📚 Year 10 Eduqas Statistics: Core Knowledge Review | Year 10 Eduqas 统计：核心知识点梳理
A solid understanding of the fundamental statistical concepts taught in Year 10 is vital for success on the Eduqas GCSE Statistics paper. This review consolidates all the essential topics – from data types and sampling techniques to correlation and index numbers – into one structured, bilingual guide. Use it to check your knowledge, fill any gaps, and boost your confidence.
扎实掌握 Year 10 所教授的基础统计概念，对于 Eduqas GCSE 统计考试的成功至关重要。这篇复习文章将所有核心主题——从数据类型和抽样技巧到相关性和指数——整合成一份结构清晰的中英双语指南。用它来检测已有知识、填补漏洞并提升自信。
1. Types of Data and Data Collection | 数据类型与数据收集
Statistical investigation starts with recognising the type of data you are dealing with. Broadly, data falls into two categories: qualitative (categorical) and quantitative (numerical). Qualitative data describes qualities or categories, such as eye colour or car brand, and is often non‑numerical. Quantitative data represents counts or measurements and is further split into discrete and continuous forms.
统计调查的第一步是认清你正在处理的数据类型。数据大致分为两类：定性（分类）数据和定量（数值）数据。定性数据描述的是性质或类别，如眼睛颜色或汽车品牌，通常是非数字的。定量数据表示计数或测量值，并进一步划分为离散型和连续型。
Discrete quantitative data can only take specific, separate values – usually whole numbers. Examples include the number of students in a class, the roll of a die, or shoe sizes (when counted in half‑size steps). You cannot have 2.3 students or 5.7 on a die, so these data are countable but not divisible beyond the recorded increments.
离散定量数据只能取特定的、分离的值——通常是整数。比如一个班级的学生人数、掷骰子的点数或鞋码（按半码计数）。你不可能有 2.3 个学生或掷出 5.7 点，因此这些数据是可数的，但不能超出所记录的最小间隔而被无限细分。
Continuous quantitative data, on the other hand, can take any value within a given range. Measurements like height, mass, temperature and time are continuous because they can be refined to ever finer subdivisions. It is important to know which type you are dealing with because it affects what graphs and calculations are appropriate.
另一方面，连续定量数据可以在给定范围内取任意值。像身高、体重、温度和时间等测量值就是连续的，因为它们可以被无限细分。认清数据类型至关重要，因为这决定了后续适合采用哪些图表和计算。
Data can also be classified by how it is collected. Primary data is collected first‑hand by the researcher for a specific purpose, for instance through experiments or questionnaires. Secondary data is information that already exists, gathered by someone else for a different purpose, such as census statistics or weather records. Both have advantages: primary data gives you control over accuracy, while secondary data saves time and costs.
数据还可以按收集方式分类。原始数据是研究者为特定目的亲手收集的，例如通过实验或问卷得到的数据。二手数据则是已存在的信息，由他人为其他目的收集的，比如人口普查统计或气象记录。两者各有优势：原始数据让你对准确性有掌控，而二手数据则节省时间和成本。
When designing data‑collection tools like questionnaires, avoid leading questions, keep language simple, and ensure response options cover all possible answers. A well‑designed questionnaire reduces bias and raises the reliability of the data you collect.
在设计问卷等数据收集工具时，避免诱导性问题，使用简单的语言，并确保回答选项覆盖所有可能答案。一份设计良好的问卷能减少偏差，提高所收集数据的可靠性。
2. Sampling Methods | 抽样方法
A population is the entire group you want to study, while a sample is a subset selected from it. Sampling is necessary because surveying a whole population is often impractical, but the sample must represent the population if conclusions are to be valid.
总体是你想要研究的整个群体，而样本是从中选出的一个子集。抽样之所以必要，是因为普查整个总体往往不现实，但要使结论有效，样本必须能代表总体。
Random sampling gives every member of the population an equal chance of being chosen, often using a random number generator or 'names out of a hat'. It is fair but can be unrepresentative by chance, especially with small samples. Systematic sampling selects individuals at regular intervals from a list – for example, every 10th name. It is quick, but if the list has a hidden pattern, the sample may be biased.
