📚 Year 12 CCEA Statistics: Key Terminology Quick Reference | Year 12 CCEA 统计：词汇术语速记指南
Mastering the language of statistics is half the battle. This quick reference unpacks the essential terms you will meet in Year 12 CCEA Statistics, from data description to hypothesis testing, with each concept explained in a crisp English paragraph immediately followed by a clear Chinese translation. Use it to build your confidence, revise efficiently, and secure those marks on exam day.
掌握统计学的语言是成功的一半。这份速记指南梳理了 Year 12 CCEA 统计课程中你会遇到的关键术语，从数据描述到假设检验，每个概念先用一段简洁的英文解释，紧接着给出清晰的中文翻译。用它来建立信心、高效复习，并在考试当天稳稳拿下分数。
1. Population, Sample, and Parameters | 总体、样本与参数
A population is the complete set of all items or individuals that we wish to study. A sample is a carefully chosen subset of the population, taken to make conclusions about the whole without having to examine every member.
总体是我们想要研究的全部对象或个体的完整集合。样本是从总体中仔细挑选出的子集，用来推断整个总体，而无需检查每一个个体。
A parameter is a fixed numerical measure that describes a characteristic of a population, such as the population mean μ or population standard deviation σ. A statistic is a numerical measure calculated from sample data, like the sample mean x̄, and is used to estimate the corresponding parameter.
参数是描述总体某一特征的固定数值度量，例如总体均值 μ 或总体标准差 σ。统计量是从样本数据中计算出的数值度量，如样本均值 x̄，用来估计相应的参数。
A census attempts to collect data from every member of the population, while a survey collects data only from a sample. Censuses can be time‑consuming and expensive, so samples are used far more often in practice.
普查试图从总体的每一个成员那里收集数据，而调查只从样本中收集数据。普查可能耗时且昂贵，因此在实践中样本的使用要普遍得多。
2. Measures of Central Tendency | 集中趋势度量
The mean, or average, is found by summing all data values and dividing by the number of values. It is the balance point of the dataset but can be heavily influenced by outliers. For grouped data, we use the midpoint of each class to estimate the mean.
均值是将所有数据值相加再除以数据个数得到的平均数。它是数据集的平衡点，但容易受到异常值的影响。对于分组数据，我们使用每个区间的组中点来估算均值。
The median is the middle value when the data are arranged in order. For an odd‑sized list it is the centre value; for an even‑sized list it is the mean of the two middle values. The median is robust to extreme values, making it useful for skewed distributions.
中位数是将数据按顺序排列后位于中间位置的值。如果数据个数为奇数，它就是最中间的那个值；如果为偶数，则是中间两个值的均值。中位数对极端值不敏感，因此在偏斜分布中格外有用。
The mode is the value that occurs most frequently. A dataset may have one mode (unimodal), two modes (bimodal), or several. For grouped data, the modal class is the interval with the highest frequency, and we can estimate the mode within it.
众数是出现频率最高的数值。一个数据集可以有一个众数（单峰）、两个众数（双峰）或多个众数。对于分组数据，众数所在的组是频数最高的区间，我们可以在此区间内估算众数的具体位置。
3. Measures of Dispersion | 离散程度度量
The range is the difference between the largest and smallest values in a dataset. It gives a quick sense of spread but completely ignores the middle 50% of the data and is highly sensitive to extreme values.
极差是数据集中最大值与最小值之差。它能快速反映数据的分布宽度，但完全忽略了中间 50% 的数据，并且对极端值非常敏感。
The interquartile range (IQR) is the difference between the upper quartile Q₃ and the lower quartile Q₁. It measures the spread of the central 50% of the data and is not affected by outliers, making it a more resilient measure of dispersion than the range.
四分位距（IQR）是上四分位数 Q₃ 与下四分位数 Q₁ 之差。它衡量了中间 50% 数据的离散程度，且不受异常值影响，因此是比极差更稳健的离散度量。
Variance σ² (population) or s² (sample) is the average of the squared deviations from the mean. Standard deviation, the square root of variance, brings the measure back to the original units. The sample variance uses n‑1 in the denominator to give an unbiased estimate of the population variance.
