📚 Year 13 OCR Statistics: Vocabulary & Terminology Memory Guide | Year 13 OCR 统计：词汇术语速记指南
Mastering statistical vocabulary is the first step towards confidence in Year 13 OCR Statistics. This guide breaks down essential terms – from probability distributions to hypothesis testing and chi‑squared tests – and provides memory hooks to help you recall definitions, formulas and conditions under exam pressure. Each term is presented in English immediately followed by its Chinese equivalent, making it easy to reinforce understanding in both languages.
掌握统计学术语是学好 Year 13 OCR 统计的第一步。本指南解析了从概率分布、假设检验到卡方检验等关键术语，并配有记忆技巧，帮助你在考试中快速回忆定义、公式和使用条件。每个术语均采用先英文后中文的配对呈现方式，让你在双语对照中巩固理解。
1. Discrete Random Variables & Probability Distributions | 离散随机变量与概率分布
gives P(X = x) for every outcome x. The sum of all probabilities must equal 1, and the cumulative distribution function (CDF) gives P(X ≤ x) by summing the PMF up to that point.
离散随机变量只能取可数个值，每个值对应一个概率。概率质量函数（PMF）给出每个结果 x 的 P(X = x)。所有概率之和必须等于 1，而累积分布函数（CDF）通过累加 PMF 得到 P(X ≤ x)。
Memory tip: Think of the PMF as the “exact probability recipe” and the CDF as a “running total”. Always check Σ p(x) = 1 as a quick validation of your table.
记忆提示：把 PMF 想象成“精确概率配方”，CDF 是“累计总和”。始终用 Σ p(x) = 1 快速检验表格是否正确。
2. Expectation and Variance | 期望与方差
The expectation E(X) is the long‑ P(X = x). Variance Var(X) measures the spread of the distribution, given by Var(X) = E(X²) – [E(X)]². The standard deviation is the square root of the variance, σ = √Var(X).
期望 E(X) 是随机变量的长期平均值，计算公式为 Σ x P(X = x)。方差 Var(X) 衡量分布的离散程度，常用公式 Var(X) = E(X²) – [E(X)]²。标准差是方差的平方根 σ = √Var(X)。
X). Remember that adding a constant shifts the mean but leaves the spread unchanged; scaling multiplies the variance by the square of the multiplier.
对于线性变换 Y = aX + b，期望为 E(Y) = aE(X) + b，方差为 Var(Y) = a² Var(X)。记住加上常数只会平移均值，不改变波动；乘上系数则方差变成原来的系数平方倍。
3. Binomial Distribution | 二项分布
A binomial distribution X ~ B(n, p) models the number of successes in n independent trials, each with constant probability p. The probability mass function is:
P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ
二项分布 X ~ B(n, p) 描述 n 次独立试验中成功的次数，每次成功概率为 p。其概率质量函数如上所示。
Conditions: fixed number of trials, independence, constant probability, and binary outcomes (success/failure). If trials are not independent or p changes, the binomial model does not apply.
条件：固定试验次数、每次试验独立、成功概率恒定、结果只有成功/失败两种。如果试验不独立或 p 变化，则不能使用二项分布。
Memory hook: B for “Binary outcomes, Binomial coefficient, Binomial tables”. Compute the mean as np and variance as np(1 – p). Use these to verify calculations quickly.
记忆点：B 代表“二元结果、二项系数、二项表”。期望为 np，方差为 np(1 – p)，可以快速核对计算结果。
4. Poisson Distribution | 泊松分布
The Poisson distribution X ~ 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. Its PMF is:
P(X = r) = e⁻λ λʳ / r!
泊松分布 X ~ Po(λ) 用于单位时间或空间内事件发生次数，事件独立且平均发生率为 λ。其概率质量函数如上。
often used to approximate a binomial B(n, p) when n is large and p is small, with λ = np.
关键性质：均值与方差都等于 λ。使用条件：事件单独、随机发生。当 n 很大而 p 很小时，常用 λ = np 的泊松分布来近似二项分布 B(n, p)。
Memory tip: “Poisson has its own rhythm – mean equals variance equals λ”. For approximation, check n > 50 and np < 5 to keep errors small.
