📚 Pre-U CIE Statistics: Core Knowledge Review | Pre-U CIE 统计：核心知识点梳理
Pre-U CIE Statistics provides a rigorous foundation in data analysis, probability, and statistical inference, preparing students for university-level work in the sciences, social sciences, and beyond. This article walks through the essential topics, clarifying key definitions, formulas, and the logical connections that tie the subject together.
Pre-U CIE 统计为学生在数据分析、概率和统计推断方面提供了扎实的基础，为大学阶段科学、社会科学等领域的学习做好准备。本文梳理核心知识点，厘清重要定义、公式以及各主题之间的逻辑联系。
1. Descriptive Statistics and Data Representation | 描述统计与数据展示
The mean, often denoted by x̄ for a sample and μ for a population, is the arithmetic average of the data. It is sensitive to extreme values.
平均数常以 x̄ 表示样本均值、μ 表示总体均值，是数据的算术平均值，对极端值敏感。
The median is the middle value when data are ordered; the mode is the most frequent value. For skewed distributions, the median is a more reliable measure of central tendency than the mean.
中位数是数据排序后中间的值；众数是出现频率最高的值。在偏态分布中，中位数的代表性优于均值。
Measures of dispersion include the range, interquartile range (IQR = Q3 − Q1), variance (s² or σ²) and standard deviation (s or σ). The standard deviation describes the average distance of observations from the mean.
离散程度的度量包括极差、四分位距（IQR = Q₃ − Q₁）、方差（s² 或 σ²）和标准差（s 或 σ）。标准差反映观测值偏离均值的平均距离。
Box plots and histograms display the shape, centre and spread of a data set. Cumulative frequency curves allow quick estimation of percentiles and the median.
箱线图和直方图展示数据的形状、中心和散布。累积频率曲线便于快速估计百分位数和中位数。
2. Fundamentals of Probability | 概率论基础
Probability is a measure on a scale from 0 to 1. For any event A, 0 ≤ P(A) ≤ 1, and the sum of probabilities of all mutually exclusive outcomes in a sample space equals 1.
概率是 0 到 1 之间的度量。对任意事件 A，0 ≤ P(A) ≤ 1，且样本空间中所有互斥结果的概率之和为 1。
The addition rule for mutually exclusive events is P(A ∪ B) = P(A) + P(B). When events are not mutually exclusive, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
互斥事件的加法法则为 P(A ∪ B) = P(A) + P(B)。当事件不互斥时，P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
Conditional probability is given by P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Events A and B are independent if P(A ∩ B) = P(A) × P(B), or equivalently P(A|B) = P(A).
条件概率定义为 P(A|B) = P(A ∩ B) / P(B)，P(B) > 0。若 P(A ∩ B) = P(A) × P(B) 或等价地 P(A|B) = P(A)，则事件 A 与 B 独立。
Bayes’ theorem links conditional probabilities: P(A|B) = [P(B|A) × P(A)] / P(B). Tree diagrams are useful for structuring multi-stage probability problems.
贝叶斯定理将条件概率关联起来：P(A|B) = [P(B|A) × P(A)] / P(B)。树状图有助于梳理多阶段概率问题。
3. Discrete Random Variables and Distributions | 离散随机变量与分布
A discrete random variable X takes countable values. Its probability distribution is described by P(X = x). The expected value E(X) = Σ x · P(X = x) represents the long-run average.
离散随机变量 X 取可数个值，概率分布用 P(X = x) 描述。期望 E(X) = Σ x · P(X = x) 表示长期平均。
The variance Var(X) = E[(X − μ)²] = E(X²) − [E(X)]², and the standard deviation is the square root of the variance.
方差 Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²，标准差为其平方根。
The binomial distribution B(n, p) models the number of successes in n independent trials each with success probability p. Its probability mass function is P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ, with mean np and variance np(1−p).
二项分布 B(n, p) 模拟 n 次独立试验中成功的次数，每次成功概率为 p。概率质量函数为 P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ，均值 np，方差 np(1−p)。
The Poisson distribution Po(λ) describes the number of events occurring in a fixed interval when events happen independently at a constant average rate λ. P(X = r) = (e⁻λ λʳ) / r!, with mean and variance both equal to λ.
泊松分布 Po(λ) 描述固定区间内独立事件发生的次数，平均发生率为 λ。P(X = r) = (e⁻λ λʳ) / r!，均值和方差均为 λ。
The geometric distribution Geo(p) models the number of trials up to and including the first success. P(X = r) = (1−p)ʳ⁻¹ p, with mean 1/p.
几何分布 Geo(p) 模拟直至首次成功所需的试验次数。P(X = r) = (1−p)ʳ⁻¹ p，均值为 1/p。
4. Continuous Random Variables and the Normal Distribution | 连续随机变量与正态分布
Continuous random variables are described by a probability density function f(x). The probability over an interval is the area under the curve: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.
连续随机变量由概率密度函数 f(x) 描述，区间上的概率为曲线下面积：P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx。
The normal distribution N(μ, σ²) is the most important continuous distribution. Its density curve is symmetric and bell-shaped. A standard normal variable Z = (X − μ) / σ follows N(0,1).