随机抽样让总体中的每个成员都有均等被选中的机会，通常使用随机数生成器或“抽签”法。这种方法公平，但可能因偶然性而不具代表性，尤其是样本量小时。系统抽样则从一个列表中每隔固定间隔抽取个体——例如每第 10 个姓名。这种方法快捷，但如果列表有隐含规律，样本可能产生偏差。
Stratified sampling divides the population into distinct groups (strata) based on a characteristic like age or gender, then takes a random sample from each group proportional to its size. This guarantees that each subgroup is represented accurately. Quota sampling is similar but non‑random: interviewers are told how many people from each category to select, which is cheaper and faster but carries a higher risk of bias.
分层抽样首先按年龄或性别等特征将总体分成不同的层（strata），然后从每一层按比例随机抽取样本。这能保证每个子群体都被准确代表。配额抽样类似但非随机：采访者被告知从各类别抽取多少人，这种方法成本更低、速度更快，但偏差风险更高。
A well‑chosen sampling method minimises bias and makes your conclusions valid for the target population. You will often be asked to identify the method used or to suggest the most suitable one for a given scenario.
选择恰当的抽样方法可以最大限度地减少偏差，让你对目标总体的结论有效。考试中你常会被要求识别所用抽样方法，或为给定情境推荐最合适的方法。
3. Presenting Data: Charts and Diagrams | 数据展示：图表与图形
Once data is collected, it must be presented clearly. Bar charts are used for discrete or categorical data, with gaps between the bars to emphasise the distinct categories. Pie charts show proportions of a whole at a glance, making them ideal for qualitative data when you want to highlight relative sizes.
数据收集后，必须清晰呈现。条形图用于离散或分类数据，条与条之间有间隔以强调不同类别。饼图能一目了然地展示各部分占整体的比例，非常适合在需要突显相对大小时的定性数据展示。
For ordered discrete data, stem‑and‑leaf diagrams combine a frequency table with the original data values, keeping all the information visible. A back‑to‑back stem‑and‑leaf plot allows you to compare two related datasets directly, which is extremely useful for side‑by‑side comparison.
针对有序离散数据，茎叶图将频数表与原始数值结合在一起，保留了所有信息。背靠背茎叶图则可直接比较两个相关数据集，这在并列对比时极为有用。
Box plots (box‑and‑whisker diagrams) summarise data by displaying the minimum, lower quartile, median, upper quartile and maximum. They are excellent for comparing the spread and central location of several batches of data. Histograms, on the other hand, are used for continuous grouped data: the area of each bar represents frequency, so bar widths can vary, and there are no gaps between bars unless a frequency class is empty.
箱线图（箱须图）通过显示最小值、下四分位数、中位数、上四分位数和最大值来概括数据。它在比较多批数据的离散程度和中心位置时非常出色。而直方图则用于连续分组数据：每个条形的面积代表频数，因此条形宽度可以变化，且条与条之间无间隙，除非某个频率组为空。
Cumulative frequency curves and frequency polygons help you visualise how data accumulates. From a cumulative frequency graph, you can estimate medians, quartiles and percentiles by reading off the values at key points on the curve.
累积频数曲线和频数多边形有助于将数据的累积过程可视化。从累积频数图中，你可以通过读取曲线上关键点的数值来估计中位数、四分位数和百分位数。
4. Measures of Central Tendency | 集中趋势的度量
The three principal averages are the mean, median and mode. Each gives a different 'centre' of the data and understanding when to use which average is a key skill. The mean is calculated by summing all values and dividing by the number of values. It uses every data point and is affected by extreme outliers.
三种主要的平均数是平均数、中位数和众数。每一种都给出了数据的不同“中心”，懂得何时使用哪种平均数是关键技能。平均数是通过将全部数值相加后除以数值的个数来计算得出的。它用到了每个数据点，并会受到极端异常值的影响。
The median is the middle value when data are ordered. It is unaffected by outliers, making it a better measure for skewed distributions. The mode is the most frequently occurring value; it is the only average that can be used for qualitative data and helps identify common categories in marketing or surveys.
