Here are the foundational definitions and rules you must recall instantly.

理解随机变量的行为是所有统计推断的基础。以下是您必须立即回忆起来的基本定义和规则。

= Σ x·P(X = x). Variance is Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]².

对于离散随机变量 X，概率质量函数为 P(X = x)：期望值 E(X) = Σ x·P(X = x)。方差 Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]²。

f(x): E(X) = ∫ x f(x) dx over the domain. Var(X) = ∫ (x − μ)² f(x) dx = ∫ x² f(x) dx − μ².

对于连续随机变量，概率密度函数为 f(x)：E(X) = ∫ x f(x) dx，积分范围为定义域。方差 Var(X) = ∫ (x − μ)² f(x) dx = ∫ x² f(x) dx − μ²。

(X) + E(Y). If independent, Var(X + Y) = Var(X) + Var(Y) and Var(X − Y) = Var(X) + Var(Y).

重要规则：E(aX + b) = aE(X) + b 且 Var(aX + b) = a² Var(X)。对于两个随机变量 X 和 Y：E(X + Y) = E(X) + E(Y)。如果独立，则 Var(X + Y) = Var(X) + Var(Y)，且 Var(X − Y) = Var(X) + Var(Y)。

Concept	Formula
Covariance (X,Y)	Cov(X,Y) = E[(X − μX)(Y − μY)] = E(XY) − E(X)E(Y)
Correlation coefficient	ρ = Cov(X,Y) / (σX σY)

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2. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in n independent Bernoulli trials, each with probability of success p.

二项分布用于模拟在 n 次独立的伯努利试验中成功的次数，每次试验成功概率为 p。

Conditions: fixed number of trials n; each trial independent; only two outcomes (success/failure); constant probability p. Notation: X ~ B(n, p).

条件：试验次数 n 固定；每次试验独立；只存在两种结果（成功/失败）；概率 p 恒定。记作：X ~ B(n, p)。

Probability mass function: P(X = r) = ⁿC_r p^r (1 − p)^n−r, for r = 0,1,2,...,n. Here ⁿC_r = n! / [r!(n−r)!].

概率质量函数：P(X = r) = ⁿC_r p^r (1 − p)^n−r，其中 r = 0,1,2,...,n。这里 ⁿC_r = n! / [r!(n−r)!]。

Mean: E(X) = np. Variance: Var(X) = np(1 − p). Standard deviation: σ = √[np(1 − p)].

均值：E(X) = np。方差：Var(X) = np(1 − p)。标准差：σ = √[np(1 − p)]。

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3. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a constant mean rate λ and independence between occurrences.

泊松分布用于模拟在固定时间或空间间隔内发生的事件次数，假设平均发生率 λ 恒定且各次发生独立。

Conditions: events occur singly; events occur randomly; events occur independently; the mean rate λ is constant. Notation: X ~ Po(λ).

条件：事件单位发生；事件随机发生；事件相互独立；平均率 λ 恒定。记作：X ~ Po(λ)。

Probability mass function: P(X = r) = (e^−λ λ^r) / r!, for r = 0,1,2,...

概率质量函数：P(X = r) = (e^−λ λ^r) / r!，其中 r = 0,1,2,...

standard deviation σ = √λ. For the Poisson distribution, the mean and variance are equal.

均值：E(X) = λ。方差：Var(X) = λ。因此标准差 σ = √λ。对于泊松分布，均值和方差相等。

Sum of independent Poisson variables: If X ~ Po(λ₁) and Y ~ Po(λ₂) independent, then X + Y ~ Po(λ₁ + λ₂).

独立泊松变量之和：如果 X ~ Po(λ₁) 且 Y ~ Po(λ₂) 独立，则 X + Y ~ Po(λ₁ + λ₂)。

ximated by Poisson with λ = np when n is large and p is small (typically n ≥ 50, np < 10).

近似：当 n 很大且 p 很小（通常 n ≥ 50, np < 10）时，二项分布 B(n, p) 可用参数 λ = np 的泊松分布近似。

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4. Continuous Uniform & Exponential Distributions | 连续均匀分布与指数分布

lly likely. PDF: f(x) = 1/(b − a) for a ≤ x ≤ b. Mean = (a + b)/2, variance = (b − a)²/12.

连续均匀分布：在区间 [a, b] 内，相同长度的区间具有相等的可能性。概率密度函数：f(x) = 1/(b − a)，a ≤ x ≤ b。均值 = (a + b)/2，方差 = (b − a)²/12。

The exponential distribution models waiting times between Poisson events. PDF: f(x) = λ e^−λx for x ≥ 0. Mean = 1/λ, variance = 1/λ². CDF: F(x) = 1 − e^−λx.

