A-Level Eduqas Statistics Formula Quick Reference | A-Level Eduqas 统计：公式定理速查手册

📚 A-Level Eduqas Statistics Formula Quick Reference | A-Level Eduqas 统计：公式定理速查手册

This quick reference handbook provides the essential formulas, theorems, and key concepts for the A-Level Eduqas Statistics specification. It is designed for efficient revision, bringing together all the critical mathematical tools you need for your exam in one place. Each section is presented in a bilingual format, with English and Chinese side by side to support deeper understanding.

本速查手册汇集了 A-Level Eduqas 统计学科所需的基本公式、定理与核心概念，旨在帮助同学们高效复习。各章节以中英对照形式呈现，方便对照理解，快速巩固考点。

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1. Descriptive Statistics | 描述统计量

For a data set x₁, x₂, …, xₙ, the sample mean and variance are fundamental measures of central tendency and spread.

数据 x₁, x₂, …, xₙ 的样本均值与方差是衡量集中趋势和离散程度的基本指标。

Sample mean: x̄ = (Σ xᵢ) / n

样本均值：x̄ = (Σ xᵢ) / n

Sample variance (unbiased estimator): s² = Σ (xᵢ − x̄)² / (n − 1)

样本方差（无偏估计量）：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) ]

For grouped data with midpoints mᵢ and frequencies fᵢ, use x̄ = Σ fᵢ mᵢ / Σ fᵢ, and s² = Σ fᵢ (mᵢ − x̄)² / (Σ fᵢ − 1).

若数据为分组形式，以组中值 mᵢ 和频数 fᵢ 计算：x̄ = Σ fᵢ mᵢ / Σ fᵢ，s² = Σ fᵢ (mᵢ − x̄)² / (Σ fᵢ − 1)。

Quartiles and percentiles: Q₁ is the 25th percentile, Q₃ the 75th. Interquartile range IQR = Q₃ − Q₁. Outliers are often defined as values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.

四分位数与百分位数：Q₁ 为第 25 百分位数，Q₃ 为第 75 百分位数；四分位距 IQR = Q₃ − Q₁。离群值通常定义为小于 Q₁ − 1.5×IQR 或大于 Q₃ + 1.5×IQR。

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2. Probability Basics | 概率基础

Probability measures the likelihood of events and obeys strict rules. For any event A, 0 ≤ P(A) ≤ 1, with P(sample space) = 1.

概率衡量事件发生的可能性，满足 0 ≤ P(A) ≤ 1，且 P(样本空间) = 1。

Complement rule: P(A’) = 1 − P(A).

互补规则：P(A’) = 1 − P(A)。

Addition rule for any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.

一般加法公式：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若事件互斥，则 P(A ∩ B) = 0。

Conditional probability: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Multiplication rule: P(A ∩ B) = P(A) × P(B|A).

条件概率：P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。乘法公式：P(A ∩ B) = P(A) × P(B|A)。

Independent events: A and B are independent if and only if P(A ∩ B) = P(A) × P(B), or equivalently P(A|B) = P(A).

独立事件：当且仅当 P(A ∩ B) = P(A) × P(B) 或 P(A|B) = P(A) 时，A 与 B 相互独立。

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3. Discrete Random Variables | 离散随机变量

A discrete random variable X takes values xᵢ with probabilities p(xᵢ) = P(X = xᵢ). The sum of all probabilities must equal 1.

离散随机变量 X 取值 xᵢ，其概率为 p(xᵢ) = P(X = xᵢ)，所有概率之和为 1。

Expected value (mean): E(X) = Σ xᵢ p(xᵢ). Variance: Var(X) = E(X²) − [E(X)]² = Σ xᵢ² p(xᵢ) − μ².

期望值（均值）：E(X) = Σ xᵢ p(xᵢ)；方差：Var(X) = E(X²) − [E(X)]² = Σ xᵢ² p(xᵢ) − μ²。

Linear transformations: For constants a and b, E(aX + b) = a E(X) + b, Var(aX + b) = a² Var(X).

线性变换：对常数 a, b，有 E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。

Expected value of sum: E(X + Y) = E(X) + E(Y). For independent X and Y, Var(X + Y) = Var(X) + Var(Y) and E(XY) = E(X)E(Y).

期望的可加性：E(X + Y) = E(X) + E(Y)；若 X 与 Y 独立，则 Var(X + Y) = Var(X) + Var(Y) 且 E(XY) = E(X)E(Y)。

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4. Binomial & Poisson Distributions | 二项分布与泊松分布

Binomial distribution B(n, p): number of successes in n independent trials, each with probability p of success.

