GCSE OCR Statistics: Core Knowledge Review | GCSE OCR 统计：核心知识点梳理

📚 GCSE OCR Statistics: Core Knowledge Review | GCSE OCR 统计：核心知识点梳理

This revision guide brings together the essential topics for GCSE OCR Statistics, from data collection and sampling to probability distributions and index numbers. Whether you are building a systematic study plan or doing last-minute review, the paired English-Chinese explanations will help you master key concepts and terminology with confidence.

这份复习指南汇集了 GCSE OCR 统计的核心主题，涵盖数据收集与抽样、概率分布与指数等内容。无论你是在制定系统复习计划还是进行考前冲刺，英中对照的讲解都能帮助你牢固掌握核心概念与术语。

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). Quantitative data is further divided into discrete data, which can only take specific values (e.g. number of students), and continuous data, which can take any value within a range (e.g. height).

数据可分为定性（类别）数据和定量（数值）数据。定量数据又分为离散数据（只能取特定值，如学生人数）和连续数据（可在某一范围内取任意值，如身高）。

Primary data is collected directly by the researcher for a specific purpose, whereas secondary data refers to information that already exists, such as government statistics or internet databases. Primary data is often more reliable but can be expensive and time-consuming to collect.

原始数据是研究者为特定目的直接收集的，二手数据则是已经存在的信息，如政府统计数据或网络数据库。原始数据通常更可靠，但收集起来可能费时费力。

When designing a data collection method, you need to consider the population, sample frame, and potential sources of bias. A pilot study can help test the procedure and refine questions before the main investigation.

在设计数据收集方法时，需要明确总体、抽样框以及可能的偏误来源。预调查有助于在大规模调查前测试流程并完善问题。

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2. Sampling Methods in Detail | 抽样方法详解

Simple random sampling gives every member of the population an equal chance of being selected, often using random number generators. It is easy to understand but may not represent subgroups well.

简单随机抽样使总体中每个个体都有均等被选中的机会，通常使用随机数生成器。此方法易于理解，但可能无法很好地代表各个子群。

Stratified sampling divides the population into distinct strata (e.g. age groups) and takes a random sample from each in proportion to the stratum size. This improves representativeness.

分层抽样将总体划分为不同的层（如年龄组），并按各层大小比例从每层中随机抽取样本。这能提高样本的代表性。

Systematic sampling selects every k-th member after a random start. It is simple to implement, but if the list has an underlying pattern, bias may occur.

系统抽样在随机起点后每隔 k 个个体抽取一个样本。实施简单，但若列表存在隐藏周期，可能导致偏误。

Quota sampling is a non-probability method where interviewers are given quotas for certain categories. It is cheap and fast but can introduce selection bias.

配额抽样是一种非概率方法，访问员需按指定类别完成配额。此方法成本低、速度快，但可能引入选择偏差。

Cluster sampling splits the population into clusters, then randomly selects whole clusters. It is useful when the population is large and spread out geographically.

整群抽样将总体分成若干群，然后随机抽取整个群。当总体庞大且地理分布广时，该方法很实用。

Method	Advantage	Disadvantage
Simple random	Unbiased, theoretically pure	Need full population list
Stratified	Reflects subgroup proportions	Requires knowledge of strata sizes
Systematic	Simple to use	Can be biased if hidden pattern exists
Quota	Quick and low cost	Not random; high risk of bias

方法对比表：简单随机抽样无偏但需完整名单；分层抽样能反映子群比例；系统抽样简单却可能遇周期偏误；配额抽样快但有选择偏差。

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3. Presenting Data: Charts and Graphs | 数据展示：图表

Bar charts are used for categorical or discrete data; each category has a bar whose length represents frequency. Bars should have equal width and a gap between them.

条形图用于分类数据或离散数据；每个类别竖条的长度表示频数。条形宽度应相等，条间留有空隙。

ortions of a whole. The angle for each sector is calculated as (frequency / total) × 360°.

饼图展示各部分占整体的比例。每个扇形的角度 = (频数 / 总数) × 360°。

ical axis is frequency density = frequency / class width. Unequal class widths are common.

直方图用于分组连续数据。面积代表频数，因此纵轴为频数密度 = 频数 / 组距。组距常常不相等。

Frequency polygons connect the midpoints of each class interval with straight lines, often superimposed on a histogram. Cumulative frequency graphs plot cumulative frequency against upper class boundaries, giving an S-shaped curve used to find medians and quartiles.

