📚 Year 13 OCR Statistics: Key Terminology Flashcard Guide | Year 13 OCR 统计：关键术语速记指南
Mastering the vocabulary of statistics is half the battle when preparing for OCR Year 13 exams. This revision article is designed as a bilingual terminology flashcard guide, covering the essential terms you need to know across hypothesis testing, probability distributions, confidence intervals, chi‑squared tests, and regression analysis. Each core concept is explained clearly in English, immediately followed by its Chinese equivalent, helping you build a robust conceptual framework that sticks.
在备考 OCR 13 年级统计考试时，掌握专业词汇是成功的一半。本文是一份双语术语速记指南，涵盖假设检验、概率分布、置信区间、卡方检验以及回归分析等核心领域。每个关键概念先用英文清晰阐述，紧接着提供对应的中文解释，帮助你构建牢固且易于记忆的概念体系。
1. Hypothesis Testing Foundations | 假设检验基础
Null hypothesis (H₀): A statement of no effect, no difference, or no association. It is assumed to be true until the sample data provide sufficient evidence to reject it. For a coin being fair, H₀: p = 0.5.
原假设 (H₀)：陈述没有效应、没有差异或没有关联的命题。在没有足够样本证据推翻它之前，我们假定它为真。例如，对于一枚公平硬币，H₀: p = 0.5。
Alternative hypothesis (H₁ or Hₐ): The statement that contradicts the null hypothesis. It represents what the investigator wants to prove. It can be one‑tailed (H₁: p > 0.5) or two‑tailed (H₁: p ≠ 0.5).
备择假设 (H₁ 或 Hₐ)：与原假设对立的陈述，代表研究者希望证明的结论。它可以是单尾的（H₁: p > 0.5）或双尾的（H₁: p ≠ 0.5）。
Test statistic: A value calculated from the sample data, used to decide whether to reject H₀. Common test statistics include Z, t, χ², and the number of successes in a binomial test.
检验统计量：由样本数据计算得出的数值，用于决定是否拒绝原假设。常见的检验统计量包括 Z、t、χ² 以及二项检验中的成功次数。
One‑tailed (directional) test: A test where the alternative hypothesis specifies a direction of interest (e.g., greater than, less than). All the significance level is placed in one tail of the sampling distribution.
单尾（定向）检验：备择假设指明了关注的方向（如大于或小于）的检验。全部显著性水平都落在抽样分布的单侧尾部。
Two‑tailed (non‑directional) test: A test where the alternative hypothesis states that the parameter is simply not equal to the hypothesised value. The significance level is split equally between the two tails.
双尾（非定向）检验：备择假设仅指出参数不等于假设值的检验。显著性水平被平分到分布的两侧尾部。
2. Significance Levels and p‑values | 显著性水平与 p 值
Significance level (α): The probability of making a Type I error – rejecting a true null hypothesis. Commonly α = 0.05 or 0.01. It is the threshold against which the p‑value is compared.
显著性水平 (α)：犯第一类错误的概率，即错误地拒绝了真实的原假设。常用 α = 0.05 或 0.01。它是判断 p 值是否足够小的基准。
p‑value: The probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true. If p‑value ≤ α, we reject H₀; if p‑value > α, we do not reject H₀.
p 值：在原假设为真的前提下，观察到至少与当前结果一样极端的检验统计量的概率。若 p 值 ≤ α，拒绝 H₀；若 p 值 > α，不拒绝 H₀。
Type I error: Rejecting H₀ when it is actually true. Its probability is controlled by α. A very small α reduces the chance of a false positive.
第一类错误 (Type I error)：原假设为真时却拒绝了它。其概率由 α 控制。非常小的 α 能降低假阳性的几率。
Type II error (β): Failing to reject H₀ when it is false. Its probability is denoted β. The power of a test is 1 − β, which reflects the ability to detect a real effect.
第二类错误 (Type II error, β)：原假设为假时却没有拒绝它。其概率用 β 表示。检验的功效为 1 − β，反映了检测真实效应的能力。
Power of a test: The probability of correctly rejecting a false null hypothesis. Power depends on sample size, α, and the true parameter value. Larger samples give higher power.