中位数是排序后居于中间位置的值。它不受异常值影响，因此对于偏态分布是更好的度量指标。众数是出现频率最高的值；它是唯一可用于定性数据的平均数，有助于在市场或调查中识别最常见的类别。
When data are grouped in a frequency table, the mean can be estimated using midpoint values. Multiply each midpoint by its frequency, sum these products, and divide by the total frequency. The modal class is the group with the highest frequency, while the median class contains the middle position.
当数据以频数表的分组形式给出时，可以用组中值来估算平均数。将每个组中值乘以该组频数，求和后再除以总频数。众数所在组是频数最高的组，而中位数所在组包含了中间位置。
Choosing the appropriate average depends on the data type and the story you want to tell. For symmetrical distributions, mean and median are similar; in skewed data, the median is often preferred.
选择合适的平均数取决于数据类型和你想要传达的信息。对于对称分布，平均数和中位数相近；在偏斜数据中，则通常优先考虑中位数。
5. Measures of Dispersion | 离散程度的度量
Dispersion tells you how spread out the data are. The simplest measure is the range, calculated as maximum minus minimum. While easy to compute, the range is very sensitive to outliers and ignores the distribution of middle values.
离散程度告诉你数据的分散情况。最简单的度量是极差，用最大值减去最小值计算。极差虽然易于计算，但对异常值非常敏感，且完全忽略了中间值的分布。
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁). It covers the middle 50% of the data, making it robust against outliers. Quartiles are often found from an ordered list, a cumulative frequency graph, or by using positions: Q₁ at (n+1)/4 th value and Q₃ at 3(n+1)/4 th value.
四分位距（IQR）是上四分位数（Q₃）与下四分位数（Q₁）之差。它覆盖了中间 50% 的数据，因此对异常值具有稳健性。四分位数通常从有序列表、累积频数图中查找，或通过位置公式确定：Q₁ 位于第 (n+1)/4 个值，Q₃ 位于第 3(n+1)/4 个值。
Standard deviation is the most comprehensive measure of spread, showing the typical distance of data points from the mean. A larger standard deviation indicates more varied data. For a sample, the formula is:
标准差是最全面的离散指标，显示了数据点与平均数之间的典型距离。标准差越大，数据变化幅度越大。对于样本，其计算公式为：
s = √[ Σ(x − x̄)² / (n − 1) ]
You may need to calculate standard deviation from a list or a grouped frequency table. Remember to square the differences, average them (using n-1 for a sample), and then take the square root. Understanding these measures allows you to describe not just the centre, but the consistency of a dataset.
你可能需要从列表或分组频数表中计算标准差。请记住顺序：计算差值平方，求其平均（样本使用 n−1），最后开平方。理解这些度量后，你不仅能描述数据的中心，也能描述其一致性。
6. Grouped Frequency Distributions | 分组频数分布
Continuous data are often summarised in grouped frequency tables where class intervals are defined. To estimate the mean, treat each midpoint as the representative value for its class, then apply the same method as for ungrouped data. This yields an estimated mean because we assume values are evenly spread within each class.
连续数据常常汇总在分组频数表中，并明确划出组距。要估算平均数，可将每个组中值视为该组的代表值，然后采用与未分组数据相同的方法。由于我们假设每组内数值均匀分布，所以得出的是一个估计平均数。
Finding the median from grouped data requires linear interpolation. Identify the median class interval, then use the formula:
要从分组数据中求出中位数，就需要线性插值。首先找到中位数所在组，然后使用以下公式：
Median = L + [ (n/2 − F) / f ] × w
where L is the lower boundary of the median class, F is the cumulative frequency before the class, f is the frequency of the median class, and w is the class width. This gives a much more precise estimate than simply stating the median class.
其中 L 是中位数所在组的下限，F 是该组之前的累积频数，f 是中位数所在组的频数，w 为组距。与单纯指出中位数组相比，这种方法给出的估计值精确得多。
ight must be adjusted using the formula frequency density = frequency ÷ class width. In cumulative frequency graphs, points are plotted at the upper class boundaries and joined smoothly, enabling estimation of medians, quartiles and percentiles without a formula.
