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

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

This quick-reference guide brings together the essential formulae, definitions and theorems required for the CCEA A-Level Statistics course. It covers descriptive statistics, probability, common distributions, confidence intervals and hypothesis tests in a clear, point-by-point format. Use it alongside your revision to check definitions and to practise applying the correct formulae under exam conditions.

本速查手册汇总了 CCEA A-Level 统计课程中必须掌握的核心公式、定义和定理。内容涵盖描述性统计、概率、常用分布、置信区间和假设检验，以清晰的要点形式呈现。复习时可随时查阅定义，并练习在考试环境中正确运用公式。

1. Describing Data – Mean, Median, Mode, Quartiles | 数据描述 —— 均值、中位数、众数、四分位数

For a raw data set x₁, x₂, …, xₙ, the sample mean is x̄ = (Σ xᵢ)/n. The median is the middle value when data are ordered; for an even number of observations it is the average of the two central values. The mode is the most frequently occurring value.

对于原始数据集 x₁, x₂, …, xₙ，样本均值 x̄ = (Σ xᵢ)/n。中位数是数据排序后位于中间的值；当观测值为偶数时，中位数为中间两个数值的平均值。众数是出现频次最高的值。

Lower quartile (Q₁) is the median of the lower half of the data; upper quartile (Q₃) is the median of the upper half. The interquartile range (IQR) = Q₃ − Q₁. For grouped data, quartiles are obtained by linear interpolation within the appropriate class interval.

下四分位数 (Q₁) 是数据下半部分的中位数；上四分位数 (Q₃) 是数据上半部分的中位数。四分位距 IQR = Q₃ − Q₁。对于分组数据，四分位数通过在相应组段内线性插值求得。

2. Measures of Dispersion – Range, IQR, Variance, Standard Deviation | 离散程度度量 —— 极差、四分位距、方差、标准差

Range = maximum − minimum. Interquartile range (IQR) = Q₃ − Q₁, giving the spread of the middle 50% of data.

极差 = 最大值 − 最小值。四分位距 IQR = Q₃ − Q₁，表示中间 50% 数据的散布范围。

Sample variance: s² = Σ (xᵢ − x̄)² / (n − 1). The equivalent computational form is s² = (Σ xᵢ² − (Σ xᵢ)² / n) / (n − 1). For a discrete frequency distribution: s² = Σ f(x − x̄)² / (Σ f − 1) or the computational version. Standard deviation s = √s².

样本方差：s² = Σ (xᵢ − x̄)² / (n − 1)。等价的计算形式为 s² = (Σ xᵢ² − (Σ xᵢ)² / n) / (n − 1)。对于离散频率分布：s² = Σ f(x − x̄)² / (Σ f − 1) 或使用计算形式。标准差 s = √s²。

3. Skewness and Box Plots | 偏度与箱线图

A distribution is positively skewed if mean > median > mode; the right tail is longer. Negative skew has mean < median < mode. A simple measure of skewness is 3(mean − median) / standard deviation.

如果均值 > 中位数 > 众数，分布为正偏（右偏），右尾较长。负偏分布有均值 < 中位数 < 众数。一个简单的偏度度量是 3(均值 − 中位数) / 标准差。

Box plots display the five-number summary: minimum, Q₁, median, Q₃, maximum. Outliers are points more than 1.5 × IQR below Q₁ or above Q₃. They are shown as individual dots.

箱线图展示五数概括：最小值、Q₁、中位数、Q₃、最大值。离群值是低于 Q₁ − 1.5×IQR 或高于 Q₃ + 1.5×IQR 的数据点，用单独的点标出。

4. Probability Rules – Addition, Multiplication, Conditional, Bayes | 概率规则 —— 加法、乘法、条件概率、贝叶斯定理

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

加法规则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若事件互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

Multiplication rule: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B). Independent events satisfy P(A ∩ B) = P(A) × P(B) and P(B|A) = P(B).

乘法规则：P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)。独立事件满足 P(A ∩ B) = P(A) × P(B) 且 P(B|A) = P(B)。

5. Discrete Distributions – Binomial and Poisson | 离散分布 —— 二项分布与泊松分布

Binomial distribution B(n, p): X ~ B(n, p). Probability mass function: P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r!(n − r)!]. Mean µ = np; variance σ² = np(1 − p). The distribution applies to a fixed number of independent trials with constant probability of success p.

