📚 Year 13 Cambridge Statistics: Vocabulary & Terminology Quick Memorization Guide | Year 13 剑桥统计：词汇术语速记指南
This guide distills the essential statistical terms, notations, and memos you will encounter in the Year 13 Cambridge A Level Statistics course. Each section pairs English explanations with their Chinese equivalents, helping you build rapid recall for examination contexts.
本文提炼了 Year 13 剑桥 A Level 统计课程中必须掌握的术语、符号与速记要点。每个小节都以英文—中文对照形式呈现，帮助你快速建立考试场景下的词汇反应。
1. Continuous Random Variables, PDF & CDF | 连续随机变量、概率密度函数与累积分布函数
A continuous random variable (CRV) can take any value within an interval. Its distribution is specified by a probability density function (pdf) f(x), where the total area under the curve equals 1. Probabilities are found as integrals: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. Since a single point has zero area, P(X = k) = 0 for any k.
连续随机变量（CRV）可以在某个区间内取任意实数。它的分布由概率密度函数 f(x) 描述，曲线下方总面积为 1。概率通过积分得到：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。由于单点面积为 0，对任何 k 都有 P(X = k) = 0。
The cumulative distribution function (cdf) is defined as F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt. F(x) is non‑decreasing, ranges from 0 to 1, and its derivative recovers the pdf: f(x) = F'(x). The median m satisfies F(m) = 0.5; the lower quartile Q₁ satisfies F(Q₁) = 0.25, and the upper quartile Q₃ satisfies F(Q₃) = 0.75.
累积分布函数（cdf）定义为 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。F(x) 单调不减，取值从 0 到 1，对其求导即可还原 pdf：f(x) = F'(x)。中位数 m 满足 F(m) = 0.5；下四分位数 Q₁ 满足 F(Q₁) = 0.25，上四分位数 Q₃ 满足 F(Q₃) = 0.75。
2. Expectation and Variance Rules | 期望与方差的性质
For a CRV, the expectation (mean) is μ = E(X) = ∫ x f(x) dx over the domain. The variance is Var(X) = E[(X − μ)²] = ∫ (x − μ)² f(x) dx. A shortcut computational form is widely used:
Var(X) = E(X²) − [E(X)]²
对于连续随机变量，期望（均值）μ = E(X) = ∫ x f(x) dx。方差 Var(X) = E[(X − μ)²] = ∫ (x − μ)² f(x) dx。常用的计算简便公式为：Var(X) = E(X²) − [E(X)]²。
Linear transformations follow clean rules: E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X). For sums of independent random variables, E(X + Y) = E(X) + E(Y) and Var(X + Y) = Var(X) + Var(Y). These identities underpin many sampling results.
线性变换规则简洁：E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。对于独立随机变量的和，有 E(X + Y) = E(X) + E(Y)，Var(X + Y) = Var(X) + Var(Y)。这些恒等式是很多抽样结论的基础。
3. The Continuous Uniform Distribution | 连续均匀分布
A uniform (rectangular) distribution on [a, b] has constant pdf: f(x) = 1/(b − a) for a ≤ x ≤ b, and 0 elsewhere. Its key summaries are:
Mean = (a + b)/2, Variance = (b − a)²/12
区间 [a, b] 上的均匀分布（矩形分布）具有常数值概率密度函数：f(x) = 1/(b − a)，a ≤ x ≤ b，其他地方为 0。其关键数字特征为：均值 = (a + b)/2，方差 = (b − a)²/12。
The cdf is linear: F(x) = (x − a)/(b − a) for a ≤ x ≤ b. To find percentiles, set F(x) equal to the required proportion and solve for x. This distribution is frequently used to model situations where all outcomes in a range are equally likely.
累积分布函数是线性的：F(x) = (x − a)/(b − a)，a ≤ x ≤ b。求百分位数时，令 F(x) 等于所需比例并解出 x 即可。均匀分布常用于建模区间内所有结果等可能的情形。
4. The Normal Distribution | 正态分布
The normal distribution N(μ, σ²) has bell‑shaped pdf f(x) = 1/(σ√(2π)) e−(x−μ)²/(2σ²). Its shape is determined entirely by the mean μ and standard deviation σ. Approximately 68% of data lies within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.
正态分布 N(μ, σ²) 的钟形概率密度函数为 f(x) = 1/(σ√(2π)) e−(x−μ)²/(2σ²)。分布形状完全由均值 μ 和标准差 σ 决定。大约 68% 的数据落在 μ ± σ，95% 落在 μ ± 2σ，99.7% 落在 μ ± 3σ。
To calculate probabilities, standardise to Z = (X − μ)/σ, which follows N(0, 1). Use the standard normal table to find Φ(z) = P(Z ≤ z). Then P(X ≤ x) = Φ((x − μ)/σ). The symmetry Φ(−z) = 1 − Φ(z) often saves time.