方差 σ²（总体）或 s²（样本）是各数据与均值离差平方的平均数。标准差是方差的平方根，将度量转回原始单位。样本方差的分母使用 n‑1，以便提供总体方差的无偏估计。
deviation using the formula √[ Σf x² / Σf − ( Σf x / Σf )² ]. The term Σf x² means each x‑value is squared, multiplied by its frequency, and then summed.
在处理频数表时，我们使用公式 √[ Σf x² / Σf − ( Σf x / Σf )² ] 来计算标准差。这里的 Σf x² 表示每个 x 值先平方，乘以其频数，然后求和。
4. Probability Basics | 概率基础
Probability measures the chance that an event will occur, expressed as a number between 0 and 1. An experiment is a repeatable process that gives rise to a set of outcomes, and an event is any collection of outcomes from that experiment.
概率衡量某个事件发生的可能性，用 0 到 1 之间的一个数表示。试验是一个可重复的过程，会产生一系列结果，而事件则是该试验结果的任意一个集合。
n(S), where n(A) is the number of outcomes in A and n(S) is the total number of outcomes.
样本空间是试验所有可能结果的集合。如果所有结果等可能发生，那么事件 A 的概率 P(A) = n(A) / n(S)，其中 n(A) 为 A 包含的结果数，n(S) 为总结果数。
clusive if they cannot happen at the same time. For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B). The complement of an event A (written A' or Aᶜ) is the event that A does not occur, and P(A') = 1 − P(A).
如果两个事件不可能同时发生，则称它们互斥。对于互斥事件 A 和 B，P(A ∪ B) = P(A) + P(B)。事件 A 的补事件（记为 A' 或 Aᶜ）表示 A 不发生，且 P(A') = 1 − P(A)。
5. Independent and Conditional Events | 独立事件与条件概率
ccurrence of one does not affect the probability of the other. For independent events, P(A ∩ A) × P(B). Independence is a crucial assumption that must be checked, not assumed lightly.
如果事件 A 的发生不影响事件 B 的发生概率，则它们独立。对于独立事件，P(A ∩ B) = P(A) × P(B)。独立性是一个关键假设，必须仔细检验，不能随意假定。
Conditional probability, P(A| s the probability that A occurs given that B has already occurred. It is calculated as P(A ∩ B) / P(B), provided P(B) > 0. Rearranging gives the multiplication rule P(A ∩ B) = P(B) × P(A|B).
条件概率 P(A|B) 表示在已知 B 已经发生的情况下 A 发生的概率。它的计算公式是 P(A ∩ B) / P(B)，前提是 P(B) > 0。移项后即可得到乘法法则 P(A ∩ B) = P(B) × P(A|B)。
Probability tree diagrams are powerful visual aids for sequential events. On each branch we write the conditional probability, and the probability of a complete path is the product of the probabilities along that path.
概率树图是处理序贯事件的强大可视化工具。我们在每条分支上写出条件概率，而一条完整路径的概率就是该路径上各分支概率的乘积。
6. Discrete Random Variables | 离散随机变量
A random variable is a variable whose numerical value is determined by the outcome of an experiment. A discrete random variable can only take a countable number of distinct values, such as the score on a die or the number of defective items in a sample.
随机变量是其数值取决于试验结果的变量。离散随机变量只能取有限个或可数无限个明确的数值，例如骰子的点数或样本中次品的件数。
sum of all probabilities must equal 1, and each probability must satisfy 0 ≤ P(X = x) ≤ 1.
离散随机变量 X 的概率分布列出了每一个可能的取值 x 及其对应的概率 P(X = x)。所有概率之和必须等于 1，且每一个概率必须满足 0 ≤ P(X = x) ≤ 1。
The expected value, E(X) or μ, is the long‑ bility: E(X) = Σ [x · P(X = x)]. The variance Var(X) = E[(X − μ)²] = Σ [x² P(X = x)] − μ².
