📚 AS Eduqas Statistics: Core Concepts Summary | AS Eduqas 统计：核心知识点梳理
Mastering the core concepts in AS Statistics is essential for tackling the Eduqas exam with confidence. This article summarises the key topics: statistical sampling, data presentation, measures of location and spread, probability, discrete random variables, the binomial distribution, and hypothesis testing. Each section is paired with concise Chinese explanations to support bilingual learning.
掌握 AS 统计学核心概念是自信应对 Eduqas 考试的关键。本文梳理重点主题：统计抽样、数据表示、位置与离散度量、概率、离散随机变量、二项分布和假设检验。每部分均配有简明中文解释，助力双语学习。
1. Statistical Sampling | 统计抽样
A population is the whole set of items of interest. A sample is a subset of the population, used to draw conclusions without surveying everyone. The sampling frame is a list of all population members. The goal is a representative sample with minimal bias.
总体是所关心的全部对象的集合。样本是总体的一个子集，无需普查即可推断结论。抽样框是所有总体成员的名单。目标是获得具有代表性的样本并使偏差最小化。
Simple random sampling ensures every member has an equal probability of selection, often using random number generators. It avoids selection bias but requires a full sampling frame and may be costly.
简单随机抽样使每个成员被选中的概率相等，通常借助随机数生成器。它避免了选择偏差，但需要完整的抽样框，且成本可能较高。
Systematic sampling selects members at regular intervals from an ordered list (e.g., every 10th person). It is quick and simple but can introduce bias if a hidden pattern coincides with the interval.
系统抽样从有序名单中按固定间隔选取成员（如每隔 9 人取第 10 人）。该方法快速简便，但若隐藏模式与间隔重合，则可能引入偏差。
Stratified sampling divides the population into strata (e.g., age groups) and randomly samples proportionally from each stratum. It guarantees representation of all groups and improves precision.
分层抽样将总体划分为若干层（如年龄组），按比例从各层随机抽样。它能保证各群体均被覆盖，并提高估计精度。
Opportunity sampling selects individuals who are easily accessible, such as people passing by. It is convenient and cheap but highly prone to selection bias and lacks generalisability.
机会抽样选取最容易接触到的个体，如路过的行人。这种方法方便且成本低，但极易产生选择偏差，泛化能力差。
Quota sampling attempts to mirror population proportions by filling quotas for certain characteristics without random selection. It can reduce some bias but interviewer bias may still distort results.
配额抽样试图通过按特征配额填充样本来反映总体结构，但不进行随机选择。它能减少部分偏差，但调查员偏见仍可能扭曲结果。
Bias in sampling can arise from a poor sampling frame, non-response, leading questions, or convenience choices. Always link the sampling method to potential sources of bias in exam answers.
偏差可能源于有缺陷的抽样框、无响应、诱导性问题或便利性选择。考试答题时，务必将抽样方法与其潜在偏差来源联系起来。
2. Data Presentation & Interpretation | 数据表示与解释
Box plots (box-and-whisker diagrams) display the five-number summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. They reveal central tendency, spread, and skewness. Outliers are usually defined as values below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR and are marked with separate points.
箱线图（盒须图）展示五数概括：最小值、下四分位数(Q₁)、中位数(Q₂)、上四分位数(Q₃)和最大值。它可揭示集中趋势、离散程度和偏态。离群值通常定义为小于 Q₁ − 1.5×IQR 或大于 Q₃ + 1.5×IQR 的数据点，并以独立标记画出。
Stem-and-leaf diagrams preserve original data while showing distribution shape. A key explains place values. They make it easy to find medians and quartiles and to identify overall pattern and gaps.
茎叶图在展示分布形状的同时保留了原始数据。图例说明数位。借助茎叶图可轻松找到中位数和四分位数，并识别整体模式和缺口。
Histograms display grouped continuous data. The area of each bar is proportional to frequency. When class widths are unequal, plot frequency density = frequency / class width. Interpreting histograms often involves estimating medians and proportions within intervals.
直方图用于展示分组连续数据。每个条形的面积与频数成正比。当组距不相等时，需绘制 频率密度 = 频数 / 组距。理解直方图通常需要估计中位数和区间内的比例。
Cumulative frequency diagrams plot cumulative frequency against the upper class boundary. Connecting points with a smooth curve allows estimation of medians, quartiles, and percentiles by reading off values at 50%, 25%, and 75% of total frequency.
累积频率图以累积频数对组上限描点。用平滑曲线连接后，可从总频数的 50%、25% 和 75% 位置读取中位数、四分位数和百分位数的估计值。
3. Measures of Location and Spread | 位置度量与离散度量
Measures of location summarise the centre of a data set. The mean, x̄, is calculated as the sum of all values divided by the count. For a frequency distribution, x̄ = Σfx / Σf. The median is the middle value when data are ordered; for n items its position is (n+1)/2. The mode is the most frequent value. Quartiles divide the sorted data into four equal parts; the interquartile range (IQR) = Q₃ − Q₁ measures the spread of the middle 50%.
位置度量概括数据集的中心。均值 x̄ 按所有数据值之和除以个数计算；对于频数分布，x̄ = Σfx / Σf。中位数是排序后居中的值，对于 n 个数据，位置为 (n+1)/2。众数是出现频率最高的值。四分位数将排序数据四等分；四分位距 IQR = Q₃ − Q₁ 衡量中间 50% 数据的散布程度。
Measures of spread describe variability. Variance and standard deviation are the most useful. For a sample, variance is s² = Σ(x − x̄)² / (n−1). For grouped data, s² = Σf(x − x̄)² / (Σf − 1). Standard deviation s is the square root of variance. A larger s indicates more dispersed data. In the exam, always state the formula used unless given in the question.
离散度量描述变异性。方差和标准差最为有用。对于样本，方差 s² = Σ(x − x̄)² / (n−1)；对于分组数据，s² = Σf(x − x̄)² / (Σf − 1)。标准差 s 是方差的平方根。s 越大说明数据越分散。考试中除非题目提供，否则应写出所用公式。
s² = Σ(x − x̄)² / (n−1) s = √[Σ(x − x̄)² / (n−1)]
s² = Σ(x − x̄)² / (n−1) s = √[Σ(x − x̄)² / (n−1)]
Choosing measures: When data are skewed or contain outliers, the median and IQR are more robust than the mean and standard deviation. Describe shape (symmetrical, positive/negative skew) before selecting the appropriate statistics to support your conclusion.
度量的选择：数据存在偏态或离群值时，中位数和 IQR 比均值和标准差更稳健。在选择恰当的统计量之前，先描述分布形状（对称、正偏或负偏），以支撑你的结论。
4. Probability | 概率
Basic probability: For an event A, P(A) = number of favourable outcomes / total number of equally likely outcomes. All probabilities lie between 0 and 1. The complement rule is P(A’) = 1 − P(A).
基本概率：对于事件 A，P(A) = 有利结果数 / 等可能结果总数。所有概率值介于 0 和 1 之间。互补规则为 P(A’) = 1 − P(A)。
Mutually exclusive events cannot occur simultaneously, so P(A ∩ B) = 0. The addition law becomes P(A ∪ B) = P(A) + P(B). For non-mutually exclusive events, the general addition law applies: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
互斥事件不能同时发生，因此 P(A ∩ B) = 0。加法法则简化为 P(A ∪
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