📚 Year 11 Cambridge Statistics: Glossary & Quick Memorisation Guide | Year 11 剑桥统计：词汇术语速记指南
Building a solid foundation in statistical terminology is the first and most crucial step towards mastering Cambridge IGCSE Statistics. This guide translates every key term into clear definitions, paired with memory hooks and examples, so you can move from confusion to confidence quickly. Use it as a revision checklist and a daily reference to anchor your understanding before tackling past papers.
在剑桥 IGCSE 统计学习中，打牢术语基础是走向成功最关键的第一步。本指南将每一个核心术语转化为清晰的定义，搭配记忆技巧和实例，帮助你迅速拨开迷雾、建立自信。你可以把它当作复习清单和日常参考书，在接触历年真题前巩固理解。
1. Core Concepts: Population vs Sample | 核心概念：总体与样本
Population refers to the entire group of individuals or items that we want to draw conclusions about. Every single member of the group is included. Sample is a subset of the population, carefully selected to represent the whole. A census collects data from every member of the population. When we measure a characteristic of a population we call it a parameter; when we calculate the same characteristic from a sample we call it a statistic. A simple memory trick: Population → Parameter, Sample → Statistic.
总体（Population） 指的是我们希望得出结论的整个个体或项目集合，包含该群体中的每一个成员。样本（Sample） 是从总体中精心抽取的一个子集，用以代表整体。普查（Census） 则从总体的每一个成员处收集数据。当我们测量总体的某一特征时，这个数值称为 参数（Parameter）；当我们从样本计算同一特征时，则称为 统计量（Statistic）。简单的记忆口诀：总体对参数，样本对统计量。
2. Data Types: Qualitative, Quantitative, Discrete, Continuous | 数据类型：定性、定量、离散、连续
Data falls into two broad families. Qualitative data (categorical) describes qualities or labels, such as eye colour, car brand, or favourite sport. It is non-numerical. Quantitative data (numerical) counts or measures something and is split further into discrete and continuous. Discrete data can only take specific values, usually integers – for example, the number of students in a class or shoe size (in half-sizes). Continuous data can take any value within a range and is obtained by measuring – height, weight, time, temperature. Remember: if you count it, it is discrete; if you measure it, it is continuous.
数据可分为两大类。定性数据（Qualitative data），也称分类数据，描述的是品质或标签，如眼睛颜色、汽车品牌或最喜欢的运动，它是非数值型的。定量数据（Quantitative data） 是对事物进行计数或测量，并进一步分为 离散型（Discrete） 和 连续型（Continuous）。离散数据只能取特定值，通常是整数，例如班级学生人数或鞋码（包括半码）。连续数据可以在一个区间内取任意值，是通过测量得到的，如身高、体重、时间和温度。记住：凡是通过计数得到的就是离散变量；凡是通过测量得到的就是连续变量。
3. Central Tendency: Mean, Median, Mode | 集中趋势：平均数、中位数、众数
The mean is the arithmetic average. For ungrouped data, mean = Σx / n, where x represents each value and n is the number of values. The mean is sensitive to outliers. The median is the middle value when data are arranged in order. If n is odd, median = the middle item; if n is even, median = the mean of the two middle items. The median is resistant to extreme values and is therefore better for skewed distributions. The mode is the value that appears most often. A data set may have one mode (unimodal), two modes (bimodal) or no mode. Memory aid: Mean listens to every voice, median sits in the middle, mode shouts the loudest.
平均数（Mean） 是算术平均值。对于未分组数据，平均数 = Σx / n，其中 x 代表每一个取值，n 是数据个数。平均数容易受异常值影响。中位数（Median） 是将所有数据按大小顺序排列后位于中间位置的值。若 n 为奇数，中位数是正中间的那一项；若 n 为偶数，中位数是中间两项的平均值。中位数对极端值不敏感，因此在偏态分布中更具代表性。众数（Mode） 是出现次数最多的值。一组数据可能有一个众数（单峰）、两个众数（双峰）或没有众数。记忆辅助：平均数倾听每一个声音，中位数稳坐中间，众数喊得最响。
Mean (x̄) = Σx / n
4. Measures of Spread: Range, IQR, Variance & Standard Deviation | 离散程度：极差、四分位距、方差与标准差
Range is the simplest measure of spread: Range = maximum value – minimum value. It tells us the total width of the data but is easily distorted by outliers. Interquartile range (IQR) is a more robust measure: IQR = upper quartile Q₃ – lower quartile Q₁. It covers the middle 50% of the data and is not influenced by extreme scores. The variance and standard deviation measure how much the data values deviate from the mean on average. For a sample, variance s² = Σ(x − x̄)² / (n − 1). The standard deviation s is the square root of the variance. Use (n − 1) for sample data (Bessel’s correction) to obtain an unbiased estimate. When describing a Normal distribution, roughly 68% of data lie within ±1 standard deviation of the mean.
