📚 Core Knowledge Refresher for Year 11 OCR Statistics | Year 11 OCR 统计：核心知识点梳理
Welcome to this comprehensive refresher guide covering the core topics of the Year 11 OCR Statistics course. Whether you are preparing for exams or consolidating your understanding, this article systematically breaks down the essential concepts, definitions, formulas and techniques you need to master, from types of data and sampling methods through to measures of spread, correlation, time series, probability and index numbers. Each section pairs English explanations with Chinese translations to support bilingual learners.
欢迎来到这份针对 Year 11 OCR 统计课程核心知识点的综合梳理指南。无论你是备考冲刺还是巩固所学，这篇文章都会系统地拆解你需要掌握的关键概念、定义、公式与技巧，从数据类型、抽样方法，一直讲到离散程度、相关性、时间序列、概率与指数。每个部分都以中英对照形式呈现，帮助双语学习者加深理解。
1. Types of Data and Data Collection | 数据类型与数据收集
Data can be classified as qualitative (categorical) or quantitative (numerical). Qualitative data are non-numerical labels, such as eye colour, gender or type of car; they can be counted but not measured. Quantitative data consist of numbers and are further split into discrete and continuous types.
数据可分为定性（分类）数据和定量（数值）数据。定性数据是非数值的标签，如眼睛颜色、性别或汽车类型，这类数据只能计数，不能测量。定量数据由数字组成，并可进一步分为离散型和连续型。
Discrete data take exact, countable values, usually whole numbers, such as the number of students in a class or the number of goals in a match. Continuous data can take any value within a range, limited only by the precision of the measuring instrument, such as height, mass, time or temperature.
离散型数据取精确、可计数的值，通常是整数，例如班级人数或一场比赛的进球数。连续型数据可以在某一区间内取任意值，其精度只受测量仪器限制，例如身高、质量、时间和温度。
Data may also be classified by how they were obtained. Primary data are collected first-hand by the researcher, for example through a questionnaire, an experiment or a survey. Secondary data already exist, such as government statistics, published reports or school records.
数据还可以按获取方式分类。一手数据由研究者直接收集，例如通过问卷、实验或调查得到。二手数据是已经存在的数据，例如政府统计数据、公开报告或学校档案。
- Primary data: tailored to your question and the method is controlled, but collection is time-consuming and expensive.
一手数据：贴合你的研究问题、方法可控，但收集耗时且成本高。 - Secondary data: cheap, quick and often large scale, but may be out of date, incomplete or originally collected for a different purpose.
二手数据：便宜、快捷、规模通常较大，但可能过时、不完整，或原本是为其他目的收集的。 - A census measures every member of a population; a sample survey measures only part of it.
普查测量总体中的每一个个体；抽样调查只测量其中的一部分。
2. Populations, Samples and Sampling Methods | 总体、样本与抽样方法
A population is the complete set of individuals or items about which you want to draw conclusions, while a sample is a subset of that population selected for study. A sampling frame is a list of all members of the population from which the sample is drawn, for example a school register, a customer database or a postcode list. Sampling saves time and money, but every conclusion carries sampling error.
总体是你想要得出研究结论的完整个体或项目集合，而样本是从中选出、用于研究的一个子集。抽样框是总体全部成员的名册，样本即从中抽取，例如学校学籍册、客户数据库或邮编列表。抽样可以节省时间和成本，但任何结论都会带有抽样误差。
OCR expects you to know five standard sampling methods and to justify which is appropriate in a given context: simple random, systematic, stratified, cluster and quota sampling, plus non-random methods such as opportunity sampling.
OCR 要求你掌握五种标准抽样方法，并能结合具体情境说明哪种方法更合适：简单随机抽样、系统抽样、分层抽样、整群抽样和配额抽样，此外还有机会抽样等非随机方法。
| Method 方法 | How it works 操作方式 | Strength / Weakness 优缺点 |
| Simple random 简单随机 | Every member has an equal chance; use a random number generator or lottery. | Unbiased, but needs a complete sampling frame. |
| Systematic 系统 | Take every kth item from the frame, where k = population ÷ sample size. | Quick and even coverage, but may hit a periodic pattern. |
| Stratified 分层 | Divide into strata (year group, gender), then sample randomly within each in proportion. | More representative; guarantees every group appears. |
| Cluster 整群 | Choose whole clusters (classes, streets) at random and survey everyone inside. | Cheap for fieldwork, but larger sampling error. |
| Quota 配额 | Interviewer fills a fixed quota in each category, no frame required. | Fast, but not random and open to interviewer bias. |
For stratified sampling, the number taken from each stratum is proportional to its size:
对于分层抽样，每个层中抽取的人数与其规模成正比：
Number from stratum = (stratum size ÷ population size) × sample size
某层抽样人数 = (该层人数 ÷ 总体人数) × 样本量
3. Presenting Data: Tables, Charts and Histograms | 数据呈现：表格、图表与直方图
Frequency tables summarise raw data into classes with tallies and frequencies, and two-way tables show the relationship between two categorical variables. Bar charts display discrete or categorical data with gaps between bars, while pie charts show proportions, where the sector angle is found from (frequency ÷ total frequency) × 360°.