二项分布 B(n, p)： n 次独立试验中成功的次数，每次成功概率为 p。

Probability mass function: P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r! (n − r)!].

概率质量函数：P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 ⁿCᵣ = n! / [r! (n − r)!]。

Mean: E(X) = np. Variance: Var(X) = np(1 − p).

均值：E(X) = np；方差：Var(X) = np(1 − p)。

Poisson distribution Po(λ): models the number of events in a fixed interval when events occur independently at a constant average rate λ.

泊松分布 Po(λ)： 单位时间内随机事件发生次数的分布，事件独立且平均发生率为 λ。

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

概率质量函数：P(X = r) = e⁻λ λʳ / r! (r = 0, 1, 2, …)。

Mean = λ, Variance = λ.

均值 = λ，方差 = λ。

Poisson approximation to binomial: When n is large, p is small, and np = λ is moderate, B(n, p) ≈ Po(np).

泊松近似二项分布：当 n 大、p 小且 np = λ 适中时，可用 Po(np) 近似 B(n, p)。

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5. Normal Distribution | 正态分布

The normal distribution N(μ, σ²) is a continuous distribution with mean μ and variance σ². Its probability density is symmetric about μ.

正态分布 N(μ, σ²) 是连续型分布，均值为 μ，方差为 σ²，概率密度关于 μ 对称。

The standard normal variable Z = (X − μ) / σ ~ N(0, 1). Use tables of Φ(z) = P(Z ≤ z) to find probabilities.

标准正态变量 Z = (X − μ) / σ ~ N(0, 1)，使用标准正态分布表 Φ(z) = P(Z ≤ z) 求概率。

Key intervals: P(μ − σ < X < μ + σ) ≈ 0.6827, P(μ − 2σ < X < μ + 2σ) ≈ 0.9545, P(μ − 3σ < X < μ + 3σ) ≈ 0.9973.

重要区间：P(μ − σ < X < μ + σ) ≈ 0.6827，P(μ − 2σ < X < μ + 2σ) ≈ 0.9545，P(μ − 3σ < X < μ + 3σ) ≈ 0.9973。

To find probabilities for X ~ N(μ, σ²), standardise: P(X ≤ a) = P(Z ≤ (a − μ)/σ).

求概率时标准化：P(X ≤ a) = P(Z ≤ (a − μ)/σ)。

Inverse normal: Given probability p, find z such that Φ(z) = p, then x = μ + zσ.

逆正态：给定概率 p，查表得 z 使 Φ(z) = p，则 x = μ + zσ。

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6. Sampling & Central Limit Theorem | 抽样与中心极限定理

Random samples from a population provide unbiased estimators. The sample mean x̄ is an unbiased estimator of the population mean μ.

从总体中随机抽样，样本均值 x̄ 是总体均值 μ 的无偏估计量。

For a sample of size n from a population with mean μ and variance σ², the sampling distribution of x̄ has mean μ and variance σ² / n. If X is normal, x̄ is exactly normal.

若总体均值为 μ、方差为 σ²，样本量为 n，则 x̄ 的抽样分布均值为 μ，方差为 σ² / n。若总体正态，x̄ 也严格服从正态分布。

Central Limit Theorem (CLT): For a large sample size (typically n ≥ 30), the distribution of the sample mean x̄ is approximately normal N(μ, σ² / n), regardless of the shape of the population.

中心极限定理 (CLT)： 当样本量足够大时（通常 n ≥ 30），无论总体分布形状如何，样本均值 x̄ 近似服从 N(μ, σ² / n)。

Standard error of the mean = σ / √n. If σ is unknown, the estimated standard error is s / √n.

均值的标准误 = σ / √n；若 σ 未知，则用估计标准误 s / √n。

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7. Confidence Intervals | 置信区间

A confidence interval gives a range of plausible values for a population parameter. For a normal population mean with known variance σ²:

置信区间给出总体参数的可能取值范围。已知方差 σ²，正态总体均值的置信区间：

x̄ ± z × (σ/√n)

where z is the critical value from the standard normal distribution for the desired confidence level (e.g. z=1.96 for 95%).

其中 z 为标准正态分布对应置信水平的临界值（如 95% 置信水平 z=1.96）。

When σ² is unknown and sample size small, use the t-distribution with n−1 degrees of freedom:

当 σ² 未知且样本量较小时，使用自由度为 n−1 的 t 分布：

x̄ ± t × (s/√n)

Confidence interval for a population proportion p (based on a large sample) uses the sample proportion p̂:

总体比例 p 的置信区间（大样本）使用样本比例 p̂：

p̂ ± z × √[ p̂(1−p̂) / n ]

Interpretation: A 95% confidence interval means that if we repeatedly took samples, 95% of the intervals would contain the true parameter.