频数多边形将各组中点用直线连接，常叠加在直方图上。累积频率图则绘制累积频数与组上限的关系，形成S形曲线，用于查找中位数与四分位数。

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

The mean (x̄ mean = Σ(fx) / Σf. For grouped data we use the midpoint of each class as an estimate of x.

平均数（x̄）等于所有数据值的总和除以数据个数。对于频数表，平均数 = Σ(fx) / Σf。分组数据则使用各组中点作为 x 的估计。

The median is the middle value when data are ordered. For n values, the position is (n + 1) / 2. The mode is the most frequent value; for grouped data we refer to the modal class.

中位数是将数据排序后位于中间的数值。有 n 个数据时，中位数的位置为 (n + 1) / 2。众数是出现最多的值；分组数据中则指众数组。

Choosing an appropriate average depends on the data distribution. The mean is sensitive to outliers, while the median is robust and better for skewed data. The mode is useful for categorical data.

选择适当的平均数据取决于数据分布。平均数对异常值敏感，而中位数稳健且更适合偏斜数据。众数适用于分类数据。

Mean for grouped data: x̄ ≈ Σ(f × midpoint) / Σf

分组数据平均数：x̄ ≈ Σ(f × 组中点) / Σf

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5. Measures of Spread and Dispersion | 离散程度的度量

= highest value - lowest value. It is easy to compute but heavily influenced by outliers.

极差 = 最大值 - 最小值。计算简便，但极易受异常值影响。

here Q1 is the lower quartile and Q3 is the upper quartile. IQR describes the spread of the middle 50% of data, making it more robust than range.

四分位距 (IQR) = Q3 - Q1，其中 Q1 为下四分位数，Q3 为上四分位数。IQR 描述中间 50% 数据的离散程度，比极差更稳健。

Standard deviation measures the average distance of each data point from the mean. For a sample, the formula is:

s = √( Σ(x_i - x̄)² / (n - 1) )

标准差衡量各数据点与平均数的平均距离。样本标准差公式如上。

Variance is s². A smaller standard deviation indicates data points are clustered closely around the mean; a larger one shows greater spread.

方差即 s²。标准差越小，数据点越聚集在平均数周围；越大则数据越分散。

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6. Cumulative Frequency and Box Plots | 累积频率与箱线图

To draw a cumulative frequency diagram, first add a cumulative frequency column to the frequency table. Plot points at each upper class boundary and join them with a smooth curve. The curve makes it possible to estimate the median, quartiles, and percentiles.

绘制累积频率图时，先在频数表中增加累积频数列。在每组上限处描点并用光滑曲线连接。曲线可用于估计中位数、四分位数和百分位数。

omparing distributions side by side and for identifying outliers using the 1.5 × IQR rule.

箱线图（盒须图）展示五数概括：最小值、Q1、中位数、Q3 和最大值。它非常适合并行比较分布，并可用 1.5 × IQR 法则识别异常值。

Lower fence = Q1 - 1.5 × IQR, Upper fence = Q3 + 1.5 × IQR

下围栏 = Q1 - 1.5 × IQR，上围栏 = Q3 + 1.5 × IQR

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7. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph displays the relationship between two variables. The independent variable is placed on the x-axis, the dependent variable on the y-axis. The pattern of points can reveal correlation.

散点图呈现两个变量之间的关系。自变量放在 x 轴，因变量放在 y 轴。点的分布形态可揭示相关性。

Positive correlation means that as one variable increases, the other tends to increase. Negative correlation means that as one increases, the other tends to decrease. No correlation indicates no clear linear pattern.

正相关意味着一个变量增大时另一个变量也倾向于增大。负相关则是一个变量增大时另一个倾向于减小。无相关表示无明显线性模式。

Correlation does not imply causation. A strong correlation might be caused by a third lurking variable. The strength can be assessed visually or by calculating the product-moment correlation coefficient (PMCC), though GCSE questions often focus on interpretation.

相关不代表因果。强相关可能源自第三个潜在变量。强度可通过目测或计算积矩相关系数 (PMCC) 评估，但 GCSE 题目多考查解读。

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8. Regression and Lines of Best Fit | 回归与最佳拟合线

+ c, where m is the gradient and c is the y-intercept. The line should pass as close as possible to all points, with roughly equal numbers of points on each side.

最佳拟合线（回归线）可目测绘制或通过计算得到。方程通常写为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。直线应尽可能靠近所有点，两侧点数量大致相等。

Once the line is drawn, it can be used to make predictions: interpolation (estimating within the data range) is usually reliable, while extrapolation (outside the range) is risky because the linear trend may not hold.