极差（Range） 是最简单的一种离散量数：极差 = 最大值 – 最小值。它告诉我们数据的总跨度，但极易受异常值影响。四分位距（Interquartile range, IQR） 是一种更为稳健的度量：IQR = 上四分位数 Q₃ – 下四分位数 Q₁。它涵盖中间 50% 的数据，不受极端值干扰。方差（Variance） 和 标准差（Standard deviation） 衡量数据的各个取值平均偏离平均数多少。对于样本，方差 s² = Σ(x − x̄)² / (n − 1)。标准差 s 是方差的平方根。样本数据使用 (n − 1) （贝塞尔校正）以得到无偏估计。在描述正态分布时，大约 68% 的数据落在距离平均数 ±1 个标准差的范围内。
s² = Σ(x − x̄)² / (n − 1) and s = √[ Σ(x − x̄)² / (n − 1) ]
5. Cumulative Frequency & Percentiles | 累积频数与百分位数
A cumulative frequency table adds up frequencies row by row so that each entry gives the total number of observations up to that point. The cumulative frequency graph (ogive) plots upper class boundaries against cumulative frequency. From this graph we can read off quartiles: Q₁ at 25% of total frequency, Q₂ (median) at 50% and Q₃ at 75%. Percentiles divide the data into 100 equal parts; the kth percentile is the value below which k% of the observations fall. The interquartile range IQR = Q₃ − Q₁ is often used together with the box-and-whisker plot to identify potential outliers (1.5 × IQR rule).
累积频数（Cumulative frequency） 表格将频数逐行累加，使每一项都给出截至该区间为止的总观测次数。累积频数图（又称折线图或穹形图）将组上限对标在横轴，累积频数对标在纵轴。从图上可以直接读取 四分位数（Quartiles）：Q₁ 位于总频数的 25% 处，Q₂（中位数）位于 50%，Q₃ 位于 75%。百分位数（Percentiles） 将数据分成 100 等份；第 k 百分位数是指有 k% 的观测值落在该值以下。四分位距 IQR = Q₃ − Q₁ 常与箱线图一起使用，并用 1.5 × IQR 法则识别潜在异常值。
6. Graphical Representations | 图形表示
Bar charts display categorical (qualitative) data with gaps between the bars. The height of each bar represents frequency or frequency density. Histograms are used for continuous quantitative data grouped into classes; there are no gaps between bars, and it is the area of each bar that is proportional to frequency. Always use frequency density = frequency ÷ class width when drawing a histogram with unequal class widths. A frequency polygon is formed by joining the midpoints of the tops of histogram bars. The box-and-whisker plot (box plot) shows the five-number summary: minimum, Q₁, median, Q₃ and maximum, making it easy to compare distributions and spot skewness. Outliers are marked as individual points beyond the whiskers.
条形图（Bar chart） 用于展示分类（定性）数据，条与条之间有间隙。每个条形的高度代表频数或频数密度。直方图（Histogram） 用于已分组的连续定量数据；条之间没有间隙，并且是用条形的 面积 来表示频数。当组距不相等时，必须使用 频数密度 = 频数 ÷ 组距 来绘制直方图。频数折线图（Frequency polygon） 是将直方图每个条形顶端的中点连接而成的折线。箱线图（Box-and-whisker plot） 展示了五数概括：最小值、Q₁、中位数、Q₃ 和最大值，可以方便地进行分布比较并识别偏态。超出触须范围的异常值会单独用点标出。
7. Correlation & Scatter Diagrams | 相关性与散点图
A scatter diagram is used to explore the relationship between two numerical variables. We call the explanatory variable the independent variable (plotted on the x-axis) and the response the dependent variable (on the y-axis). Correlation describes the strength and direction of a linear relationship. It can be positive (as one increases, the other tends to increase), negative (as one increases, the other tends to decrease) or zero (no linear pattern). The line of best fit can be drawn by eye or calculated using the least-squares method, and its equation is used to make predictions. Crucially, correlation does not imply causation; a strong association may be due to a lurking variable. Use the phrase ‘relationship, not reason’ to remember this.