正态分布 N(μ, σ²) 是最重要的连续分布，密度曲线对称呈钟形。标准化变量 Z = (X − μ) / σ 服从 N(0,1)。
Probabilities for normal variables are found using tables or technology. The 68–95–99.7 rule states that approximately 68% of data lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.
正态变量的概率使用表格或技术求得。68–95–99.7 规则表明约 68% 的数据落在 μ ± 1σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。
Normal approximations to binomial or Poisson distributions are valid when conditions are met (e.g. np ≥ 5 and n(1−p) ≥ 5 for binomial; λ > 10 for Poisson). A continuity correction is applied.
在满足条件时可将二项或泊松分布近似为正态（如二项要求 np ≥ 5 且 n(1−p) ≥ 5；泊松要求 λ > 10），并需进行连续性校正。
5. Sampling and the Central Limit Theorem | 抽样与中心极限定理
A sample statistic is a random variable whose distribution is called the sampling distribution. The sample mean x̄ is an unbiased estimator of the population mean μ, with standard error σ/√n.
样本统计量是随机变量，其分布称为抽样分布。样本均值 x̄ 是总体均值 μ 的无偏估计，标准误为 σ/√n。
The Central Limit Theorem (CLT) states that, for sufficiently large sample size n (typically n ≥ 30), the sampling distribution of x̄ is approximately normal, regardless of the shape of the population distribution. This underpins most inference procedures.
中心极限定理指出，当样本量 n 足够大（通常 n ≥ 30）时，无论总体分布形状如何，x̄ 的抽样分布都近似正态。这是大多数推断程序的基础。
CLT: x̄ ~ N(μ, σ² / n) approximately, for large n.
中心极限定理：大样本下 x̄ 近似服从 N(μ, σ² / n)。
For the sample proportion p̂, the sampling distribution under repeated sampling is approximately N( p, p(1−p) / n ) when both np ≥ 5 and n(1−p) ≥ 5.
对于样本比例 p̂，在 np ≥ 5 且 n(1−p) ≥ 5 时，重复抽样下其抽样分布近似为 N( p, p(1−p) / n )。
6. Parameter Estimation and Confidence Intervals | 参数估计与置信区间
A point estimate is a single value used to estimate a population parameter. An interval estimate gives a range of plausible values, accompanied by a confidence level, such as 95%.
点估计是用单个数值估计总体参数。区间估计给出一个可信的数值范围，并附有置信水平，如 95%。
A confidence interval for the population mean μ, when σ is known, is x̄ ± z* × (σ/√n), where z* is the critical value from N(0,1). When σ is unknown, we use the t-distribution with n−1 degrees of freedom: x̄ ± t* × (s/√n).
当 σ 已知时，总体均值 μ 的置信区间为 x̄ ± z* × (σ/√n)，z* 为 N(0,1) 的临界值。当 σ 未知时，使用自由度为 n−1 的 t 分布：x̄ ± t* × (s/√n)。
A confidence interval for a population proportion p is p̂ ± z* × √[ p̂(1−p̂) / n ]. The margin of error decreases as the sample size increases.
总体比例 p 的置信区间为 p̂ ± z* × √[ p̂(1−p̂) / n ]。误差幅度随样本量增大而减小。
The formula for the minimum sample size required to achieve a given margin of error E for a mean is n ≥ (z*σ / E)²; for a proportion, use n ≥ (z*² p̂(1−p̂)) / E² with a conservative p̂ = 0.5 if no prior estimate is available.
要达到指定的均值误差幅度 E，所需最小样本量为 n ≥ (z*σ / E)²；对于比例，使用 n ≥ (z*² p̂(1−p̂)) / E²，若无可用的先验估计则取保守值 p̂ = 0.5。
7. Hypothesis Testing: Means and Proportions | 假设检验：均值和比例
A hypothesis test assesses whether sample data provide enough evidence to reject a null hypothesis H₀ in favour of an alternative hypothesis H₁. The significance level α (often 0.05) is the probability of a Type I error (rejecting H₀ when it is true).
假设检验评估样本数据是否提供足够的证据拒绝原假设 H₀ 而支持备择假设 H₁。显著性水平 α（常取 0.05）是第一类错误（H₀ 为真时拒绝 H₀）的概率。
The p-value is 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.
p 值是在 H₀ 为真的前提下，得到至少与观测值同样极端的检验统计量的概率。若 p 值 ≤ α，则拒绝 H₀。
For a test about a single mean when σ is known, use the z-test: z = (x̄ − μ₀) / (σ/√n). When σ is unknown, apply the one-sample t-test: t = (x̄ − μ₀) / (s/√n), with df = n−1.
σ 已知时单样本均值检验用 z 检验：z = (x̄ − μ₀) / (σ/√n)。σ 未知时用单样本 t 检验：t = (x̄ − μ₀) / (s/√n)，自由度 df = n−1。
Two-sample t-tests compare the means of two independent groups. The pooled t-test assumes equal population variances; the unpooled version (Welch’s t-test) does not. For paired data, use the paired t-test based on the differences.