记忆点：“泊松有自己的韵律 – 均值等于方差等于 λ”。近似时检查 n > 50 且 np < 5，可将误差控制在较小范围。
5. Normal Distribution & Standardisation | 正态分布与标准化
The normal distribution X ~ N(μ, σ²) is a continuous, bell‑shaped curve defined by its mean μ and variance σ². Any normal random variable can be converted to a standard normal Z ~ N(0, 1) using the z‑score:
Z = (X – μ) / σ
正态分布 X ~ N(μ, σ²) 是由均值 μ 和方差 σ² 决定的连续型钟形曲线。任何正态变量都可借助 z 分数转化为标准正态 Z ~ N(0, 1)，公式如上所示。
obabilities. For inverse problems, you locate the probability in the body of the table to read off the z‑value, then back‑ o apply continuity corrections when approximating a discrete distribution with the normal.
使用标准正态表查找概率。反向问题时，先在表内找到对应概率读出 z 值，再反算 X = μ + z σ。用正态近似离散分布时，务必进行连续性校正。
Memory tip: “Z is the link between all normals.” Sketch the bell curve and shade the area – a picture is worth ten formula manipulations.
记忆提示：“Z 是联通所有正态分布的桥梁。”画图并涂鸦阴影区域 – 一幅图胜过十次公式推导。
6. Sampling Distributions & the Central Limit Theorem | 抽样分布与中心极限定理
A sampling distribution is the probability distribution of a statistic (like the sample mean) calculated from many samples of the same size. The standard error measures how much the statistic varies from sample to sample. For the sample mean X̄ from a population with variance σ², the standard error is σ / √n.
抽样分布是指从多个相同容量的样本中计算出的统计量（如样本均值）的概率分布。标准误衡量该统计量在不同样本间的波动程度。对于总体方差 σ²，样本均值 X̄ 的标准误为 σ / √n。
g distribution of the mean will be approximately normal regardless of the shape of the original population. This underpins confidence intervals and hypothesis tests for means.
中心极限定理（CLT）指出，当样本量足够大（通常 n ≥ 30）时，无论原始总体形状如何，样本均值的抽样分布都近似正态。这是均值的置信区间与假设检验的理论基础。
Memory phrase: “Mean of many tends to normal.” normal, the sampling distribution is normal for any n; otherwise rely on CLT when n ≥ 30.
记忆口诀：“多者之平均趋向正态。”务必检查条件：若原始总体本身正态，则任何 n 下抽样分布皆正态；否则当 n ≥ 30 时依靠 CLT。
7. Confidence Intervals for Mean & Proportion | 均值和比例的置信区间
A confidence interval provides a range of plausible values for a population parameter. For a population mean μ with known σ, the 95% confidence interval is:
x̄ ± z × (σ / √n)
where z = 1.96 for 95% confidence. When σ is unknown and estimated by s, use the t‑distribution and t‑value in place of z.
置信区间给出总体参数的可能取值范围。对于已知 σ 的总体均值 μ，95% 置信区间如上，其中 z = 1.96。当 σ 未知用 s 估计时，改用 t 分布及相应的 t 值。
For a population proportion p, the approximate confidence interval uses p̂ ± z × √[p̂(1 – p̂) / n], provided np̂ and n(1 – p̂) are both greater than 5. Interpretation: we are 95% confident that the interval captures the true parameter – it does not mean there is a 95% probability that the parameter lies in this particular interval.
对于总体比例 p，近似置信区间为 p̂ ± z × √[p̂(1 – p̂) / n]，前提是 np̂ 和 n(1 – p̂) 均大于 5。解释：我们有 95% 的信心认为该区间包含了真实参数 – 这并不代表该具体区间包含参数的概率是 95%。
Memory trigger: “Z‑multiples make boundaries.” ze and confidence level. A larger sample narrows the interval; higher confidence widens it.
记忆触发：“z 乘数定边界。”宽度 2 × z × SE 依赖于样本量和置信水平。样本越大区间越窄，置信水平越高区间越宽。
8. Hypothesis Testing – Core Concepts | 假设检验 – 核心概念
A hypothesis test assesses evidence against a null hypothesis H₀ in favour of an alternative hypothesis H₁. The significance level α (commonly 0.05) is the threshold probability of rejecting H₀ when it is true. The test statistic is calculated from sample data and compared to a critical value, or its p‑value is compared to α.
假设检验评价反对原假设 H₀ 而支持备择假设 H₁ 的证据。显著性水平 α（常用 0.05）是当 H₀ 为真时拒绝 H₀ 的概率门槛。检验统计量由样本数据算出，与临界值比较，或将其 p 值与 α 比较。
error occurs when H₀ is false but not rejected (probability = β). The power of a test is 1 – β. To reduce both errors, increase the sample size.