二项分布 B(n, p)：X ~ B(n, p)。概率质量函数：P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 ⁿCᵣ = n! / [r!(n − r)!]。均值 µ = np；方差 σ² = np(1 − p)。该分布适用于固定次数的独立试验，每次成功概率恒为 p。

Poisson distribution Po(λ): X ~ Po(λ). P(X = r) = e⁻λ λʳ / r! for r = 0, 1, 2, … . Mean = λ; variance = λ. The Poisson approximates the binomial when n is large and p is small with λ = np.

泊松分布 Po(λ)：X ~ Po(λ)。P(X = r) = e⁻λ λʳ / r!，r = 0, 1, 2, … 。均值 = λ；方差 = λ。当 n 很大、p 很小时，可用 λ = np 的泊松分布近似二项分布。

6. Continuous Distributions – Normal, t, Chi-squared, F | 连续分布 —— 正态、t、卡方、F 分布

Normal distribution N(µ, σ²): probability density function f(x) = (1/√(2π σ²)) exp(−(x − µ)²/(2σ²)). Standard normal Z ~ N(0, 1) with Z = (X − µ)/σ. About 68% of data lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 (empirical rule).

正态分布 N(µ, σ²)：概率密度函数 f(x) = (1/√(2π σ²)) exp(−(x − µ)²/(2σ²))。标准正态 Z ~ N(0,1)，Z = (X − µ)/σ。大约 68% 的数据落在均值 1 个标准差内，95% 落在 2 个标准差内，99.7% 在 3 个标准差内（经验法则）。

Student’s t-distribution with ν degrees of freedom is symmetric and heavier-tailed than normal. As ν → ∞ it approaches N(0,1). Used when population standard deviation is unknown.

自由度为 ν 的学生 t 分布对称且尾部比正态更重。当 ν → ∞ 时趋近于 N(0,1)。在总体标准差未知时使用。

Chi-squared distribution χ²(ν): sum of squares of ν independent standard normal variables. Skewed right; mean = ν, variance = 2ν. F-distribution F(ν₁, ν₂): ratio of two independent χ² variables each divided by their degrees of freedom.

卡方分布 χ²(ν)：ν 个独立标准正态变量的平方和。右偏；均值 = ν，方差 = 2ν。F 分布 F(ν₁, ν₂)：两个独立的卡方变量分别除以各自自由度的比值。

7. Sampling Distributions – Central Limit Theorem and Standard Error | 抽样分布 —— 中心极限定理与标准误

The central limit theorem (CLT) states that for a large sample size (n ≥ 30), the sampling distribution of the sample mean x̄ is approximately normal with mean µ and variance σ²/n, regardless of the shape of the population distribution. Standard error of the mean = σ/√n.

中心极限定理 (CLT) 指出，对于大样本量 (n ≥ 30)，无论总体分布形状如何，样本均值 x̄ 的抽样分布近似正态，均值为 µ，方差为 σ²/n。均值的标准误 = σ/√n。

For the sample proportion p̂, mean = p, standard error = √[p(1 − p)/n] under simple random sampling. When σ is unknown, the standard error is estimated using s/√n for means.

对于样本比例 p̂，在简单随机抽样下均值 = p，标准误 = √[p(1 − p)/n]。当 σ 未知时，用 s/√n 估计均值的标准误。

8. Confidence Intervals – for Mean, Proportion, Difference | 置信区间 —— 均值、比例、差异

A (1 − α) × 100% confidence interval for a population mean µ when σ is known: x̄ ± z_{α/2} × σ/√n. When σ is unknown and n < 30 (or data normal), use t-distribution: x̄ ± t_{ν,α/2} × s/√n with ν = n − 1. For large n, t ≈ z.

当 σ 已知时，总体均值 µ 的 (1 − α)×100% 置信区间：x̄ ± z_{α/2} × σ/√n。当 σ 未知且 n < 30（或数据正态），使用 t 分布：x̄ ± t_{ν,α/2} × s/√n，ν = n − 1。大样本下 t ≈ z。

Confidence interval for a population proportion p: p̂ ± z_{α/2} × √[p̂(1 − p̂)/n], provided n is large enough that np̂ ≥ 5 and n(1 − p̂) ≥ 5.