频数表把原始数据按组别汇总为划记与频数，双向表则展示两个分类变量之间的关系。条形图用带间隙的柱子表示离散型或分类数据，饼图表示比例，其中扇形圆心角由 (频数 ÷ 总频数) × 360° 求得。
Stem-and-leaf diagrams are useful because they keep the original values while showing shape, and they make the median and quartiles easy to read. For continuous data grouped into classes of unequal width, a bar chart would be misleading, so a histogram with frequency density on the vertical axis is used instead.
茎叶图的优点在于保留原始数值的同时还能显示分布形状，中位数与四分位数也一目了然。对于组距不相等的连续型分组数据，用条形图会产生误导，因此应改用纵轴为频率密度的直方图。
Frequency density = frequency ÷ class width
频率密度 = 频数 ÷ 组距
- In a histogram it is the area of each bar, not the height, that represents frequency.
在直方图中，代表频数的是每个矩形的面积，而不是高度。 - Use a line graph for time series data and a comparative bar chart for comparing two distributions.
时间序列数据用折线图，比较两个分布时可用对比条形图。 - Always label axes with the variable and its units, and give the chart a clear title.
图表要标明变量名称与单位，并给出清晰标题。
4. Measures of Central Tendency | 集中趋势的度量
The three measures of central tendency are the mean, the median and the mode. The mean uses every value in the data set and is calculated by adding all values and dividing by how many there are. From a frequency table the mean is found using the midpoints of each class.
集中趋势的三个度量是平均数、中位数和众数。平均数用到了数据中的每一个值，计算方法为全部数值之和除以个数。由频数表求平均数时，使用每个组区间的组中值。
x̄ = Σx ÷ n x̄ = Σfx ÷ Σf
The median is the middle value when the data are arranged in order; for an even number of values it is the mean of the two middle values. The mode is the most frequently occurring value, and for grouped data we refer to the modal class, the class with the highest frequency.
中位数是数据按大小排序后位于中间的值；当数据个数为偶数时，取中间两个值的平均数。众数是出现次数最多的值；对于分组数据，我们称频数最高的组为众数组。
- Mean: uses all the data, suitable for roughly symmetric distributions, but strongly affected by outliers.
平均数：利用全部数据，适合大致对称的分布，但极易受异常值影响。 - Median: resistant to outliers and skew, so it is preferred for house prices and incomes.
中位数：不受异常值和偏态影响，因此房价、收入类数据更常用它。 - Mode: the only average that works for qualitative data and is useful for clothing sizes.
众数：唯一适用于定性数据的平均量，在服装尺码等问题中很有用。 - A weighted mean is calculated as x̄ = Σwx ÷ Σw, where w is the weight of each value.
加权平均数按 x̄ = Σwx ÷ Σw 计算，其中 w 是每个数值的权重。
5. Measures of Dispersion: Range, IQR and Standard Deviation | 离散程度的度量：极差、四分位距与标准差
t value minus the lowest, and the interquartile range, IQR = Q₃ − Q₁, covers the middle 50% of the data and ignores extreme values.
只有平均数并不能完整描述一组数据；两个分布可以有相同的平均数，但离散程度差别很大。极差等于最大值减最小值；四分位距 IQR = Q₃ − Q₁，覆盖中间 50% 的数据，不受极端值影响。
The standard deviation measures the typical distance of each value from the mean. It is the square root of the variance, which is the mean of the squared deviations. A large standard deviation means the data are widely spread; a small one means they are clustered around the mean.
标准差衡量每个数值与平均数之间的典型距离。它等于方差的平方根，而方差是偏差平方的平均值。标准差大表示数据分散；标准差小表示数据集中于平均数附近。
σ = √( Σ(x − μ)² ÷ n )
s = √( Σ(x − x̄)² ÷ (n − 1) )
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