分组数据的直方图条形之间无间隙，每个条形的面积与频数成正比。当组距不相等时，必须用公式“频数密度 = 频数 ÷ 组距”来调整高度。在累积频数图中，点标在组距上界，并圆滑连接，这样无需公式即可估算中位数、四分位数和百分位数。
7. Introduction to Probability | 概率基础
Probability measures how likely an event is, expressed as a number between 0 (impossible) and 1 (certain). Relative frequency is an estimate of probability obtained from experimental or historical data: it is calculated as frequency of the event divided by total number of trials.
概率衡量一个事件发生的可能性，用 0（不可能）和 1（必然发生）之间的数字表示。相对频率是通过实验或历史数据得到的概率估计值，它用事件发生的频数除以试验总次数来计算。
For equally likely outcomes, probability is simply number of favourable outcomes divided by total number of possible outcomes. This classical definition works well for coins and dice, but for real‑world events you rely on relative frequency or theoretical models. Sample space diagrams list all possible outcomes and are essential for calculating probabilities of combined events.
对于等可能的结果，概率简单地等于有利结果数除以所有可能结果的总数。这种古典定义适用于掷硬币和骰子，但在现实事件中你需要依赖相对频率或理论模型。样本空间图列出了所有可能的结果，对于计算复合事件的概率至关重要。
Tree diagrams are a powerful tool for illustrating sequences of events. Multiply probabilities along branches for combined events, and add probabilities of branches that lead to the desired outcome. When events are independent, the outcome of one does not affect the other. For mutually exclusive events, they cannot happen at the same time.
树图是展示事件序列的强大工具。沿着分支相乘概率可以得到复合事件的概率，再将通向目标结果的分支概率相加。当事件相互独立时，一个事件的结果不影响另一个。互斥事件则不能同时发生。
grams help visualise intersections, unions and conditional probabilities. The notation P(A| occurring given that B has already occurred. Conditional probability can be found from two‑way tables or using the formula P(A|B) = P(A ∩ B) / P(B). These concepts form the backbone of data‑based chance questions.
维恩图有助于可视化交集、并集和条件概率。符号 P(A|B) 表示在 B 已发生的情况下 A 发生的概率。条件概率可以从二维表中获取，或使用公式 P(A|B) = P(A ∩ B) / P(B) 计算。这些概念构成了基于数据的概率问题的基础。
8. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (or scatter plot) displays the relationship between two numerical variables. Each point represents a pair of values. By examining the pattern of points, you can describe the correlation: positive if both variables increase together, negative if one decreases as the other increases, and zero if there is no visible pattern.
散点图（散点图）展示的是两个数值变量之间的关系。每个点代表一对数据值。通过观察点的分布模式，你可以描述相关性：若两个变量同时增加则为正相关，若一个增加另一个减少则为负相关，若无清晰模式则为零相关。
Correlation does not imply causation. A strong correlation between ice‑cream sales and sunburn does not mean one causes the other; both are influenced by a third factor – hot weather. When asked to interpret, always look for plausible causation or common underlying variables.
相关性并不意味着因果关系。冰激凌销量与晒伤之间的强相关并不意味着一者导致另一者；两者都受到第三个因素——炎热天气的影响。在读图解释时，始终要寻找合理的因果关系或潜在的共同变量。
The strength of correlation can be judged visually, but a numerical measure gives precision. Spearman's rank correlation coefficient (rₛ) is a common non‑parametric measure. It is calculated by ranking both variables, finding the differences d in ranks, and using the formula:
相关的强度可以直观判断，但数值度量能提供精确性。斯皮尔曼等级相关系数（rₛ）是一种常用的非参数度量。它对两个变量分别排序，求出等级差值 d，并使用以下公式：
rₛ = 1 − (6 Σ d²) / [n(n² − 1)]
where n is the number of pairs. The value of rₛ ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). A value close to 0 suggests weak or no correlation.
其中 n 是数据对数。rₛ 的值介于 −1（完全负相关）和 +1（完全正相关）之间。接近 0 则表明相关性强或不存在。
traight line of the form y = a + bx, the gradient b quantifies the rate of change. Predict
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