解释：95% 置信区间表示若重复随机抽样，有 95% 的区间会包含真值。

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8. Hypothesis Testing (Means & Proportions) | 假设检验（均值与比例）

A hypothesis test assesses evidence against a null hypothesis H₀. For a population mean μ with known σ, the test statistic is:

假设检验用于评判反对零假设 H₀ 的证据强度。若 σ 已知，检验均值 μ 的统计量为：

Z = (x̄ − μ₀) / (σ/√n)

For unknown σ, use the t-statistic with n−1 degrees of freedom:

若 σ 未知，使用自由度 n−1 的 t 统计量：

t = (x̄ − μ₀) / (s/√n)

For a proportion p, test statistic (large sample):

检验比例 p 的统计量（大样本）：

Z = (p̂ − p₀) / √[ p₀(1−p₀) / n ]

Critical region: determined by the significance level α. p-value: the probability of observing a test statistic at least as extreme as the one obtained, assuming H₀ is true. Reject H₀ if p-value < α.

拒绝域：由显著性水平 α 决定。p 值：在 H₀ 成立时观察到当前及更极端统计量的概率。若 p 值 < α，拒绝 H₀。

For a two-tailed test at 5% level, critical z-values are ±1.96. For one-tailed, use z = 1.645 or −1.645.

双侧检验 5% 水平，临界 z 值为 ±1.96；单侧检验使用 ±1.645。

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9. Chi-Squared Tests | 卡方检验

The chi-squared (χ²) test is used for categorical data. Test statistic:

卡方 (χ²) 检验用于分类数据。检验统计量：

χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ

where Oᵢ are observed frequencies and Eᵢ are expected frequencies.

其中 Oᵢ 为观测频数，Eᵢ 为期望频数。

Degrees of freedom (goodness-of-fit): ν = k − 1 − m, where k = number of categories, m = number of estimated parameters from data.

自由度（拟合优度）：ν = k − 1 − m，k 为类别数，m 为样本估计的参数个数。

For contingency tables (test of association): degrees of freedom = (r − 1)(c − 1), where r = rows, c = columns. Expected frequency for cell (i, j): Eᵢⱼ = (row i total × column j total) / grand total.

列联表（独立性检验）自由度：ν = (r − 1)(c − 1)，其中 r 为行数，c 为列数。单元格 (i, j) 的期望频数 Eᵢⱼ = (第 i 行合计 × 第 j 列合计) / 总计。

Conditions: All expected frequencies should be ≥ 1; no more than 20% of expected frequencies should be < 5.

使用条件：所有期望频数 ≥ 1，且少于 20% 的期望频数 < 5。

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10. Correlation & Regression | 相关与回归

Product moment correlation coefficient (Pearson’s r): measures linear correlation between two variables X and Y.

积矩相关系数 (Pearson’s r)： 衡量两变量 X 与 Y 间的线性相关程度。

r = Sₓᵧ / √(Sₓₓ Sᵧᵧ)

where Sₓᵧ = Σ (xᵢ − x̄)(yᵢ − ȳ), Sₓₓ = Σ (xᵢ − x̄)², Sᵧᵧ = Σ (yᵢ − ȳ)². −1 ≤ r ≤ 1.

其中 Sₓᵧ = Σ (xᵢ − x̄)(yᵢ − ȳ)，Sₓₓ = Σ (xᵢ − x̄)²，Sᵧᵧ = Σ (yᵢ − ȳ)²。−1 ≤ r ≤ 1。

Spearman’s rank correlation coefficient uses ranks when data is not linear or is ordinal:

斯皮尔曼等级相关系数 基于秩次，适用于非线性或顺序数据：

rₛ = 1 − (6 Σ dᵢ²) / [n(n² − 1)]

where dᵢ = difference between the ranks of each pair.

其中 dᵢ 为每对数据的秩次差。

Least squares regression line: y = a + bx, where

最小二乘法回归直线： y = a + bx，其中

b = Sₓᵧ / Sₓₓ

a = ȳ − b x̄

Residuals: eᵢ = yᵢ − ŷᵢ. The sum of residuals is zero, and the line minimises the sum of squared residuals.

残差：eᵢ = yᵢ − ŷᵢ；残差和为零，回归直线使残差平方和最小。

Interpolation is prediction within the range of data; extrapolation is outside that range and may be unreliable.

内插法是在数据点范围内预测，外推法超出范围可能不可靠。

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