📚 Common Statistical Methods and Their Typical Applications | 常用统计方法与典型应用
Statistics is the science of collecting, organising, analysing, and interpreting data. In A-level mathematics, understanding core statistical methods equips students with the tools to make informed decisions in real-world contexts, from scientific research to business forecasting.
统计学是收集、整理、分析和解释数据的科学。在 A-level 数学中，掌握核心统计方法能帮助学生掌握在现实情境中做出理性决策的工具，无论是科学研究还是商业预测。
1. Descriptive Statistics | 描述性统计
Descriptive statistics summarise and describe the main features of a dataset. They provide a clear picture of the data’s central tendency, spread, and shape without making inferences about a wider population.
描述性统计用于概括和描述数据集的主要特征。它们能清晰地呈现数据的集中趋势、离散程度和分布形态，而不对更大的总体进行推断。
- Key measures include mean, median, mode, range, variance, and standard deviation.
- 核心指标包括平均数、中位数、众数、极差、方差和标准差。
- Graphical tools such as histograms, box plots, and stem-and-leaf diagrams are also part of descriptive statistics.
- 直方图、箱线图和茎叶图等图形工具也属于描述性统计的范畴。
2. Measures of Central Tendency | 集中趋势度量
Central tendency identifies the centre of a dataset. The mean is the arithmetic average, the median is the middle value when data are ordered, and the mode is the most frequent value.
集中趋势用于确定数据集的中心位置。平均数（均值）是算术平均值，中位数是将数据排序后的中间值，众数是出现频率最高的数值。
Mean = Σx / n Median = middle value Mode = most frequent value
For grouped data, the mean can be estimated using mid-points of class intervals. The median is often preferred when outliers are present because it is not affected by extreme values.
对于分组数据，可以使用组中值来估算平均数。当数据中存在异常值时，中位数通常更受青睐，因为它不受极端值的影响。
| Measure | Best used when |
| Mean | Data are symmetric and no outliers |
| Median | Data are skewed or contain outliers |
| Mode | Data are categorical or need the most common value |
集中趋势度量选择原则：数据对称且无异常值时用平均数；数据偏斜或含异常值时用中位数；数据为分类数据或需要最常见值时用众数。
3. Measures of Dispersion | 离散程度度量
Dispersion describes how spread out the data are. The range is the difference between the maximum and minimum values, but it is sensitive to outliers.
离散程度描述数据的分散程度。极差是最大值与最小值之差，但它对异常值十分敏感。
The variance and standard deviation measure the average squared deviation from the mean. A larger standard deviation indicates greater variability.
方差和标准差衡量数据相对平均数的平均平方偏差。标准差越大，表示数据变异性越大。
Variance σ² = Σ(xᵢ − μ)² / N Standard deviation σ = √(σ²)
For sample data, the divisor becomes n − 1 to provide an unbiased estimate of the population variance. The interquartile range (IQR) is another robust measure, defined as Q₃ − Q₁.
对于样本数据，除数变为 n − 1，以提供总体方差的无偏估计。四分位距（IQR）是另一种稳健的度量，定义为 Q₃ − Q₁（上四分位数减下四分位数）。
4. Probability Concepts and Distributions | 概率概念与分布
Probability is the foundation of statistical inference. A probability distribution assigns probabilities to the possible outcomes of a random variable.
概率是统计推断的基础。概率分布将概率分配给随机变量的所有可能结果。
Common discrete distributions include the binomial distribution, which models the number of successes in n independent Bernoulli trials with constant probability p.
常见的离散分布包括二项分布，它用于建模 n 次独立伯努利试验中成功次数，其中每次成功概率 p 保持不变。
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ
The normal distribution is the most important continuous distribution. Its probability density function is symmetric about the mean μ, and its shape is controlled by the standard deviation σ. Standardising gives the Z-score:
正态分布是最重要的连续分布。其概率密度函数关于均值 μ 对称，形状由标准差 σ 决定。标准化得到 Z 分数：
Z = (X − μ) / σ
Z-scores allow us to find probabilities using standard normal tables and are widely used in hypothesis testing.
Z 分数使我们能够通过标准正态分布表查找概率，并广泛应用于假设检验中。
5. Sampling Methods | 抽样方法
In statistics, we often cannot study an entire population, so we select a sample. The way we sample determines how well the sample represents the population.
在统计学中，我们通常无法研究整个总体，因此需要选择样本。抽样方式决定了样本对总体的代表性。
- Simple random sampling: every member has an equal chance of selection; often done using random number tables or software.
- 简单随机抽样：每个成员被选中的概率相等；通常使用随机数表或软件实现。
- Stratified sampling: the population is divided into strata, and proportional samples are drawn from each.
- 分层抽样：将总体划分为若干层，从每层按比例抽取样本。
- Systematic sampling: select every k-th member after a random start.
- 系统抽样：随机起点后每隔 k 个成员抽取一个。
- Cluster sampling: divide the population into clusters, then randomly select whole clusters.
- 整群抽样：将总体划分为若干群，然后随机抽取整个群。
Stratified sampling ensures that minority groups are represented, while cluster sampling is cost-effective for large, geographically spread populations.