双样本 t 检验比较两组独立样本的均值。合并方差 t 检验假设总体方差相等；不合并方差版本（Welch t 检验）无需此假设。配对数据则使用基于差值的配对 t 检验。
Hypothesis tests for a single proportion use z = (p̂ − p₀) / √[ p₀(1−p₀) / n ]. Comparisons of two proportions also use a normal approximation, with the pooled estimate of p under H₀.
单样本比例检验使用 z = (p̂ − p₀) / √[ p₀(1−p₀) / n ]。双样本比例比较同样使用正态近似，并在 H₀ 下使用合并比例估计。
8. Chi-Squared Tests | 卡方检验
The chi-squared (χ²) test is used for categorical data. The goodness-of-fit test compares observed frequencies with expected frequencies under a specified model. The test statistic is χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ.
卡方（χ²）检验用于分类数据。拟合优度检验比较观测频数与特定模型下的期望频数。检验统计量为 χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ。
Degrees of freedom for a goodness-of-fit test are (number of categories − 1 − number of estimated parameters). The test requires all expected frequencies to be at least 5; categories may need to be combined.
拟合优度检验的自由度为（类别数 − 1 − 估计参数的个数）。检验要求所有期望频数不低于 5，必要时需合并类别。
The chi-squared test for independence assesses whether two categorical variables are associated in a contingency table. Expected frequencies are calculated under the assumption of independence: E = (row total × column total) / grand total. Degrees of freedom = (rows − 1) × (columns − 1).
卡方独立性检验评估列联表中两个分类变量是否有关联。期望频数在独立假设下计算：E = （行合计 × 列合计）/ 总计。自由度 = （行数 − 1）×（列数 − 1）。
9. Correlation and Regression Analysis | 相关与回归分析
The Pearson product-moment correlation coefficient r measures the strength and direction of a linear relationship between two numerical variables. It ranges from −1 (perfect negative) to +1 (perfect positive).
Pearson 积矩相关系数 r 衡量两个数值变量之间线性关系的强度和方向，取值范围从 −1（完全负相关）到 +1（完全正相关）。
The formula is r = S_xy / √(S_xx S_yy), where S_xy = Σ(x − x̄)(y − ȳ), S_xx = Σ(x − x̄)² and S_yy = Σ(y − ȳ)². Spearman’s rank correlation coefficient r_s is used when the relationship is monotonic but not necessarily linear, or when outliers are present.
计算公式为 r = S_xy / √(S_xx S_yy)，其中 S_xy = Σ(x − x̄)(y − ȳ)，S_xx = Σ(x − x̄)²，S_yy = Σ(y − ȳ)²。Spearman 等级相关系数 r_s 用于单调但不一定是线性的关系，或存在异常值时。
A simple linear regression model has the form y = a + bx, where b = S_xy / S_xx and a = ȳ − b x̄. The least squares method minimises the sum of squared residuals Σ(yᵢ − ŷᵢ)².
简单线性回归模型的形式为 y = a + bx，其中 b = S_xy / S_xx，a = ȳ − b x̄。最小二乘法使残差平方和 Σ(yᵢ − ŷᵢ)² 最小化。
The coefficient of determination R² = 1 − (SS_residual / SS_total) indicates the proportion of variation in y explained by the linear model. Residual plots help check assumptions of linearity, constant variance and independence.
判定系数 R² = 1 − (残差平方和 / 总平方和) 表示线性模型解释的 y 变异比例。残差图有助于检查线性、等方差和独立性假设。
10. Non-parametric Tests | 非参数检验
Non-parametric tests do not assume a specific population distribution. They are useful when the normality assumption is violated or when data are ordinal. The Sign test is a simple test for the median of a single sample, based on the number of positive and negative signs relative to a hypothesised median.
非参数检验不假设特定的总体分布，当正态性假设不成立或数据为顺序尺度时很有用。符号检验是基于观测值与假设中位数的正负符号个数对单样本中位数进行的一种简单检验。
The Wilcoxon signed-rank test uses both the sign and the magnitude of differences, making it more powerful than the sign test for symmetric distributions. It is applied to paired data or a single sample median test.
Wilcoxon 符号秩检验同时利用差值的符号和大小，对对称分布比符号检验更有效。可用于配对数据或单样本中位数检验。
The Mann–Whitney U test (or Wilcoxon rank-sum test) compares two independent samples by ranking all observations together. It tests whether the two populations have the same distribution or whether one tends to produce larger values than the other.
Mann–Whitney U 检验（或 Wilcoxon 秩和检验）通过对所有观测值统一排序来比较两个独立样本，检验两个总体分布是否相同或其中一组倾向于取更大的值。
These tests are robust against outliers and do not require equal variances, but they test for differences in medians or overall distributions rather than means.
这些检验对异常值稳健，且不要求方差相等，但它们检验的是中位数或整体分布的差异，而非均值。
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