计算概率时，先转化为标准正态：Z = (X − μ)/σ ~ N(0, 1)。查阅标准正态表可得 Φ(z) = P(Z ≤ z)。于是 P(X ≤ x) = Φ((x − μ)/σ)。利用对称性 Φ(−z) = 1 − Φ(z) 常能节省时间。
5. Normal Approximations to Binomial and Poisson | 对二项分布与泊松分布的正态逼近
When np and nq are both >5, Binomial B(n, p) can be approximated by N(np, npq) where q = 1 − p. For Poisson Po(λ), when λ is large (typically λ > 15), use N(λ, λ) as an approximation. Always apply a continuity correction (see next section) because you are approximating a discrete distribution with a continuous one.
当 np 与 nq 均大于 5 时，二项分布 B(n, p) 可用正态分布 N(np, npq) 逼近，其中 q = 1 − p。对于泊松分布 Po(λ)，当 λ 较大（一般 λ > 15）时，使用 N(λ, λ) 作为近似。必须施加连续性校正（见下一节），因为是用连续分布去逼近离散分布。
Checklist: identify the original distribution, compute μ and σ² for the approximating normal, then work with probabilities including a ± 0.5 correction. For a binomial P(X < k) becomes P(X < k − 0.5) under the normal curve.
检查清单：识别原始分布，计算逼近正态的 μ 和 σ²，然后计算带有 ± 0.5 校正的概率。例如二项分布的 P(X < k) 转换为正态曲线下的 P(X < k − 0.5)。
6. Continuity Correction in Detail | 连续性校正详解
The continuity correction adjusts the boundary by 0.5 to align a discrete bar with the continuous normal curve. Common adjustments: P(X ≤ k) → P(X < k + 0.5), P(X ≥ k) → P(X > k − 0.5), P(X = k) → P(k − 0.5 < X < k + 0.5).
连续性校正将边界调整 0.5，使离散的条形与连续正态曲线对准。常见调整：P(X ≤ k) → P(X < k + 0.5)，P(X ≥ k) → P(X > k − 0.5)，P(X = k) → P(k − 0.5 < X < k + 0.5)。
Always apply the correction when using a normal approximation to a discrete distribution unless the question explicitly permits a rough estimate. Failure to include it is a common loss of marks in hypothesis testing.
在用正态分布逼近离散分布时务必使用校正，除非题目明确允许粗略估计。遗漏连续性校正是假设检验中常见的失分点。
7. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A sample mean X̄ from a population with mean μ and variance σ² has expected value E(X̄) = μ and variance Var(X̄) = σ²/n. The standard error is σ/√n. If the population is normal, X̄ is exactly normal. The Central Limit Theorem (CLT) states that for sufficiently large n (usually n ≥ 30), X̄ is approximately N(μ, σ²/n) regardless of the population shape.
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。标准误为 σ/√n。若总体正态，则 X̄ 精确服从正态分布。中心极限定理（CLT）指出，对于足够大的样本量（通常 n ≥ 30），无论总体形状如何，X̄ 近似服从 N(μ, σ²/n)。
For a sample proportion p̂ from a binomial population, E(p̂) = p, Var(p̂) = p(1−p)/n, and for large n the approximate distribution is normal.
对于来自二项总体的样本比例 p̂，有 E(p̂) = p，Var(p̂) = p(1−p)/n，且当 n 大时近似服从正态分布。
8. Confidence Intervals | 置信区间
A 95% confidence interval for the population mean μ (when σ is known) is given by:
x̄ ± 1.96 × σ/√n
The value 1.96 is the z‑score capturing the central 95% of the standard normal distribution. For other confidence levels, replace 1.96 with the corresponding critical value zα/2. The interpretation: if we repeated the sampling many times, 95% of the intervals constructed would contain the true μ.
当总体标准差 σ 已知时，总体均值 μ 的 95% 置信区间为：x̄ ± 1.96 × σ/√n。1.96 是标准正态分布中间 95% 对应的 z 值。对于其他置信水平，用对应的临界值 zα/2 替换 1.96。置信区间的解释：如果多次重复抽样，构造的区间中有 95% 会包含真实的 μ。
For a population proportion p, a 95% confidence interval uses p̂ ± 1.96 × √(p̂(1−p̂)/n). These intervals are widely used to assess estimation precision.