期望值 E(X) 或称 μ，是随机变量 X 的长期平均值，通过将每个值乘以其概率再求和得到：E(X) = Σ [x · P(X = x)]。方差 Var(X) = E[(X − μ)²] = Σ [x² P(X = x)] − μ²。
7. The Binomial Distribution | 二项分布
A binomial setting arises when we have a fixed number of independent trials, n, each with two possible outcomes (success or failure), and the probability of success, p, is constant. The random variable X counts the number of successes in n trials.
当满足以下条件时，我们就遇到二项分布：固定次数的独立试验 n，每次试验只有两种可能结果（成功或失败），且成功概率 p 保持不变。随机变量 X 计算 n 次试验中成功的次数。
The probability of exactly r successes is given by P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, where ⁿCᵣ = n! / [r!(n − r)! efficient. The notation B(n, p) indicates a binomial distribution with parameters n and p.
恰好 r 次成功的概率由 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ 给出，其中 ⁿCᵣ = n! / [r!(n − r)!] 是二项式系数。记号 B(n, p) 表示参数为 n 和 p 的二项分布。
s Var(X) = np(1 − p). These formulas only apply when the four binomial conditions are met.
二项分布的均值为 E(X) = np，方差为 Var(X) = np(1 − p)。这些公式只有在四个二项条件都满足时才成立。
8. The Normal Distribution | 正态分布
A continuous random variable can take any value in an interval. The normal distribution is a symmetric, bell‑shaped continuous distribution defined by two parameters: the mean μ and the standard deviation σ. We write X ~ N(μ, σ²).
连续随机变量可以在一个区间内取任意值。正态分布是一种对称的钟形连续分布，由两个参数定义：均值 μ 和标准差 σ。我们记作 X ~ N(μ, σ²)。
The standard normal distribution Z ~ s allows us to use standard normal tables to find probabilities for any normal distribution.
标准正态分布 Z ~ N(0, 1²) 的均值为 0，标准差为 1。任何正态变量 X 都可以通过 z = (x − μ) / σ 转换为标准正态变量 Z。这使得我们可以使用标准正态表来查找任意正态分布的概率。
The total area under the normal curve is 1. Probabilities are found as areas: P(a < X < b) is the area under the curve between a and b. Because the distribution is continuous, endpoints do not affect the area.
正态曲线下的总面积等于 1。概率即为面积：P(a < X < b) 是曲线下方 a 与 b 之间的面积。因为分布是连续的，端点不影响面积大小。
9. Correlation and Regression | 相关与回归
Correlation measures the strength and direction of a linear relationship between two variables. The product‑moment correlation coefficient, r, always lies between −1 and +1. Values close to +1 indicate strong positive correlation, values close to −1 indicate strong negative correlation, and values near 0 suggest weak or no linear correlation.
相关度量了两个变量之间线性关系的强度和方向。积矩相关系数 r 始终介于 −1 与 +1 之间。数值接近 +1 表示强正相关，接近 −1 表示强负相关，接近 0 则表示线性相关较弱或无线性相关。
Spearman's rank correlation coefficient, rₛ, is used when data are ranks rather than precise measurements or when the relationship is monotonic but not necessarily linear. It evaluates how well the relationship can be described by a monotonic function.
斯皮尔曼秩相关系数 rₛ 用于数据是以秩次出现而非精确测量值时，或者当关系单调但不一定是线性时。它评估的是变量间关系能够被一个单调函数描述的程度。
Least‑ minimises the sum of squared residuals. The gradient b = Sₓₚ / Sₓₓ and the intercept a = ȳ − b x̄ be used to predict y for a given x, but extrapolation beyond the data range is unreliable.
最小二乘法回归是找出最佳拟合直线 y = a + bx，这条直线使残差平方和达到最小。斜率 b = Sₓₚ / Sₓₓ，截距 a = ȳ − b x̄。回归直线可用于由给定的 x 预测 y，但超出数据范围的外推是不可靠的。
10. Index Numbers | 指数
onally set to 100. A simple price index for a single item is (current price / base price) × 100.