得到直线后，可用于预测：内插（在数据范围内估计）通常可靠，而外推（超出范围）有风险，因线性趋势可能不再成立。

If a calculator is used, the regression equation is obtained from summary statistics. The slope m and intercept c are given by complicated formulas, but in the exam you will often be provided with the equation or use pre-calculated values.

若使用计算器，回归方程可从汇总统计中求得。斜率 m 与截距 c 有复杂公式，但考试中常直接给出方程或提供计算后的数值。

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9. Probability Fundamentals | 概率基础

event A is P(A) = number of favourable outcomes / total number of equally likely outcomes.

概率是事件发生可能性的度量，范围从 0（不可能）到 1（必然）。事件 A 的概率 P(A) = 有利结果数 / 等可能结果总数。

trials). The more trials, the closer the estimate tends to the true probability (the law of large numbers).

相对频率是通过实验得到的概率估计：P(A) ≈ A的发生频数 / 试验总数。试验次数越多，估计越接近真实概率（大数定律）。

Sample space diagrams and Venn diagrams help visualise all possible outcomes and relationships between events (union, intersection, complement). Tree diagrams are essential for multi-stage events, allowing calculation of conditional probabilities by multiplying along branches.

样本空间图和文氏图有助于可视化所有可能结果及事件之间的关系（并、交、补）。树状图对多阶段事件至关重要，通过沿分支相乘可计算条件概率。

P(A and B) = P(A) × P(B | A) = P(B) × P(A | B)

P(A 且 B) = P(A) × P(B | A) = P(B) × P(A | B)

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10. Probability Distributions | 概率分布

bilities. For a discrete random variable X, the probabilities must satisfy Σ P(X = x) = 1.

概率分布列出随机变量所有可能的取值及其对应概率。对于离散随机变量 X，所有概率之和必须满足 Σ P(X = x) = 1。

ability p) and failure (probability q = 1 - p). The probability of exactly r successes is:

P(X = r) = ⁿC_r p^r q^{n - r}

二项分布适用于固定试验次数 n 且每次试验相互独立、只有成功（概率 p）和失败（概率 q = 1 - p）两种结果的情形。恰好成功 r 次的概率公式如上。

ou might use binomial tables or tree diagrams for small n rather than calculating the formula directly.

二项分布的期望值 E(X) = np，方差 Var(X) = npq。在 GCSE 统计中，对于较小的 n，你可能使用二项分布表或树状图而非直接代入公式计算。

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11. Time Series and Moving Averages | 时间序列与移动平均

A time series is a set of data recorded over time, e.g. monthly sales. It often contains four components: trend, seasonal variation, cyclic variation, and random (irregular) fluctuations.

时间序列是一组随时间记录的数据，如月度销售额。常包含四个分量：趋势、季节变动、循环变动和随机（不规则）波动。

A moving average smooths out short-term fluctuations to reveal the underlying trend. For a 3-point moving average, replace each data point (except the first and last) by the mean of itself and its two neighbours.

移动平均可平滑短期波动，揭示潜在趋势。对于三点移动平均，将每个数据点（首尾除外）替换为它自身与相邻两个点的均值。

Seasonal variation can be estimated by subtracting the moving average (trend) from the original data. These seasonal effects can then be averaged across corresponding seasons to produce typical seasonal adjustments for forecasting.

通过从原始数据中减去移动平均（趋势），可估算季节变动。随后对各对应季节的效应取平均，得出典型的季节调整量，用于预测。

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12. Index Numbers | 指数

period is usually set to 100. Simple price relative = (current price / base price) × 100.

指数用于衡量变量相对于基期值随时间的变化。基期通常设为 100。简单价比 = (现价 / 基价) × 100。

A weighted index considers the importance (weight) of different items. The Retail Price Index (RPI) and Consumer Price Index (CPI) are key economic indicators. They show how the average price of a basket of goods and services changes over time.

加权指数考虑不同项目的重要性（权重）。零售价格指数 (RPI) 与消费价格指数 (CPI) 是重要的经济指标，显示一篮子商品和服务的平均价格随时间的变化情况。

Chain base index links comparisons from one period to the next, rather than always referring back to a fixed base. This is useful when the composition of the basket changes regularly.

链基指数逐期连接比较，而不总是回溯到固定基期。当篮子构成经常变化时，该方法很有用。

Chain index for period t = (Value in t / Value in t-1) × previous chain index

t 期链指数 = (t 期值 / t-1 期值) × 上一期链指数

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Published by TutorHao | Statistics Revision Series | aleveler.com

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