📚 CAIE IGCSE Statistics: Quick Guide to Key Vocabulary | CAIE IGCSE 统计：核心词汇速记指南
Mastering the language of statistics is half the battle. For CAIE IGCSE Statistics, a solid grasp of essential terms helps you interpret questions accurately, present data clearly, and draw valid conclusions. This guide provides concise definitions and quick memory aids for the most important terms you will meet in your Year 11 exam, from basic measures to probability and data display.
掌握统计学的语言是成功的一半。在 CAIE IGCSE 统计考试中，扎实掌握核心术语能帮助你准确理解题意、清晰展示数据，并得出有效结论。本指南针对 Year 11 考试中最重要的术语，提供简洁定义与快速记忆技巧，涵盖从基本度量到概率与数据呈现的全部内容。
1. Population and Sample | 总体和样本
The population is the entire group of individuals or items that we are interested in studying. For example, all students in a school or every light bulb produced by a factory.
总体是我们感兴趣研究的全部个体或项目的集合。例如，一所学校的所有学生，或某工厂生产的所有灯泡。
A sample is a subset of the population selected to represent it. We use samples because it is often impractical or too expensive to collect data from every member of the population. A good sample should be random and large enough to avoid bias.
样本是从总体中选出的、用以代表总体的一个子集。我们使用样本，是因为从总体的每个成员收集数据往往不切实际或成本过高。一个好的样本应当随机选取，且足够大以避免偏差。
2. Measures of Central Tendency: Mean, Median, Mode | 集中趋势度量：平均数、中位数、众数
Mean (x̄) is the arithmetic average. For ungrouped data, x̄ = Σx / n, where Σx is the sum of all values and n is the number of values. For grouped data, use the midpoints of class intervals.
平均数（x̄）是算术平均值。未分组数据公式为 x̄ = Σx / n，其中 Σx 是所有数值之和，n 是数值个数。分组数据需使用组中值计算。
Median is the middle value when the data are arranged in order. For n odd, median is the (n+1)/2 th value; for n even, it is the average of the n/2 th and (n/2 +1)th values. In a cumulative frequency diagram, it is the value at the 50th percentile.
中位数是将数据按大小排序后位于中间的值。若 n 为奇数，中位数是第 (n+1)/2 个值；若 n 为偶数，则是第 n/2 和 (n/2+1) 个值的平均数。在累积频率图中，中位数对应 50% 位置处的数值。
Mode is the value that occurs most frequently. A data set can have one mode (unimodal), two modes (bimodal), or more. For grouped data, the modal class is the class interval with the highest frequency.
众数是出现次数最多的值。一组数据可以有一个众数（单峰）、两个众数（双峰）或更多。对于分组数据，众数所在组是频率最高的那个组距。
| Measure | Advantage | Disadvantage |
|---|---|---|
| Mean | Uses all data values | Affected by extreme values (outliers) |
| Median | Not affected by outliers | Ignores most actual values |
| Mode | Easy to find; can be used for non-numeric data | May not exist or may not be unique; ignores data |
Comparison illustrated with a simple data set: 2, 3, 3, 5, 25. Mean = 7.6, Median = 3, Mode = 3. The mean is pulled upward by the outlier 25.
用一个简单数据集来对比：2, 3, 3, 5, 25。平均数 = 7.6，中位数 = 3，众数 = 3。异常值 25 把平均数拉高了。
3. Measures of Spread: Range and Interquartile Range | 分散度量：极差和四分位数间距
Range = maximum value − minimum value. It gives the total spread but is sensitive to outliers.
极差 = 最大值 − 最小值。它反映整体散布情况，但容易受异常值影响。
Interquartile range (IQR) = upper quartile (Q3) − lower quartile (Q1). It measures the spread of the middle 50% of data and is resistant to outliers. Q1 is the median of the lower half, Q3 is the median of the upper half.