分层抽样能确保少数群体得到代表，而整群抽样对于地域分布广泛的大规模总体更为经济高效。
6. Hypothesis Testing | 假设检验
Hypothesis testing is a formal procedure for deciding whether a claim about a population parameter is supported by sample data. We start with a null hypothesis H₀ and an alternative hypothesis H₁.
假设检验是一种正式程序，用于判断关于总体参数的声明是否得到样本数据的支持。我们从零假设 H₀ 和备择假设 H₁ 开始。
The test statistic measures the distance between the sample result and the value claimed in H₀. If this statistic falls in the critical region, we reject H₀ at the chosen significance level α.
检验统计量衡量样本结果与 H₀ 声称值之间的距离。如果该统计量落在临界区域内，我们在选定的显著性水平 α 下拒绝 H₀。
P-value < α → reject H₀ P-value ≥ α → do not reject H₀
The significance level is the probability of making a Type I error, i.e. rejecting a true null hypothesis. A Type II error occurs when we fail to reject a false null hypothesis.
显著性水平是犯第一类错误的概率，即拒绝真实零假设的概率。第二类错误发生在未能拒绝错误的零假设时。
7. Correlation and Linear Regression | 相关与线性回归
Correlation measures the strength and direction of a linear relationship between two quantitative variables. Pearson’s product-moment correlation coefficient r ranges from −1 to +1.
相关分析衡量两个定量变量之间线性关系的强度和方向。皮尔逊积矩相关系数 r 的取值范围为 −1 到 +1。
- r = +1: perfect positive linear correlation
- r = +1：完全正线性相关
- r = −1: perfect negative linear correlation
- r = −1：完全负线性相关
- r = 0: no linear correlation
- r = 0：无线性相关
Regression analysis finds the line of best fit, usually by the least squares method. The regression line of y on x is written as y = a + bx, where b is the slope and a is the intercept.
回归分析通过最小二乘法找到最佳拟合线。y 对 x 的回归线写作 y = a + bx，其中 b 是斜率，a 是截距。
b = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)² a = ȳ − b·x̄
Correlation does not imply causation; a strong correlation may be due to a lurking variable or pure coincidence.
相关并不等于因果；强相关可能是由潜在变量或纯粹巧合造成的。
8. Chi-Square Test | 卡方检验
The chi-square (χ²) test is used for categorical data to assess how well observed frequencies fit expected frequencies. Two common applications are the goodness-of-fit test and the test for independence.
卡方（χ²）检验适用于分类数据，用于评估观测频数与期望频数的拟合程度。两种常见应用是拟合优度检验和独立性检验。
The test statistic is calculated as:
检验统计量的计算公式为：
χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ
where Oᵢ is the observed frequency and Eᵢ is the expected frequency under the null hypothesis. The degrees of freedom for a contingency table with r rows and c columns is (r − 1)(c − 1).
其中 Oᵢ 为观测频数，Eᵢ 为零假设下的期望频数。对于 r 行 c 列的列联表，自由度为 (r − 1)(c − 1)。
The test requires that expected frequencies are generally at least 5. If this condition is violated, categories may need to be combined.
该检验要求期望频数通常至少为 5。如果违反该条件，可能需要合并类别。
9. t-Test and Analysis of Variance | t 检验与方差分析
The t-test is used when the population standard deviation is unknown and the sample size is small. The one-sample t-test checks whether the sample mean differs significantly from a known value.
当总体标准差未知且样本量较小时，使用 t 检验。单样本 t 检验用于检验样本均值是否与已知值存在显著差异。
t = (x̄ − μ₀) / (s / √n)
An independent-samples t-test compares the means of two separate groups, while a paired t-test compares two related measurements on the same subjects.
独立样本 t 检验比较两个独立组的均值，而配对 t 检验比较同一组受试者的两次相关测量。
Analysis of Variance (ANOVA) extends the t-test to compare three or more group means simultaneously. It uses the F-statistic, which is the ratio of between-group variance to within-group variance.
方差分析（ANOVA）将 t 检验扩展到同时比较三个或更多组的均值。它使用 F 统计量，即组间方差与组内方差的比率。
F = MS(between) / MS(within)
A significant F-test indicates that at least one group mean differs, but a follow-up post-hoc test is needed to identify which groups differ.
显著的 F 检验表明至少有一个组均值存在差异，但需要进一步的事后检验来确定哪些组之间存在差异。
10. Typical Applications in Real Life | 现实生活中的典型应用
Statistical methods permeate almost every field. In medicine, hypothesis testing determines whether a new drug is more effective than a placebo.
统计方法渗透到几乎所有领域。在医学中，假设检验用于判断新药是否比安慰剂更有效。
In business, regression analysis forecasts sales based on advertising spend, and quality control uses control charts to monitor production processes.
在商业中，回归分析根据广告投入预测销售额，质量控制使用控制图监控生产过程。
In environmental science, chi-square tests analyse the association between pollution levels and health outcomes. In education, ANOVA compares test scores across different teaching methods.
在环境科学中，卡方检验分析污染水平与健康结果之间的关联。在教育领域，方差分析比较不同教学方法下的测试成绩。
Understanding the assumptions and limitations of each method is crucial. Proper data collection, random sampling, and awareness of bias ensure that conclusions drawn are reliable and meaningful.
理解每种方法的假设和局限性至关重要。正确的数据收集、随机抽样以及对偏倚的警惕能够确保得出的结论可靠且有意义。
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