A-Level CAIE Statistics: Terminology Memory Guide | A-Level CAIE 统计：词汇术语速记指南

📚 A-Level CAIE Statistics: Terminology Memory Guide | A-Level CAIE 统计：词汇术语速记指南

This guide gathers the most frequently examined statistical terms in CAIE A-Level Statistics, grouped by syllabus topic. Each term is paired with a short definition and a memory hook to help you recall it quickly in timed questions.

本指南汇集了 CAIE A-Level 统计考试中最常出现的统计术语，按大纲主题分组。每个术语都配有简短定义和一条速记提示，帮助你在限时答题中快速回忆。

1. Data Types and Representation | 数据类型与图表表示

Variables are first classified as qualitative (categorical) or quantitative (numerical). Quantitative data are discrete if values come from counting, such as the number of students, and continuous if they come from measuring, such as height or time.

变量首先分为定性变量（类别变量）或定量变量（数值变量）。定量数据如果来自计数，如学生人数，则是离散型；如果来自测量，如身高或时间，则是连续型。

A population is the entire group being studied, while a sample is a subset of it. A parameter describes a population, and a statistic describes a sample. Memory hook: ‘P’ for population and ‘P’ for parameter go together.

总体是研究的整个群体，样本是总体的一个子集。参数描述总体，统计量描述样本。速记提示：字母 P 代表 population（总体），也代表 parameter（参数），两者配对。

Common diagrams in CAIE Statistics include bar charts, pie charts, histograms, cumulative frequency curves, box-and-whisker plots, and stem-and-leaf diagrams. Histograms use area proportional to frequency, while bar charts use equal-width bars for categories.

CAIE 统计中常见图表包括条形图、饼图、直方图、累计频率曲线、箱线图和茎叶图。直方图用面积表示频率，而条形图对各类别使用等宽条形。

Term	Meaning	Memory Hook
Class width	Upper bound minus lower bound	Width = end minus start
Midpoint	(Lower bound + upper bound) ÷ 2	Middle of the interval
Cumulative frequency	Running total of frequencies	Add as you go

For grouped data, use class midpoints to estimate the mean. For the median, use linear interpolation within the median class. Memory hook: ‘Midpoints for mean, boundaries for median.’

对于分组数据，用组中点估算均值。对于中位数，在中位数所在组内使用线性插值。速记提示：均值用组中点，中位数用组边界。

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2. Measures of Central Tendency | 集中趋势度量

The mean is the arithmetic average of all values. It uses every data point, so it can be strongly affected by outliers. The median is the middle value when data are ordered, and the mode is the most frequent value.

均值是所有数值的算术平均数。它使用了所有数据点，因此受离群值影响很大。中位数是数据排序后的中间值，众数是出现频率最高的值。

For ungrouped data, x̄ = Σx ÷ n. For grouped data, x̄ = Σfx ÷ Σf, where x is the midpoint and f is the frequency. Memory hook: ‘Mean = total of values divided by total count.’

对于未分组数据，x̄ = Σx ÷ n。对于分组数据，x̄ = Σfx ÷ Σf，其中 x 为组中点，f 为频率。速记提示：均值等于所有值之和除以总个数。

The weighted mean is Σwx ÷ Σw, where each value x is multiplied by its weight w. This is useful when some observations count more heavily than others.

加权平均值为 Σwx ÷ Σw，其中每个值 x 乘以它的权重 w。当某些观测值的权重更大时，加权平均值很有用。

For a perfectly symmetric distribution, mean = median = mode. For positive skew, mean > median > mode. For negative skew, mean < median < mode. Memory hook: the mean is pulled in the direction of the tail.

对于完全对称的分布，均值 = 中位数 = 众数。正偏态时，均值 > 中位数 > 众数。负偏态时，均值 < 中位数 < 众数。速记提示：均值被尾部方向拉走。

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3. Measures of Dispersion | 离散程度度量

Dispersion measures how spread out the data are. The range is maximum minus minimum. It is simple but ignores most data. The interquartile range, IQR = Q3 − Q1, is more robust against outliers.

离散程度度量数据分布的广度。极差等于最大值减最小值。它计算简单，但忽略了大部分数据。四分位距 IQR = Q₃ − Q₁，对离群值更稳健。

Variance is the average of squared deviations from the mean. Standard deviation is its square root and has the same units as the data. Memory hook: ‘Standard deviation = square root of mean squared deviation.’

方差是各数据与均值之差的平方的平均数。标准差是方差的平方根，单位与数据相同。速记提示：标准差等于均方偏差的平方根。

Variance = Σ(x − x̄)² ÷ n or Σx² ÷ n − x̄²

For grouped data, use class midpoints in the variance formula. The sample variance used for estimation divides by n − 1 instead of n, which makes it an unbiased estimator.