指数分布用于模拟泊松事件之间的等待时间。概率密度函数：f(x) = λ e^−λx，x ≥ 0。均值 = 1/λ，方差 = 1/λ²。累积分布函数：F(x) = 1 − e^−λx。

Memoryless property of exponential: P(X > s + t | > t). Only the exponential distribution has this property among continuous distributions.

指数分布的无记忆性：P(X > s + t | X > s) = P(X > t)。在连续分布中，只有指数分布具有此性质。

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5. Normal Distribution & Central Limit Theorem | 正态分布与中心极限定理

The normal distribution N(μ, σ²) is fundamental. Its PDF is f(x) = (1/(σ√2π)) exp(−(x − μ)²/(2σ²)). Standard normal Z ~ N(0, 1) has mean 0, variance 1.

正态分布 N(μ, σ²) 是基础。其概率密度函数为 f(x) = (1/(σ√2π)) exp(−(x − μ)²/(2σ²))。标准正态分布 Z ~ N(0, 1)，均值为 0，方差为 1。

ion: Z = (X − μ) / σ. Use Z-tables to find probabilities Φ(z). Symmetry: Φ(−z) = 1 − Φ(z).

标准化：Z = (X − μ) / σ。使用 Z 表查找概率 Φ(z)。对称性：Φ(−z) = 1 − Φ(z)。

Linear combinations of normal variables: if X ~ N(μX, σX²) and Y ~ N(μY, σY²) independent, then aX + bY ~ N(aμX + bμY, a²σX² + b²σY²).

正态变量的线性组合：若 X ~ N(μX, σX²) 且 Y ~ N(μY, σY²) 独立，则 aX + bY ~ N(aμX + bμY, a²σX² + b²σY²)。

ndom sample of size n from any distribution with mean μ and variance σ², the sample mean X̄ σ²/n) for large n (usually n ≥ 30). This allows normal approximations for sums and means.

中心极限定理：对于来自任意均值为 μ、方差为 σ² 的分布的随机样本，当样本量 n 较大（通常 n ≥ 30）时，样本均值 X̄ 近似服从 N(μ, σ²/n)。这使得可以用正态近似来处理和与均值。

Continuity correction: When approximating a discrete distribution (binomial/Poisson) by normal, add or subtract 0.5 from the discrete value to adjust for continuity.

连续性校正：当用正态分布近似离散分布（二项/泊松）时，对离散值加减 0.5 以调整连续性。

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6. Sampling Distributions & Estimation | 抽样分布与估计

estimator is a statistic used to estimate a population parameter. Unbiased estimator: E(θ̂) = θ. Sample mean X̄ is an unbiased estimator of μ. Sample variance s² = Σ(x − X̄)²/(n − 1) is unbiased for σ².

估计量是用来估计总体参数的统计量。无偏估计量：E(θ̂) = θ。样本均值 X̄ 是 μ 的无偏估计量。样本方差 s² = Σ(x − X̄)²/(n − 1) 是 σ² 的无偏估计量。

Standard error of the mean: SE(X̄) = σ/√n. If σ unknown, estimated standard error = s/√n.

均值的标准误差：SE(X̄) = σ/√n。若 σ 未知，估计标准误差 = s/√n。

Confidence interval for population mean μ (σ known, or large sample using CLT): X̄ ± z* × σ/√n. When σ unknown and normally distributed data, use t-distribution: X̄ ± t* × s/√n with df = n − 1.

总体均值 μ 的置信区间（σ 已知，或大样本使用 CLT）：X̄ ± z* × σ/√n。当 σ 未知且数据服从正态分布时，使用 t 分布：X̄ ± t* × s/√n，自由度 df = n − 1。

Parameter	Confidence Interval
Mean μ (known σ)	X̄ ± zα/2 × (σ/√n)
Mean μ (unknown σ)	X̄ ± tn−1,α/2 × (s/√n)
Difference of two means (independent, known variances)	(X̄₁ − X̄₂) ± zα/2 × √σ₁²/n₁ + σ₂²/n₂
Difference of two means (pooled variance, equal variances unknown)	(X̄₁ − X̄₂) ± tdf,α/2 × sp × √1/n₁ + 1/n₂, df = n₁ + n₂ − 2

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7. Hypothesis Testing – One Sample | 假设检验 – 单样本

Hypothesis testing involves: stating null hypothesis H₀ and alternative H₁, choosing significance level α (commonly 5% or 1%), calculating a test statistic, and comparing to critical value or using p-value.