总体比例 p 的置信区间：p̂ ± z_{α/2} × √[p̂(1 − p̂)/n]，要求 n 足够大使得 np̂ ≥ 5 且 n(1 − p̂) ≥ 5。

For difference of two independent means (σ₁, σ₂ known): (x̄₁ − x̄₂) ± z_{α/2} × √(σ₁²/n₁ + σ₂²/n₂). If variances unknown but assumed equal, use pooled variance: sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2), then (x̄₁ − x̄₂) ± t_{ν,α/2} × sₚ × √(1/n₁ + 1/n₂) with ν = n₁+n₂−2.

两个独立总体均值差（σ₁, σ₂ 已知）的置信区间：(x̄₁ − x̄₂) ± z_{α/2} × √(σ₁²/n₁ + σ₂²/n₂)。若方差未知但假设相等，使用合并方差：sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²]/(n₁+n₂−2)，则 (x̄₁ − x̄₂) ± t_{ν,α/2} × sₚ × √(1/n₁ + 1/n₂)，ν = n₁+n₂−2。

9. Hypothesis Testing – One-Sample and Two-Sample Tests | 假设检验 —— 单样本与双样本检验

Steps: state null hypothesis H₀ and alternative H₁, choose significance level α, compute test statistic, find critical value or p‑value, then decide. For a one‑sample z‑test (σ known): z = (x̄ − µ₀)/(σ/√n). For a t‑test (σ unknown): t = (x̄ − µ₀)/(s/√n) with ν = n − 1.

检验步骤：提出原假设 H₀ 和备择假设 H₁，选择显著性水平 α，计算检验统计量，确定临界值或 p 值，然后作出判断。单样本 z 检验（σ 已知）：z = (x̄ − µ₀)/(σ/√n)。t 检验（σ 未知）：t = (x̄ − µ₀)/(s/√n)，ν = n − 1。

Two‑sample test for difference of means (independent samples, variances known): z = (x̄₁ − x̄₂ − ∆₀)/√(σ₁²/n₁ + σ₂²/n₂). If variances unknown but equal, pooled t‑test: t = (x̄₁ − x̄₂ − ∆₀)/[sₚ × √(1/n₁ + 1/n₂)], ν = n₁+n₂−2. Paired t‑test uses differences dᵢ: t = (d̄ − ∆₀)/(s_d / √n), ν = n − 1.

双样本均值差检验（独立样本，方差已知）：z = (x̄₁ − x̄₂ − ∆₀)/√(σ₁²/n₁ + σ₂²/n₂)。若方差未知但相等，合并方差 t 检验：t = (x̄₁ − x̄₂ − ∆₀)/[sₚ × √(1/n₁ + 1/n₂)]，ν = n₁+n₂−2。配对 t 检验使用差值 dᵢ：t = (d̄ − ∆₀)/(s_d / √n)，ν = n − 1。

For a test of a single proportion, use z = (p̂ − p₀)/√[p₀(1 − p₀)/n] under H₀, assuming large n.

单比例检验使用 z = (p̂ − p₀)/√[p₀(1 − p₀)/n]（在原假设下），要求大样本。

10. Chi-Squared Tests for Independence and Goodness-of-Fit | 卡方检验 —— 独立性与拟合优度

Chi-squared statistic: χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ, where Oᵢ is observed frequency and Eᵢ is expected frequency. Degrees of freedom = (number of categories − 1) for goodness‑of‑fit, or (rows − 1)(columns − 1) for a contingency table testing independence.

卡方统计量：χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ，Oᵢ 为观测频数，Eᵢ 为期望频数。拟合优度检验的自由度为（分类数 − 1）；用于独立性检验的列联表自由度为（行数 − 1）×（列数 − 1）。

Goodness‑of‑fit test compares observed frequencies to those expected under a specified distribution. For a test of independence in an r×c table, expected frequencies are Eᵢⱼ = (row total × column total) / grand total. The test is valid when at least 80% of Eᵢ ≥ 5 and no Eᵢ < 1. For a 2×2 table, apply Yates’ correction (subtract 0.5 from each |O − E|) when any expected frequency is small.

拟合优度检验将观测频数与指定分布下的期望频数进行比较。在 r×c 表的独立性检验中，期望频数 Eᵢⱼ =（行合计 × 列合计）/ 总计。当至少 80% 的 Eᵢ ≥ 5 且没有 Eᵢ < 1 时，检验有效。对于 2×2 表，当任一期望频数较小时，应施行 Yates 校正（每个 |O − E| 减去 0.5）。

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