检验功效：正确拒绝错误原假设的概率。功效取决于样本量、α 和真实参数值。样本越大，功效越高。
3. Critical Regions and Rejection Rules | 临界区域与拒绝规则
Critical value: The boundary value(s) that separate the rejection region from the non‑rejection region. For a Z‑test at the 5 % two‑tailed level, the critical values are approximately ±1.96.
临界值：将拒绝域与非拒绝域分开的边界值。对于 5 % 双尾 Z 检验，临界值约为 ±1.96。
Critical (rejection) region: The set of values of the test statistic that lead to the rejection of H₀. If the test statistic falls inside the critical region, we reject H₀.
临界区域（拒绝域）：导致拒绝原假设的检验统计量的取值集合。若检验统计量落入该区域，则拒绝 H₀。
Acceptance region: The range of values where H₀ is not rejected. It is the complement of the critical region. Note that we never “accept” H₀, we only fail to reject it.
接受域：不拒绝原假设的取值范围，是临界区域的补集。需注意，我们从不“接受”H₀，只能说未能拒绝它。
Rejection rule: A statement that specifies when to reject H₀. For example: “Reject H₀ if the test statistic X ≥ 8 in a binomial test with n = 20, α = 0.05.”
拒绝规则：指明何时拒绝 H₀ 的陈述。例如：“在 n = 20，α = 0.05 的二项检验中，若检验统计量 X ≥ 8，则拒绝 H₀。”
4. Probability Distributions Overview | 概率分布概览
Random variable: A variable whose value is a numerical outcome of a random phenomenon. It can be discrete (taking countable values) or continuous (taking any value in an interval).
随机变量：其数值由随机现象的结果决定的变量。它可以是离散的（取可数个值）或连续的（在某区间内取任意值）。
Probability mass function (PMF): A function that gives the probability that a discrete random variable is exactly equal to some value. For a binomial distribution, P(X = k) = ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ.
概率质量函数 (PMF)：给出离散随机变量恰好等于某个值的概率的函数。二项分布的 PMF 为 P(X = k) = ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ。
Probability density function (PDF): A function whose integral over an interval gives the probability that a continuous random variable falls within that interval. The height of the PDF at a single point is not a probability.
概率密度函数 (PDF)：其积分给出连续随机变量落在某区间内的概率的函数。单个点上的 PDF 高度并不是概率。
Cumulative distribution function (CDF): The function F(x) = P(X ≤ x), giving the probability that the random variable is less than or equal to x. It is used to find p‑values and critical values.
累积分布函数 (CDF)：函数 F(x) = P(X ≤ x)，给出随机变量小于或等于 x 的概率。常用于查找 p 值和临界值。
5. Binomial Distribution Parameters | 二项分布参数
Binomial distribution B(n, p): Models the number of successes in a fixed number n of independent Bernoulli trials, each with constant probability of success p.
二项分布 B(n, p)：对固定次数 n 的独立伯努利试验中成功次数的建模，每次试验的成功概率 p 恒定。
Number of trials (n): The fixed total number of observations. For ten coin tosses, n = 10. The binomial coefficient ⁿCₖ counts the number of ways to obtain k successes.
试验次数 (n)：固定的观测总次数。如投掷十次硬币，n = 10。二项式系数 ⁿCₖ 用来计算得到 k 次成功的方法数。
Probability of success (p): The probability that a single trial results in a success. In a fair die roll for a “six”, p = 1/6. The expected number of successes is np.
成功概率 (p)：单次试验得到成功的概率。掷一个公平骰子得“六”时，p = 1/6。成功的期望次数为 np。
Mean and variance: For X ~ B(n, p), the mean is μ = np, and the variance is σ² = np(1 − p). These are essential for forming Z‑approximations to the binomial.
均值与方差：对于 X ~ B(n, p)，均值为 μ = np，方差为 σ² = np(1 − p)。这些是进行二项分布正态近似的重要基础。
Assumptions: Trials are independent, each trial has exactly two outcomes (success or failure), and p remains constant for all trials.
假设条件：试验相互独立，每次试验只有两种结果（成功或失败），且 p 在所有试验中保持不变。
6. Poisson Distribution and Approximations | 泊松分布与近似
Poisson distribution Po(λ): Models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate λ. P(X = k) = (e⁻λ λᵏ) / k!.