假设检验包括：陈述原假设 H₀ 和备择假设 H₁，选择显著性水平 α（通常 5% 或 1%），计算检验统计量，并与临界值比较或使用 p 值。

Test for a mean (σ known / large sample): Z = (X̄ − μ₀) / (σ/√n). For small sample with unknown σ, use t-test: t = (X̄ − μ₀) / (s/√n) with n − 1 degrees of freedom.

对均值检验（σ 已知/大样本）：Z = (X̄ − μ₀) / (σ/√n)。对于小样本且 σ 未知，使用 t 检验：t = (X̄ − μ₀) / (s/√n)，自由度 n − 1。

Test for a binomial proportion p: Under H₀: p = p₀, test statistic Z = (p̂ − p₀) / √[p₀(1 − p₀)/n], where p̂ = X/n. Requires np₀ ≥ 5 and n(1 − p₀) ≥ 5 for normal approximation.

对二项比例 p 的检验：在 H₀: p = p₀ 下，检验统计量 Z = (p̂ − p₀) / √[p₀(1 − p₀)/n]，其中 p̂ = X/n。需要满足 np₀ ≥ 5 和 n(1 − p₀) ≥ 5 以采用正态近似。

ability α). Type II error: failing to reject H₀ when false (probability β). Power = 1 − β.

第一类错误：当 H₀ 为真时拒绝它（概率 α）。第二类错误：当 H₀ 为假时未拒绝它（概率 β）。功效 = 1 − β。

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8. Hypothesis Testing – Two Samples | 假设检验 – 双样本

Comparing two means from independent samples: test statistic t = (X̄₁ − X̄₂) / √s₁²/n₁ + s₂²/n₂ when variances not assumed equal (Welch's t-test). If equal variances assumed, use pooled standard deviation s_p² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2) and t = (X̄₁ − X̄₂) / [s_p√1/n₁ + 1/n₂] with df = n₁+n₂−2.

比较两个独立样本的均值：当未假设方差相等时（Welch t 检验），检验统计量 t = (X̄₁ − X̄₂) / √s₁²/n₁ + s₂²/n₂。如果假设方差相等，使用合并标准差 s_p² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2)，且 t = (X̄₁ − X̄₂) / [s_p√1/n₁ + 1/n₂]，自由度 df = n₁+n₂−2。

Paired t-test: for paired data, compute differences dᵢ = xᵢ − yᵢ. Then test H₀: μ_d = 0 using t = d̄ / (s_d/√n) with n−1 df, where d̄ is the mean difference and s_d is the standard deviation of differences.

配对 t 检验：对于配对数据，计算差值 dᵢ = xᵢ − yᵢ。然后检验 H₀: μ_d = 0，使用 t = d̄ / (s_d/√n)，自由度 n−1，其中 d̄ 是差值的均值，s_d 是差值的标准差。

Comparing two proportions: Z = (p̂₁ − p̂₂) / √[p̂(1−p̂)(1/n₁ + 1/n₂)], where p̂ = (x₁ + x₂)/(n₁ + n₂) is the pooled proportion under H₀.

比较两个比例：Z = (p̂₁ − p̂₂) / √[p̂(1−p̂)(1/n₁ + 1/n₂)]，其中 p̂ = (x₁ + x₂)/(n₁ + n₂) 是 H₀ 下的合并比例。

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9. Chi-Squared Tests | 卡方检验

ees of freedom = k − 1 − m, where k is number of categories, m is number of estimated parameters from data.

卡方拟合优度检验：检验观察频数是否符合期望分布。检验统计量：χ² = Σ [(Oᵢ − Eᵢ)² / Eᵢ]，自由度 = k − 1 − m，其中 k 为类别数，m 为由数据估计的参数个数。

ndependence (contingency table): Test statistic same formula, with expected frequencies Eᵢⱼ = (row total × column total) / grand total. Degrees of freedom = (rows − 1)(columns − 1).

卡方独立性检验（列联表）：检验统计量公式相同，期望频数 Eᵢⱼ = (行合计 × 列合计) / 总计。自由度 = (行数 − 1)(列数 − 1)。

Conditions for validity: all expected frequencies should be at least 1, and no more than 20% of expected frequencies below 5. If violated, merge categories.