对于分组数据，在方差公式中使用组中点。用于估计的样本方差除以 n − 1 而不是 n，使其成为无偏估计量。

Outliers are often identified by fences: lower fence = Q₁ − 1.5 × IQR, upper fence = Q₃ + 1.5 × IQR. Values outside these fences are flagged as outliers.

离群值通常用界限来识别：下界 = Q₁ − 1.5 × IQR，上界 = Q₃ + 1.5 × IQR。超出这些界限的值被标记为离群值。

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4. Permutations and Combinations | 排列与组合

A permutation is an arrangement where order matters, while a combination is a selection where order does not matter. Memory hook: ‘Permutation has position, combination only chooses.’

排列是有顺序要求的安排，组合是不考虑顺序的选择。速记提示：排列强调位置，组合只做选择。

The number of ways to arrange n distinct objects is n! = n × (n − 1) × … × 1. If some objects repeat, divide by the factorials of the repeated counts.

排列 n 个不同物体的方法数为 n! = n × (n − 1) × … × 1。如果有重复物体，要除以重复个数的阶乘。

For selecting r objects from n distinct objects, the combination formula is ⁿCₖ = n! ÷ (r!(n − r)!). This is used in binomial probabilities. Memory hook: combinations choose the positions of successes.

从 n 个不同物体中选出 r 个的组合公式为 ⁿCₖ = n! ÷ (r!(n − r)!)。这在二项概率中使用。速记提示：组合决定成功出现的位置。

When there are n₁ ways to do one task and n₂ ways to do another, the total number of ways for both tasks is n₁ × n₂. This is the multiplication principle.

如果做一件事有 n₁ 种方法，做另一件事有 n₂ 种方法，那么两件事的总方法数为 n₁ × n₂。这就是乘法原理。

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5. Probability Basics | 概率基础

An experiment has outcomes, and the sample space is the set of all possible outcomes. An event is any subset of the sample space. Probability P(A) must satisfy 0 ≤ P(A) ≤ 1.

一次试验会产生结果，样本空间是所有可能结果的集合。事件是样本空间的任意子集。概率 P(A) 必须满足 0 ≤ P(A) ≤ 1。

Two events are mutually exclusive if they cannot happen together, so P(A ∩ B) = 0. They are independent if the occurrence of one does not affect the other, so P(A ∩ B) = P(A) × P(B).

两个事件互斥意味着它们不能同时发生，因此 P(A ∩ B) = 0。两个事件独立意味着一个事件发生不影响另一个事件，因此 P(A ∩ B) = P(A) × P(B)。

The addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, it simplifies to P(A ∪ B) = P(A) + P(B).

加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。对于互斥事件，它简化为 P(A ∪ B) = P(A) + P(B)。

Conditional probability is P(A | B) = P(A ∩ B) ÷ P(B), read as ‘probability of A given B’. The multiplication rule follows: P(A ∩ B) = P(B) × P(A | B).

条件概率为 P(A | B) = P(A ∩ B) ÷ P(B)，读作“在 B 发生的条件下 A 的概率”。乘法法则由此得出：P(A ∩ B) = P(B) × P(A | B)。

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6. Discrete Random Variables | 离散随机变量

A discrete random variable takes a countable number of values. Its probability distribution lists each value with its probability. The total probability must sum to 1.

离散随机变量取可数个值。它的概率分布列出每个值及其概率。总概率之和必须为 1。

The expected value E(X) is the long-run average, calculated as E(X) = Σ x p(x). Memory hook: expectation is the probability-weighted average.

期望值 E(X) 是长期平均值，计算公式为 E(X) = Σ x p(x)。速记提示：期望就是概率加权平均。

Var(X) = E(X²) − [E(X)]² = Σ x² p(x) − μ²

For a linear transformation, E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X). Adding a constant shifts the mean but does not change the variance.

对于线性变换，E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。加上一个常数只平移均值，不改变方差。

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7. Binomial, Geometric and Poisson Distributions | 二项分布、几何分布与泊松分布

The binomial distribution applies when there are a fixed number n of independent trials, each with only two outcomes and constant success probability p. We write X ~ B(n, p).

二项分布适用于固定 n 次独立试验，每次只有两种结果，且成功概率 p 恒定。记作 X ~ B(n, p)。

P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ, where q = 1 − p

For X ~ B(n, p), the mean is np and the variance is npq. Memory hook: binomial needs a fixed number of trials and constant p.

对于 X ~ B(n, p)，均值为 np，方差为 npq。速记提示：二项分布需要固定试验次数和恒定概率 p。

The geometric distribution counts the number of trials until the first success. If X ~ Geo(p), then P(X = k) = p qᵏ⁻¹, mean = 1 ÷ p, and variance = q ÷ p².