散点图（Scatter diagram） 用来探索两个数值变量之间的关系。其中解释变量称为 自变量（Independent variable），画在 x 轴上；响应变量称为 因变量（Dependent variable），画在 y 轴上。相关性（Correlation） 描述线性关系的强度和方向。相关性可以是 正相关（一个增大另一个也趋于增大）、负相关（一个增大另一个趋于减小）或 零相关（无线性模式）。最佳拟合线（Line of best fit） 可以通过目测或最小二乘法求出，其方程可用于预测。关键一点：相关性并不意味着因果性；强关联可能由潜藏变量导致。请记住：“有关系不代表有原因”。
8. Sampling Methods | 抽样方法
A simple random sample gives every member of the population an equal chance of being selected, usually using a random number generator. It is unbiased but may not cover all subgroups well. Stratified sampling divides the population into distinct strata (e.g., year groups) and takes a random sample from each stratum proportional to its size. This guarantees representation of all key groups. Systematic sampling selects members at regular intervals from an ordered list (e.g., every 10th name). It is quick but can introduce bias if a hidden pattern exists. Bias occurs when a sample is not representative of the population. Common sources include convenience sampling, voluntary response and non-response bias. Remember: ‘A good sample is like a mini‑population.’
简单随机抽样（Simple random sample） 让总体中的每一个成员都有相等的机会被选中，通常借助随机数生成器实现。它无偏但可能无法很好地覆盖所有子群。分层抽样（Stratified sampling） 先将总体分成互不重叠的层（如年级），然后在每一层内按其规模比例随机抽取样本。这能确保所有重要群体都有代表。系统抽样（Systematic sampling） 从一个有序列表中以固定间隔选取成员（例如每第 10 个名字）。它执行快捷，但如果存在隐藏的周期性模式可能引入偏差。偏差（Bias） 是指样本不能代表总体。常见来源包括便利抽样、自愿回应和无回应偏差。请记住：“一个好的样本就像总体的微缩模型。”
9. Probability Terminology | 概率术语
An experiment is any process that produces outcomes. A single outcome is one possible result. The sample space (often denoted S) is the set of all possible outcomes. An event is a subset of the sample space. Theoretical probability assumes all outcomes are equally likely: P(A) = number of favourable outcomes / total number of outcomes. Relative frequency (experimental probability) is found by repeating an experiment many times: relative frequency = number of successful trials / total number of trials. As the number of trials increases, relative frequency tends to approach theoretical probability. Complementary events: P(not A) = 1 – P(A). Use tree diagrams for multi-stage experiments.
试验（Experiment） 是产生结果的过程。一个 结果（Outcome） 是某一种可能发生的结局。样本空间（Sample space）（常记作 S）是所有可能结果的集合。事件（Event） 是样本空间的一个子集。理论概率（Theoretical probability） 假设所有结果等可能：P(A) = 有利结果数 / 总结果数。相对频数（Relative frequency），又称实验概率，是将试验重复多次后得到的比例：相对频数 = 成功次数 / 试验总次数。随着试验次数增加，相对频数趋于理论概率。对立事件满足：P(非A) = 1 – P(A)。多阶段试验可以使用树状图来求概率。
10. Key Notation & Symbols at a Glance | 关键符号速览
Σ (capital sigma) – Summation: adds up all the values that follow.
Σ（大写西格玛） – 求和符号：将其后所有数值相加。
x̄ (x‑bar) – Sample mean: the average of a set of sample data.
x̄（x‑bar） – 样本平均数：一组样本数据的平均值。
μ (mu) – Population mean: the average of an entire population.
μ（缪） – 总体平均数：整个总体的平均值。
σ (sigma) – Population standard deviation.
σ（西格玛） – 总体标准差。
s – Sample standard deviation.
s – 样本标准差。
s² – Sample variance.
s² – 样本方差。
n – Sample size (number of observations in a sample).
n – 样本容量（样本中观测值的个数）。
N – Population size.
N – 总体大小。
Q₁, Q₂, Q₃ – First (lower) quartile, second quartile (median), third (upper) quartile.
Q₁, Q₂, Q₃ – 第一（下）四分位数、第二四分位数（中位数）、第三（上）四分位数。
IQR – Interquartile range = Q₃ − Q₁.
IQR – 四分位距 = Q₃ − Q₁。
P(A) – Probability of event A occurring.
P(A) – 事件 A 发生的概率。
fd – Frequency density = frequency / class width.
fd – 频数密度 = 频数 ÷ 组距。
r – Pearson’s product-moment correlation coefficient (strength and direction of a linear relationship).
r – 皮尔逊积差相关系数（衡量线性关系的强度和方向）。
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