Statistics: Fundamentals and Applications | 统计学基本原理与应用

📚 Statistics: Fundamentals and Applications | 统计学基本原理与应用

Statistics is the science of collecting, organising, analysing, interpreting and presenting data. It provides a framework for making decisions under uncertainty, from scientific research to everyday life.

统计学是收集、整理、分析、解释和呈现数据的科学。它为在不确定性下做出决策提供了框架，从科学研究到日常生活无处不在。

1. Population and Sample | 总体与样本

In statistics, a population is the complete set of all individuals or objects of interest, while a sample is a subset of the population selected for study. For example, if we want the average height of all students in a school, the population is every student; a sample might be 50 randomly chosen students.

在统计学中，总体是所关注的所有个体或对象的完整集合，而样本是从总体中选取用于研究的子集。例如，若要了解某校全体学生的平均身高，总体就是每一位学生；样本则可能是随机挑选的50名学生。

Census: collecting data from the entire population.

普查：从整个总体收集数据。
Sampling: collecting data from a part of the population, which is faster and cheaper than a census.

抽样：从总体的一部分收集数据，比普查更快、成本更低。
Random sampling helps reduce bias and makes the sample more representative.

随机抽样有助于减少偏差，使样本更具代表性。

2. Parameter vs. Statistic | 参数与统计量

A parameter is a numerical characteristic of a population, such as the population mean (μ). A statistic is a numerical characteristic of a sample, such as the sample mean (x̄). Since we rarely measure whole populations, we use statistics to estimate parameters.

参数是总体的数值特征，如总体均值 μ；统计量是样本的数值特征，如样本均值 x̄。由于我们很少能测量整个总体，因此用统计量来估计参数。

μ = population mean, x̄ = sample mean, σ² = population variance, s² = sample variance

字母希腊符号通常表示总体参数，拉丁字母加符号通常表示样本统计量，这是考试中重要的区分点。

3. Types of Data | 数据的类型

Data can be classified as categorical or numerical. Numerical data can be further split into discrete and continuous data.

数据可分为分类数据和数值数据；数值数据又可分为离散数据和连续数据。

Type / 类型	Example / 示例
Categorical / 分类	Colour, gender, blood type / 颜色、性别、血型
Discrete / 离散	Number of cars, count of students / 汽车数量、学生人数
Continuous / 连续	Height, time, temperature / 身高、时间、温度

Knowing data type determines which charts and statistical tests are appropriate.

了解数据类型有助于选择恰当的图表和统计检验方法。

4. Measures of Central Tendency | 集中趋势的度量

Central tendency describes the centre of a dataset. The three main measures are the mean, median and mode.

集中趋势描述数据集的中心位置，主要有三个度量：均值、中位数和众数。

Mean: the arithmetic average, calculated as the sum of values divided by the number of values.

均值：算术平均数，即各数值之和除以数值个数。
Median: the middle value when data are ordered; it is robust to outliers.

中位数：数据排序后位于中间的值；它对异常值不敏感。
Mode: the most frequently occurring value; suitable for categorical data.

众数：出现次数最多的值；适用于分类数据。

Mean x̄ = (∑x) / n

选择哪个度量取决于数据分布和是否存在极端值。

5. Measures of Dispersion | 离散程度的度量

Dispersion measures the spread of data. Common measures include range, interquartile range (IQR), variance and standard deviation.

离散程度度量数据的波动范围。常用指标包括极差、四分位距（IQR）、方差和标准差。

Range = maximum − minimum, but it is strongly affected by outliers.

极差 = 最大值 − 最小值，但受异常值影响很大。
Interquartile range = Q₃ − Q₁, describing the middle 50% of data.

四分位距 = Q₃ − Q₁，描述中间50%数据的范围。
Variance and standard deviation measure average squared deviation from the mean.

方差和标准差度量数据偏离均值的平均平方距离。

Sample variance s² = ∑(xᵢ − x̄)² / (n − 1)

使用 n−1 作为分母可以更准确地估计总体方差，这是重要考点。

6. Probability Basics | 概率基础

Probability quantifies uncertainty. The probability of an event A is P(A), with values from 0 to 1. The sum of probabilities of all possible outcomes in a sample space equals 1.

概率用于量化不确定性。事件A的概率记为 P(A)，取值在0到1之间；样本空间中所有可能结果的概率之和等于1。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

For mutually exclusive events, P(A ∩ B) = 0; for independent events, P(A ∩ B) = P(A) × P(B).

对于互斥事件，P(A ∩ B) = 0；对于独立事件，P(A ∩ B) = P(A) × P(B)。

7. Random Variables and Distributions | 随机变量与分布

A random variable assigns a numerical value to each outcome of a random experiment. It can be discrete or continuous. Its distribution describes the probabilities of its possible values.

