📚 Year 13 SQA Statistics: Glossary Memorisation Guide | Year 13 SQA 统计：词汇术语速记指南
A solid grasp of statistical vocabulary can dramatically improve your performance in the SQA Advanced Higher Statistics exam. This guide breaks down essential terminology into bilingual, bite-sized explanations with memory hooks, covering everything from data collection to inference.
牢固掌握统计术语能显著提升你在 SQA Advanced Higher Statistics 考试中的表现。本指南将核心术语拆分为中英对照的小模块，配合记忆技巧，覆盖从数据收集到推断的所有重点。
1. Populations, Samples and Parameters | 总体、样本与参数
A population is the entire collection of individuals or items we wish to study. A sample is a subset drawn from the population. A parameter is a numerical summary describing a population (e.g., population mean μ, population standard deviation σ), while a statistic describes a sample (e.g., sample mean x̄, sample standard deviation s).
总体是我们想研究的全部对象，样本是从中抽取的子集。参数是描述总体的数字特征（如总体均值 μ、总体标准差 σ），而统计量则描述样本（如样本均值 x̄、样本标准差 s）。
A quick memory fix: the initial letter of Parameter (P) matches Population, and Statistic (S) matches Sample.
快速记忆：Parameter（参数）与 Population（总体）都以字母 P 开头，Statistic（统计量）与 Sample（样本）都以 S 开头。
2. Measures of Centre and Spread | 中心与离散度量
Centre is captured by the mean, median and mode. The mean (x̄ e median is the middle value when data are ordered; the mode is the most frequent value. In skewed distributions, the median is more resistant to extreme values than the mean.
中心趋势通过均值、中位数和众数描述。均值（x̄ = Σx / n）是算术平均，中位数是排序后的中间值，众数是出现频率最高的值。在偏态分布中，中位数比均值更不容易被极端值拉偏。
₁, variance σ² or s², and standard deviation σ or s. Variance is the average of squared deviations from the mean; standard deviation is its square root, returning to the original units.
离散度用极差（最大值−最小值）、四分位距 IQR = Q₃ − Q₁、方差 σ² 或 s² 以及标准差 σ 或 s 衡量。方差是离差平方的平均值，标准差是方差的平方根，使单位与原始数据一致。
s² = Σ(x − x̄)² / (n − 1) and s = √[Σ(x − x̄)² / (n − 1)]
Remember: 'Standard deviation is the root of variance, bringing the spread back to scale.'
记住：'标准差是方差的平方根，让离散度回到原本的量纲。'
3. Random Variables and Probability Distributions | 随机变量与概率分布
A random variable X assigns a number to each outcome of a random experiment. Discrete random variables take countable values (e.g., 0, 1, 2, ...), while continuous random variables can assume any value within an interval.
随机变量 X 为随机试验的每个结果赋予一个数值。离散型随机变量取可数个值（如 0, 1, 2, ...），连续型随机变量可在某个区间内取任意值。
The probability distribution lists possible values and their probabilities for discrete variables; for continuous variables it is described by a probability density function (PDF).
概率分布列出离散变量的可能取值及其概率；连续变量的分布则用概率密度函数（PDF）描述。
The expected value E(X) = Σ x·P(x) is the long‑run average. Variance Var(X) = E(X²) − [E(X)]² measures the spread.
期望值 E(X) = Σ x·P(x) 是长期平均，方差 Var(X) = E(X²) − [E(X)]² 度量离散程度。
4. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in n independent trials, each with the same probability of success p. We write X ~ B(n, p).
二项分布用于描述 n 次独立试验中成功的次数，每次成功概率为 p。记作 X ~ B(n, p)。
k successes is P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, where C(n, k) is the binomial coefficient 'n choose k'.
恰好 k 次成功的概率为 P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ，其中 C(n, k) 是组合数 n 选 k。
² = np(1 − p). Conditions: fixed n, independent trials, two possible outcomes, constant p.
均值 μ = np，方差 σ² = np(1 − p)。使用条件：n 固定、试验独立、每次只有两种结果、p 恒定。
Think 'Binomial counts successes in repeated coin flips with an unfair coin.'