第一类错误：H₀ 真但被拒绝（概率 = α）。第二类错误：H₀ 假但未被拒绝（概率 = β）。检验功效为 1 – β。要减少两类错误，增加样本量即可。
Memory: “Guilty until proven innocent? Think of H₀ as the defendant.” You only reject H₀ if the data provide sufficient evidence beyond reasonable doubt. Write a conclusion in context: “There is sufficient/insufficient evidence to suggest that…”.
记忆：“有罪直到证明无罪？把 H₀ 想象成被告。”仅当数据提供超出合理怀疑的证据时才拒绝 H₀。在上下文中写出结论：“有/没有充分证据表明……”。
9. Correlation & Regression | 相关与回归
The product moment correlation coefficient (PMCC), r, measures the strength and direction of a linear relationship between two variables. Its value lies between –1 and +1. Spearman’s rank correlation coefficient, rₛ, measures monotonic association (not necessarily linear) and is calculated on ranked data.
乘积矩相关系数（PMCC）r 衡量两个变量间线性关系的强度和方向，取值在 –1 到 +1 之间。Spearman 等级相关系数 rₛ 测量单调关联（不要求线性），基于排序数据计算。
Regression gives the line of best fit y = a + bx, where b = Sxy / Sxx and a = ȳ – b x̄. The residual e = y – ŷ cal deviation of a point from the line; examining residuals helps check model assumptions.
回归得到最佳拟合直线 y = a + bx，其中 b = Sxy / Sxx，a = ȳ – b x̄。残差 e = y – ŷ 是点与直线的纵向偏离；检查残差可帮助判断模型假设是否合理。
ply causation. Always consider lurking variables. In hypothesis tests for ρ = 0, use the t‑test for the slope or the correlation test statistic.
强相关不意味着因果。始终要考虑潜在变量。在检验 ρ = 0 时，使用对斜率的 t 检验或相关系数检验统计量。
10. Chi‑Squared Tests for Independence | 独立性卡方检验
The chi‑squared test for independence (or association) analyses contingency tables to see if two categorical variables are related. The test statistic is:
χ² = Σ (O – E)² / E
dependence. The expected frequency for a cell is (row total × column total) / grand total.
独立性（或关联性）卡方检验分析列联表以检验两个分类变量是否相关。检验统计量如上，其中 O 为观测频数，E 为原假设独立下的期望频数。某个格子的期望频数为 (行合计 × 列合计) / 总合计。
Degrees of freedom = (rows – 1) × (columns – percentages; categories are mutually exclusive. Compare χ² to the critical value from chi‑squared tables at the chosen significance level.
自由度 = (行数 – 1) × (列数 – 1)。条件：所有期望频数应至少为 5；数据为频数而非百分比；类别互斥。将 χ² 与卡方表中选定显著性水平下的临界值比较。
Memory trick: “Observed minus Expected, squared over Expected – sum it up.” If χ² is large, the observed pattern deviates significantly from what independence would predict, leading to rejection of H₀.
记忆口诀：“观减期，平方除期，再求和。”若 χ² 很大，说明观测模式与独立预期显著偏离，从而拒绝 H₀。
11. Non‑parametric Tests – The Sign Test | 非参数检验 – 符号检验
When the normality assumption is questionable, non‑parametric methods like the sign test can be used. The sign test tests for a population median difference in paired data by counting the number of positive differences. It ignores the magnitude of differences, using only their direction.
当正态性假设存疑时，可采用非参数方法，如符号检验。符号检验通过计算配对数据中正差的个数，检验总体中位数差值。它忽略差值大小，只使用其方向。
Under the null hypothesis of no difference, the number of positive signs follows a binomial distribution B(n, 0.5). The test compares the observed count of the less frequent sign to the critical value from the binomial distribution at significance level α (or computes a two‑tailed p‑value).
在原假设“没有差异”下，正号个数服从二项分布 B(n, 0.5)。将出现次数较少的符号的观测数与显著性水平 α 下的二项分布临界值进行比较（或计算双侧 p 值）。
Memory phrase: “Sign test cares about direction, not distance.” Ties (zero differences) are usually discarded, and the sample size n is reduced accordingly. This test is simple but less powerful than parametric alternatives when normality holds.
记忆语：“符号检验重方向不重大小。”平局（差值为零）通常剔除，样本量 n 相应减少。该检验简单，但在正态性成立时效能低于参数检验。
Published by TutorHao | Statistics Revision Series | aleveler.com
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