泊松分布 Po(λ)：对固定时间或空间区间内事件发生次数建模，当事件以恒定的平均速率 λ 独立发生时使用。P(X = k) = (e⁻λ λᵏ) / k!。
Rate parameter λ: The mean number of events in the interval. For a Poisson distribution, both the mean and the variance equal λ. This property makes it “equi‑dispersed”.
速率参数 λ：区间内事件发生的平均数。泊松分布的均值与方差都等于 λ，这一性质使其具有“等离散度”。
Poisson approximation to the binomial: When n is large and p is small (typically n ≥ 50, np ≤ 10), B(n, p) can be approximated by Po(λ) with λ = np. The approximation simplifies probability calculations.
泊松近似二项分布：当 n 大、p 小（通常 n ≥ 50，np ≤ 10）时，可用 Po(λ) 近似 B(n, p)，其中 λ = np。这种近似能简化概率计算。
Conditions for Poisson modelling: Events occur randomly and independently, and the rate λ is constant. The probability of two events happening in an infinitesimally small interval is negligible.
泊松建模的条件：事件随机且独立发生，速率 λ 恒定。在无穷小的时间段内发生两个事件的概率可以忽略不计。
7. The Normal Distribution in Inferences | 正态分布与推断
Normal distribution N(μ, σ²): A continuous, bell‑shaped distribution defined by its mean μ and variance σ². It is symmetric about μ and the total area under the curve equals 1.
正态分布 N(μ, σ²)：一种由均值 μ 和方差 σ² 定义的连续、钟形分布。曲线关于 μ 对称，且总面积等于 1。
Standard normal distribution Z ~ N(0, 1): The transformation Z = (X − μ) / σ converts any normal variable X into a standard score. Tables of Φ(z) = P(Z ≤ z) are used to find probabilities and critical values.
标准正态分布 Z ~ N(0, 1)：变换 Z = (X − μ) / σ 可将任意正态变量 X 转化为标准得分。概率表 Φ(z) = P(Z ≤ z) 用来查找概率和临界值。
Central Limit Theorem (CLT): For a large sample size (n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the population’s shape, with mean μ and variance σ²/n.
中心极限定理 (CLT)：当样本容量较大（n ≥ 30）时，样本均值的抽样分布近似正态，无论总体形状如何，其均值为 μ，方差为 σ²/n。
Continuity correction: When approximating a discrete distribution (binomial or Poisson) with the normal, we add or subtract 0.5 from the discrete value to improve accuracy. For P(X ≤ 5) we use P(X ≤ 5.5) in the normal approximation.
连续性校正：用正态分布近似离散分布（二项或泊松）时，在离散值上加减 0.5 以提高精度。例如，P(X ≤ 5) 在正态近似中用 P(X ≤ 5.5)。
8. Chi‑Squared Tests for Association | 卡方独立性检验
Chi‑squared (χ²) statistic: A measure of the discrepancy between observed and expected frequencies: χ² = Σ (O − E)² / E, where O is the observed frequency and E is the expected frequency under H₀.
卡方 (χ²) 统计量：度量观测频数与期望频数之间差异的指标：χ² = Σ (O − E)² / E，其中 O 为观测频数，E 为 H₀ 下的期望频数。
Contingency table: A table displaying the frequency distribution of two categorical variables. The rows represent one variable, the columns the other. Expected values are calculated as (row total × column total) ÷ grand total.
列联表：展示两个分类变量频数分布的表格，行代表一个变量，列代表另一个。期望频数计算为（行合计 × 列合计）÷ 总计。
Degrees of freedom (df): For a test of association in an r × c table, df = (r − 1)(c − 1). The shape of the χ² distribution depends on df; larger df produce a more symmetric curve.
自由度 (df)：对于 r × c 列联表的独立性检验，df = (r − 1)(c − 1)。χ² 分布的形状依赖于自由度；自由度越大，曲线越对称。
Yates’ correction: Applied to a 2 × 2 table, this correction subtracts 0.5 from each |O − E| before squaring, making the χ² approximation more accurate for small expected counts.