想象一下：'二项分布数成功，就像不停投掷一枚不公正的硬币。'
5. Poisson Distribution | 泊松分布
The Poisson distribution models the number of rare events occurring in a fixed interval of time or space. It has a single parameter, the mean rate λ. We write X ~ Po(λ).
泊松分布模拟固定时间或空间区间内稀有事件的发生次数。它只有一个参数：平均发生率 λ。记作 X ~ Po(λ)。
The probability mass function is P(X = k) = e⁻λ λᵏ / k! . Its mean and variance both equal λ.
概率质量函数为 P(X = k) = e⁻λ λᵏ / k!。期望和方差均等于 λ。
Conditions: events occur independently, singly and at a constant average rate. If n is large and p is small, a binomial B(n, p) can be approximated by Po(np).
条件：事件独立发生、单个出现、平均速率恒定。若 n 很大而 p 很小，二项分布 B(n, p) 可用泊松分布 Po(np) 近似。
Mnemonic: 'Poisson for rare events per unit – think calls per hour, not your favourite coffee order!'
巧记：'泊松分布计稀少，就像每小时的来电数，而不是你每天点的咖啡！'
6. Normal Distribution | 正态分布
The normal distribution is a symmetric, bell‑shaped continuous distribution. Notation: X ~ N(μ, σ²). The standard normal Z ~ N(0, 1) is obtained by z = (x − μ) / σ.
正态分布是对称的钟形连续分布，记作 X ~ N(μ, σ²)。标准正态 Z ~ N(0, 1) 通过 z = (x − μ) / σ 得到。
Approximately 68% of data fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is the empirical rule.
约 68% 的数据落在均值 ± 1σ 内，约 95% 落在 ± 2σ 内，约 99.7% 在 ± 3σ 内，这就是经验法则。
The Central Limit Theorem states that for a sufficiently large sample size n, the sampling distribution of the sample mean is approximately normal, regardless of the population's shape.
中心极限定理指出：当样本量 n 足够大时，样本均值的抽样分布近似正态，无论总体形状如何。
Keep in mind: 'Normal is the bell, CLT makes the bell ring for sample means!'
牢记：'正态就是那口钟，中心极限定理让抽样均值的钟声响起！'
7. Sampling and Estimation | 抽样与估计
A point estimate is a single number calculated from sample data to estimate a population parameter (e.g., x̄ estimates μ). A confidence interval provides a range of plausible values for the parameter.
点估计是用样本数据算出的单一数值，用以估计总体参数（如用 x̄ 估计 μ）。置信区间则给出参数的可能取值范围。
For the mean of a normal population with known σ, a 95% confidence interval is x̄ ± 1.96 × σ/√n. For a population proportion p, a 95% CI is p̂ ± 1.96 × √[p̂(1 − p̂)/n].
当总体正态且 σ 已知时，均值的 95% 置信区间为 x̄ ± 1.96 × σ/√n。总体比例 p 的 95% 置信区间为 p̂ ± 1.96 × √[p̂(1 − p̂)/n]。
The margin of error is the half‑width of the confidence interval. As sample size increases, the margin of error decreases.
误差边际是置信区间宽度的一半。样本量越大，误差边际越小。
8. Hypothesis Testing | 假设检验
Hypothesis testing starts with a null hypothesis H₀ (status quo) and an alternative H₁ (research claim). A test statistic is computed from the sample, and its p‑value is the probability of obtaining a result at least as extreme as the observed one, assuming H₀ is true.
假设检验从零假设 H₀（通常表示无效应）和备择假设 H₁（研究声明）开始。根据样本计算检验统计量，其 p 值是在 H₀ 为真的前提下，得到当前及更极端结果的概率。
Compare the p‑ (often 0.05). If p ≤ α, reject H₀; otherwise, there is insufficient evidence to reject H₀.
将 p 值与显著性水平 α（常取 0.05）比较。若 p ≤ α，拒绝 H₀；否则，没有足够证据拒绝 H₀。
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