几何分布计算直到第一次成功所需的试验次数。如果 X ~ Geo(p)，则 P(X = k) = p qᵏ⁻¹，均值为 1 ÷ p，方差为 q ÷ p²。

The Poisson distribution models the number of independent events occurring in a fixed interval of time or space. If X ~ Po(λ), then P(X = k) = e⁻λ λᵏ ÷ k!, mean = λ, variance = λ.

泊松分布模拟固定时间或空间区间内独立事件的发生次数。如果 X ~ Po(λ)，则 P(X = k) = e⁻λ λᵏ ÷ k!，均值 = λ，方差 = λ。

Memory hook for Poisson: ‘Poisson has the same mean and variance, and λ is the average rate.’ Use Poisson when events occur independently at a constant average rate.

泊松分布速记：均值和方差相同，λ 是平均发生率。当事件独立且以恒定平均速率发生时使用泊松分布。

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8. Continuous Random Variables and Normal Distribution | 连续随机变量与正态分布

A continuous random variable takes values in an interval. Its probability density function (pdf) f(x) must be non-negative and integrate to 1 over the whole domain. Probability is the area under the curve.

连续随机变量在区间内取值。其概率密度函数 f(x) 必须非负，并且在整个定义域上积分为 1。概率等于曲线下的面积。

The cumulative distribution function (cdf) is F(x) = P(X ≤ x) = ∫ f(t) dt from the lower bound to x. It increases from 0 to 1.

累积分布函数为 F(x) = P(X ≤ x)，即从下界到 x 的 f(t) 积分。它从 0 递增到 1。

The normal distribution is a symmetric bell-shaped curve with mean μ and variance σ², written N(μ, σ²). To standardise, use z = (X − μ) ÷ σ.

正态分布是对称的钟形曲线，均值为 μ，方差为 σ²，记作 N(μ, σ²)。标准化公式为 z = (X − μ) ÷ σ。

z = (X − μ) ÷ σ

The standard normal distribution has mean 0 and variance 1. Use the normal table to find probabilities after converting to z. Memory hook: ‘z tells how many standard deviations X is from the mean.’

标准正态分布的均值为 0，方差为 1。转换为 z 后查正态分布表求概率。速记提示：z 表示 X 距离均值有多少个标准差。

For normal approximation to binomial, mean = np and variance = npq. For Poisson, mean = λ and variance = λ. Apply a continuity correction because a discrete distribution is approximated by a continuous one.

二项分布的正态近似中，均值 = np，方差 = npq。泊松分布中，均值 = λ，方差 = λ。因为是用连续分布近似离散分布，所以要应用连续性校正。

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9. Sampling and Estimation | 抽样与估计

A statistic is a random quantity calculated from a sample. An estimator is unbiased if its expected value equals the population parameter. The sample mean x̄ is an unbiased estimator of μ.

统计量是由样本计算出的随机量。如果估计量的期望值等于总体参数，则它是无偏的。样本均值 x̄ 是 μ 的无偏估计量。

The sampling distribution of the mean has mean μ and standard error σ ÷ √n. By the Central Limit Theorem, for large samples, x̄ is approximately normally distributed even if the population is not normal.

样本均值的抽样分布均值为 μ，标准误为 σ ÷ √n。根据中心极限定理，对于大样本，即使总体不是正态分布，x̄ 也近似服从正态分布。

Standard error = σ ÷ √n

A confidence interval for the mean is found by x̄ ± z × standard error. The z value is the critical value from the standard normal distribution for the chosen confidence level.

均值的置信区间由 x̄ ± z × 标准误计算。z 值是所选置信水平对应的标准正态分布临界值。

Memory hook: ‘The standard error shrinks as the sample size grows, because dividing by √n reduces random error.’ A larger sample gives a narrower confidence interval.

速记提示：标准误随样本量增大而减小，因为除以 √n 会降低随机误差。样本量越大，置信区间越窄。

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10. Hypothesis Testing | 假设检验

A hypothesis test begins with a null hypothesis H₀ and an alternative hypothesis H₁. H₀ is assumed true unless the evidence against it is strong enough. Memory hook: ‘H₀ is innocent until proven guilty.’

假设检验从原假设 H₀ 和备择假设 H₁ 开始。除非对 H₀ 不利的证据足够强，否则先假定 H₀ 为真。速记提示：H₀ 在证明有罪之前是无罪的。

The significance level α is the maximum probability of rejecting H₀ when it is true. A one-tailed test looks for a change in one direction, while a two-tailed test uses both tails.

显著性水平 α 是 H₀ 为真时拒绝 H₀ 的最大概率。单尾检验

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