随机变量将随机试验的每个结果赋予一个数值，可分为离散型和连续型。其分布描述了各可能取值的概率。

Discrete uniform distribution: all outcomes equally likely.

离散均匀分布：所有结果等可能。
Binomial distribution: number of successes in n independent trials with probability p.

二项分布：n次独立试验中成功次数，每次成功概率为p。
Normal distribution: symmetric bell-shaped curve described by mean μ and standard deviation σ.

正态分布：由均值μ和标准差σ描述的对称钟形曲线。

Binomial: P(X = k) = ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ

掌握分布类型是进行统计推断的基础。

8. Sampling Distribution and Central Limit Theorem | 抽样分布与中心极限定理

The sampling distribution of a statistic is the distribution of that statistic over many samples from the same population. The central limit theorem states that for a sufficiently large sample size (n ≥ 30), the sample mean x̄ will be approximately normally distributed, regardless of the population shape.

统计量的抽样分布是指从同一总体中反复抽样后该统计量的分布。中心极限定理指出：当样本量足够大（n ≥ 30）时，样本均值 x̄ 近似服从正态分布，无论总体分布形态如何。

x̄ ~ N(μ, σ²/n)

This theorem justifies using normal-based methods even when the population is not normal.

该定理使得即使总体不服从正态分布，我们仍能使用基于正态分布的方法进行推断。

9. Estimation and Confidence Intervals | 估计与置信区间

Point estimation gives a single value, such as using x̄ to estimate μ. Interval estimation provides a range of plausible values. A confidence interval is constructed with a confidence level, usually 95%.

点估计给出一个数值，例如用 x̄ 估计 μ；区间估计给出一个可能取值的范围。置信区间以一定的置信水平构建，通常为95%。

95% CI: x̄ ± 1.96 × σ/√n

If σ is unknown, use the sample standard deviation s and the t-distribution for small samples.

当σ未知且样本较小时，应使用样本标准差s和t分布。

10. Hypothesis Testing | 假设检验

Hypothesis testing is a formal procedure to decide between two claims: the null hypothesis (H₀) and the alternative hypothesis (H₁). We compute a test statistic and compare it to a critical value or p-value.

假设检验是在两个主张之间做出决定的正式程序：原假设 H₀ 与备择假设 H₁。我们计算检验统计量，并将其与临界值或p值比较。

H₀ usually states no effect or no difference; H₁ states the opposite.

H₀ 通常表示无效应或无差异；H₁ 则相反。
Type I error: rejecting H₀ when it is true (probability α).

第一类错误：H₀为真时拒绝它（概率为α）。
Type II error: failing to reject H₀ when it is false (probability β).

第二类错误：H₀为假时未拒绝它（概率为β）。

z = (x̄ − μ₀) / (σ/√n)

p值小于显著性水平（如0.05）时拒绝H₀。

11. Correlation and Regression | 相关与回归

Correlation measures the strength and direction of a linear relationship between two variables, summarised by Pearson’s correlation coefficient r (from −1 to 1). Regression finds the best-fit line for prediction.

相关性度量两个变量之间线性关系的强度和方向，常用皮尔逊相关系数 r 表示（取值-1到1）。回归则找出用于预测的最佳拟合线。

r value / r值	Interpretation / 解释
r = 1	Perfect positive linear correlation / 完全正线性相关
r = −1	Perfect negative linear correlation / 完全负线性相关
r ≈ 0	No linear correlation / 无线性相关

Regression line: y = a + bx, where b = r × (sᵧ/ₛₓ)

相关不代表因果，这一点在解释结果时极其重要。

12. Common Pitfalls in Statistics | 统计中的常见误区

Statistics can be misleading if applied incorrectly. Recognising common pitfalls is essential for critical thinking.

如果使用不当，统计可能误导人。识别常见误区对批判性思维至关重要。

Confusing correlation with causation: two variables may move together without one causing the other.

混淆相关与因果：两个变量可能共同变化，但并非一个导致另一个。
Bias in sampling: a non-random sample can make results unrepresentative.

抽样偏差：非随机样本会使结果失去代表性。
Ignoring outliers: extreme values can heavily distort the mean and range.

忽略异常值：极端值会严重扭曲均值和极差。
Misinterpreting p-values: a small p-value does not prove a large effect; it merely indicates evidence against H₀.

误解p值：小的p值并不证明效应很大，只是表示反对H₀的证据。

Careful design, proper sampling and correct interpretation are the pillars of valid statistical analysis.

精心设计、合理抽样和正确解读是有效统计分析的三大支柱。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Statistics: Fundamentals and Applications | 统计学基本原理与应用