耶茨校正：用于 2 × 2 表格，在平方前将每个 |O − E| 减去 0.5，使期望频数较小时的 χ² 近似更准确。
Assumptions: Observations are independent, and no more than 20 % of expected frequencies are below 5, with no expected frequency less than 1. If violated, categories should be combined.
假设条件：观测值独立，且期望频数小于 5 的单元格不超过总数的 20 %，并且所有期望频数均不低于 1。若违反，应合并类别。
9. Correlation vs. Regression | 相关与回归
Correlation coefficient (r): A measure of the strength and direction of a linear relationship between two variables. Pearson’s r ranges from −1 (perfect negative) to +1 (perfect positive). r = 0 indicates no linear correlation.
相关系数 (r)：衡量两个变量之间线性关系强度与方向的指标。皮尔逊 r 的取值范围为 −1（完全负相关）到 +1（完全正相关）。r = 0 表示无线性关系。
Spearman’s rank correlation (rₛ): A non‑parametric correlation based on ranked data. It is used when the relationship is monotonic but not necessarily linear, or when outliers are present.
斯皮尔曼秩相关系数 (rₛ)：基于排序数据的非参数相关度量。当关系单调但不一定是线性，或存在异常值时使用。
Hypothesis test for ρ: We test H₀: ρ = 0 (no population correlation) against H₁: ρ ≠ 0. The test statistic t = r √(n − 2) / √(1 − r²) follows a t‑distribution with n − 2 df.
对 ρ 的假设检验：检验 H₀: ρ = 0（总体无相关）对 H₁: ρ ≠ 0。检验统计量 t = r √(n − 2) / √(1 − r²) 服从自由度为 n − 2 的 t 分布。
Least squares regression line: The line ŷ = a + bx that minimises the sum of the squared vertical distances (residuals) from the data points. b = Sₓᵧ / Sₓₓ and a = ȳ − b x̄.
最小二乘回归直线：使数据点到直线的垂直距离平方和（残差平方和）最小的直线 ŷ = a + bx。b = Sₓᵧ / Sₓₓ，a = ȳ − b x̄。
Residual: The difference between an observed y‑value and the value predicted by the regression line: e = y − ŷ. Residual plots help check linearity and constant variance assumptions.
残差：观测值 y 与回归直线预测值之差：e = y − ŷ。残差图有助于检验线性与等方差假设。
10. Confidence Intervals Demystified | 置信区间解析
Confidence interval (CI): An interval estimate for an unknown population parameter, constructed from sample data. A 95 % CI has a 0.95 probability of containing the true parameter value in repeated sampling.
置信区间 (CI)：根据样本数据构造的未知总体参数的区间估计。95 % 置信区间意味着在重复抽样中，有 95 % 的概率包含真实参数值。
Confidence level: The long‑run success rate of the method. For a 95 % CI, if we took many samples, about 95 % of the constructed intervals would capture the parameter.
置信水平：该方法在长期中的成功率。对于 95 % CI，如果我们抽取大量样本，约 95 % 的区间会捕获参数。
Margin of error: The half‑width of a confidence interval. For a mean with known σ, it is z* × (σ / √n). The margin of error decreases when n increases or when the confidence level is lowered.
误差范围：置信区间的一半宽度。当 σ 已知时，对于均值有 z* × (σ / √n)。增大 n 或降低置信水平可使误差范围减小。
CI for a population proportion: Using the normal approximation, p̂ ± z* √(p̂(1 − p̂)/n). It requires np̂ ≥ 5 and n(1 − p̂) ≥ 5 for the approximation to be valid.
总体比例的置信区间：使用正态近似，p̂ ± z* √(p̂(1 − p̂)/n)。需要 np̂ ≥ 5 且 n(1 − p̂) ≥ 5 以保证近似的有效性。
CI for a population mean (t‑interval): When σ is unknown, we use the t‑distribution: x̄ ± t* (s / √n) with n − 1 degrees of freedom. This is more reliable for small samples from a normal population.
总体均值的置信区间（t 区间）：当 σ 未知时，使用 t 分布：x̄ ± t* (s / √n)，自由度为 n − 1。对于来自正态总体的小样本，该方法更为可靠。
Published by TutorHao | Statistics Revision Series | aleveler.com
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