有效性条件：所有期望频数至少为 1，且低于 5 的期望频数不超过 20%。若违反，则合并类别。

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10. Correlation & Regression | 相关与回归

Pearson's product-moment correlation coefficient: r = S_xy / √S_xx S_yy, where S_xx = Σ(x − X̄)², S_yy = Σ(y − Ȳ)², S_xy = Σ(x − X̄)(y − Ȳ). r measures the strength and direction of linear relationship, −1 ≤ r ≤ 1.

皮尔逊积矩相关系数：r = S_xy / √S_xx S_yy，其中 S_xx = Σ(x − X̄)²，S_yy = Σ(y − Ȳ)²，S_xy = Σ(x − X̄)(y − Ȳ)。r 衡量线性关系的强度和方向，取值范围 −1 ≤ r ≤ 1。

Hypothesis test for correlation: H₀: ρ = 0. Test statistic t = r√n − 2 / √1 − r², with df = n − 2. Or refer to Spearman’s rank correlation table if non-parametric.

相关性假设检验：H₀: ρ = 0。检验统计量 t = r√n − 2 / √1 − r²，自由度 df = n − 2。若为非参数，可参考斯皮尔曼秩相关系数表。

Regression line of y on x: y = a + bx, where b = S_xy / S_xx and a = Ȳ − b X̄ zes sum of squared residuals. Interpretation: for each unit increase in x, y changes by b.

y 关于 x 的回归线：y = a + bx，其中 b = S_xy / S_xx，a = Ȳ − b X̄。这将残差平方和最小化。解释：x 每增加一个单位，y 变化 b 个单位。

Coefficient of determination: R² = (S_xy)² / (S_xx S_yy) = r². It represents the proportion of variability in y explained by the linear model.

决定系数：R² = (S_xy)² / (S_xx S_yy) = r²。它表示 y 的变异中可由线性模型解释的比例。

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11. Spearman’s Rank Correlation | 斯皮尔曼秩相关系数

Spearman’s rank correlation coefficient, ρ_s or r_s, measures the monotonic relationship between two variables, not requiring linearity or normal distributions. Calculate by ranking each variable separately, then use r_s = 1 − (6 Σ dᵢ²) / [n(n² − 1)], where dᵢ = difference in ranks for each pair.

斯皮尔曼秩相关系数 ρ_s 或 r_s，衡量两个变量之间的单调关系，不要求线性或正态分布。分别对每个变量排序，然后使用 r_s = 1 − (6 Σ dᵢ²) / [n(n² − 1)]，其中 dᵢ 为每对数据的秩差。

Hypothesis test for Spearman’s rank: For small n (e.g., n ≤ 30), compare calculated r_s with critical value from Spearman’s table. For larger n, use t = r_s √n − 2 / √1 − r_s² with n−2 df, approximating the same as Pearson’s test.

斯皮尔曼秩检验的假设检验：对于小样本（如 n ≤ 30），将计算得到的 r_s 与斯皮尔曼表临界值比较。对于较大样本，使用 t = r_s √n − 2 / √1 − r_s²，自由度为 n − 2，近似于皮尔逊检验。

When tied ranks occur, assign the average rank to tied values. The formula above is approximate; exact calculation involves Pearson correlation on ranks.

当存在相同秩次时，对相同值分配平均秩次。上述公式为近似值；精确计算需对秩次进行皮尔逊相关。

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12. Non-Parametric Tests – Wilcoxon Tests | 非参数检验 – 威尔科克森检验

Wilcoxon signed-rank test (one sample or paired): Used when data are not normally distributed. For paired data, compute differences, ignore zeros, rank absolute differences, sum ranks of positive differences (T⁺) and negative differences (T⁻). Test statistic is the smaller of T⁺ and T⁻. Compare with critical values from Wilcoxon table.

威尔科克森符号秩检验（单样本或配对）：用于数据不服从正态分布时。对于配对数据，计算差值，忽略零，对差值的绝对值排序，分别求正差值的秩和 (T⁺) 与负差值的秩和 (T⁻)。检验统计量为 T⁺ 与 T⁻ 中较小者。与威尔科克森表临界值比较。

anks for the smaller sample, R. Test statistic U = n₁n₂ + n₁(n₁+1)/2 − R (or equivalent). Compare to critical values.

威尔科克森秩和检验（曼-惠特尼 U 检验）：用于两个独立样本。合并两个样本，对所有观测值排序，计算较小样本的秩和 R。检验统计量 U = n₁n₂ + n₁(n₁+1)/2 − R（或等效形式）。与临界值比较。

These non-parametric tests do not assume normality, and are based on ranks, making them robust to outliers.

这些非参数检验不假设正态性，且基于秩次，因此对异常值具有稳健性。

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