对于总体比例 p，95% 置信区间为 p̂ ± 1.96 × √(p̂(1−p̂)/n)。置信区间广泛用于评估估计精度。
9. Hypothesis Testing Key Terms | 假设检验关键术语
Null hypothesis H₀: the status‑quo statement to be tested (often “no effect” or a specific parameter value). Alternative hypothesis H₁: what we suspect might be true instead (two‑tailed or one‑tailed). The test statistic is computed from the sample and compared with a critical value. The significance level α is the probability of rejecting H₀ when it is true; commonly 5% or 1%.
原假设 H₀：需要检验的现况陈述（常为“无效果”或某个特定参数值）。备择假设 H₁：猜测可能成立的陈述（双尾或单尾）。检验统计量由样本计算得出，并与临界值比较。显著性水平 α 是原假设为真时拒绝它的概率；常用 5% 或 1%。
The p‑value is the probability of obtaining a result at least as extreme as the one observed, assuming H₀ is true. If p‑value < α, reject H₀. The critical (rejection) region is the set of values of the test statistic that lead to rejection. One‑tailed tests use one side; two‑tailed tests split α across both tails.
p 值是在 H₀ 为真时，得到至少与观测结果同样极端的概率。若 p 值 < α，拒绝 H₀。拒绝域（临界域）是导致拒绝 H₀ 的检验统计量取值集合。单尾检验只使用一侧，双尾检验将 α 平分到两侧。
10. Type I/II Errors and the Power of a Test | 第一类/第二类错误与检验的功效
Type I error: rejecting H₀ when it is actually true. The probability of a Type I error equals α. Type II error: failing to reject H₀ when H₁ is true; its probability is denoted β. The power of the test is 1 − β, the probability of correctly rejecting a false H₀.
第一类错误：H₀ 为真时却拒绝它，其概率等于 α。第二类错误：H₁ 为真时未能拒绝 H₀，其概率记为 β。检验的功效 = 1 − β，表示正确拒绝一个错误原假设的概率。
To compute power in a normal test, specify the true parameter under H₁, find the critical boundary, and calculate the probability that the test statistic falls in the rejection region. Increasing sample size boosts power for the same α.
计算正态检验的功效时，先设定 H₁ 下的真实参数值，确定临界边界，再计算检验统计量落入拒绝域的概率。增大样本量能在相同 α 下提高功效。
11. Chi‑Squared (χ²) Tests | 卡方检验
The χ² test is used for goodness‑of‑fit (comparing observed frequencies with theoretical ones) and for testing independence in contingency tables. The test statistic is:
χ² = Σ (O − E)² / E
where O are observed and E are expected frequencies. The degrees of freedom ν depend on the context: for goodness‑of‑fit to a known distribution, ν = (number of cells) − 1 − (number of estimated parameters). For an r×c contingency table, ν = (r − 1)(c − 1).
卡方检验用于拟合优度（比较观测频数与理论频数）和列联表的独立性检验。检验统计量为 χ² = Σ (O − E)² / E，其中 O 为观测频数，E 为期望频数。自由度 ν 视情境而定：对已知分布的拟合优度，ν = 单元格数 − 1 − 估计参数个数；对 r×c 列联表，ν = (r − 1)(c − 1)。
Conditions: expected frequencies should generally be ≥ 5. Combine adjacent classes if needed. A large χ² value indicates poor fit or association. Compare against the critical χ² value for the chosen significance level.
条件：期望频数一般应 ≥ 5，必要时合并相邻组。较大的 χ² 值表示拟合差或存在关联。与所选显著性水平对应的 χ² 临界值比较即可得出结论。
12. Correlation & Regression Recap | 相关与回归要点
Pearson’s product‑moment correlation coefficient r measures the strength and direction of a linear relationship between two variables, ranging from −1 to +1. The equation of the least‑squares regression line is y = a + bx, where b = Sxy / Sxx and a = ȳ − b x̄. Here Sxy = Σ (x − x̄)(y − ȳ) and Sxx = Σ (x − x̄)².
皮尔逊积矩相关系数 r 衡量两个变量之间线性关系的强度和方向，取值在 −1 到 +1 之间。最小二乘回归线方程为 y = a + bx，其中 b = Sxy / Sxx，a = ȳ − b x̄。Sxy = Σ (x − x̄)(y − ȳ)，Sxx = Σ (x − x̄)²。
Residual = observed y − predicted y. The regression line minimises the sum of squared residuals. Interpolation (predicting within the observed x‑range) is trustworthy; extrapolation is dangerous. The coefficient of determination r² gives the proportion of variability in y explained by x.
残差 = 观测 y − 预测 y。回归线使残差平方和最小。在观测 x 范围内进行内插是可靠的，外推则较危险。决定系数 r² 表示 y 的变异中可由 x 解释的比例。
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