指数衡量某个变量（如价格或数量）相对于基期的变化。基期指数通常被设定为 100。单项商品的简单价格指数为（当前价格 / 基期价格）× 100。
A weighted aggregate index combines the changes of several items, giving more weight to those of greater importance. The Laspeyres index uses base‑period quantities as weights, while the Paasche index uses current‑period quantities.
加权综合指数将多种商品的变化汇总起来，并对较重要的商品赋予更大的权重。拉氏指数使用基期数量作为权重，而帕氏指数使用当期数量作为权重。
The Retail Prices Index (RPI) and Consumer Prices Index (CPI) are real‑world examples of weighted price indices used to track inflation. They differ in the items included and the way the weights are calculated.
零售价格指数（RPI）和消费价格指数（CPI）是用于跟踪通胀的加权价格指数的现实例子。它们在所涵盖的商品项目以及权重计算方式上有所不同。
11. Time Series and Moving Averages | 时间序列与移动平均
A time series is a sequence of data points recorded at successive points in time, usually at regular intervals (e.g., monthly sales). Its components include trend, seasonal variation, cyclical fluctuation, and random (irregular) variation.
时间序列是按时间顺序记录的一串数据点，通常以固定间隔采集（如月度销售额）。它的组成要素包括趋势、季节变动、周期波动和随机（不规则）变动。
A moving average smooths out short‑term fluctuations to reveal the underlying trend. For n‑point moving averages, we average each set of n consecutive values and centre them where necessary. A 4‑point moving average often needs to be centred to align with time points.
移动平均能够抚平短期波动，从而揭示背后的趋势。对 n 点移动平均，我们每次取连续的 n 个值求平均，并在必要时进行中心化。四点移动平均通常需要中心化，以使其与时间点对齐。
Seasonal variation refers to regular patterns that repeat over a specific period, such as higher ice‑cream sales in summer. Seasonal indices can be calculated by dividing each time‑series value by its centred trend value and then averaging out the random noise.
季节变动是指在一个特定周期内有规律地重复出现的模式，例如夏季冰淇淋销量更高。季节指数可以通过将每个时间序列值除以其中心化趋势值，然后剔除随机噪音后平均得到。
12. Hypothesis Testing Terminology | 假设检验术语
A hypothesis test is a formal procedure for deciding between two competing claims about a population parameter. The null hypothesis H₀ usually represents the status quo and contains an equals sign. The alternative hypothesis H₁ (or Hₐ) represents what we hope to find evidence for.
假设检验是一种在关于总体参数的两种对立说法中做出选择的正式程序。原假设 H₀ 通常代表现状，并包含等号。备择假设 H₁（或 Hₐ）代表我们希望找到证据支持的说法。
The significance level α (commonly 5% or 1%) is the probability of rejecting H₀ when it is actually true. This is a Type I error. A Type II error occurs when we fail to reject a false null hypothesis. The power of a test is 1 − P(Type II error).
显著性水平 α（通常为 5% 或 1%）是原假设为真时却拒绝它的概率，称为 I 类错误。II 类错误发生在原假设为假时我们未能拒绝它的情况下。检验的功效等于 1 − P(II 类错误)。
The test statistic is a value calculated from sample data and compared with a critical value or used to find the p‑value. The p‑value is the probability, under H₀, of obtaining a result at least as extreme as the observed one; a small p‑value provides evidence against H₀.
检验统计量是由样本数据计算出的一个数值，用来与临界值比较或用于求 p 值。p 值是在 H₀ 成立的前提下，得到至少与观测结果同样极端的结果的概率；较小的 p 值提供了不利于原假设的证据。
l test, we calculate exact probabilities using the binomial distribution. We compare the p‑ ormal approximation, we may use the continuity correction to improve the accuracy of the test.
对于二项检验，我们使用二项分布计算精确的概率。我们将 p 值与 α 对比，若 p ≤ α 则拒绝 H₀。在使用正态近似时，我们可以采用连续性校正来提高检验的准确性。
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