四分位数间距 (IQR) = 上四分位数 (Q₃) − 下四分位数 (Q₁)。它衡量中间 50% 数据的散布程度，且不受异常值影响。Q₁ 是下半部分的中位数，Q₃ 是上半部分的中位数。
Quick trick: Think of IQR as the range of the ‘middle box’ in a box-and-whisker plot.
记忆窍门：把 IQR 想象成箱线图中“中间箱子”的宽度。
4. Standard Deviation and Variance | 标准差和方差
Variance is the average of the squared differences from the mean. Standard deviation (σ for a population, s for a sample) is the square root of variance. It tells you, on average, how far each value lies from the mean.
方差是各数据与平均数之差的平方的平均值。标准差（总体用 σ，样本用 s）是方差的平方根。它告诉你每个数据值平均来说离平均数有多远。
For ungrouped population data:
σ = √[ Σ(x – x̄)² / n ]
未分组总体数据公式：
σ = √[ Σ(x – x̄)² / n ]
For a sample, divide by (n − 1) instead of n: s = √[ Σ(x – x̄)² / (n − 1) ]. CAIE often provides the formula, but knowing when to use each matters.
样本标准差分母用 (n − 1)：s = √[ Σ(x – x̄)² / (n − 1) ]。CAIE 通常会给出公式，但知道何时用哪个仍然重要。
A larger standard deviation indicates greater spread. A small standard deviation means data points are clustered near the mean.
标准差越大，数据越分散；标准差小，数据点密集在平均数附近。
5. Frequency Tables and Cumulative Frequency | 频率表与累积频率
A frequency table shows how often each value (or each class interval) occurs. It is the foundation for calculating mean, median, mode, and drawing charts.
频率表显示每个数值（或每个组距）出现的次数。它是计算平均数、中位数、众数和绘制图表的基础。
Cumulative frequency is the running total of frequencies. Adding the frequencies step by step gives the cumulative frequency. The final cumulative frequency equals the total number of observations.
累积频率是频率的逐次累加。逐步将频率相加便得到累积频率，最终的累积频率等于观测总数。
We use a cumulative frequency diagram (a smooth curve, not a polygon) to estimate the median, quartiles, and percentiles. Plot the upper class boundary against cumulative frequency, then read values from the curve.
我们用累积频率图（光滑曲线，非折线）来估计中位数、四分位数和百分位数。绘制时用每个组的上限对累积频率描点，然后从曲线上读取对应的值。
6. Data Display: Histogram, Bar Chart, Pie Chart | 数据展示：直方图、条形图、饼图
Bar chart: Used for categorical or discrete data. Bars are separated by equal gaps; bar height represents frequency. The order of bars can be changed.
条形图：用于分类数据或离散数据。条形之间有相等的间隔；条形高度代表频率。条形的顺序可以调换。
Histogram: Used for continuous data grouped into class intervals. There are no gaps between bars. The area of each bar is proportional to frequency. If class widths are unequal, frequency density = frequency / class width must be plotted on the vertical axis.
直方图：用于已分组的连续数据。条形之间没有间隔。每个条形的面积与频率成正比。若组距宽度不等，纵轴必须用频率密度（频率密度 = 频率 / 组距宽度）来表示。
Pie chart: Shows proportions of a whole. Each sector angle = (category frequency / total frequency) × 360°.
饼图：显示各部分占整体的比例。每个扇形的角度 = (类别频数 / 总数) × 360°。
Key exam pitfall: Confusing histograms and bar charts. Always check the horizontal axis and whether there are gaps.
常见考试陷阱：混淆直方图和条形图。务必检查横轴以及条形之间是否有间隔。
7. Scatter Diagrams and Correlation | 散点图与相关性
A scatter diagram displays paired numerical data (x, y). It helps to see whether there is a relationship between two variables.
散点图展示成对的数值数据 (x, y)。它帮助观察两个变量之间是否存在关系。
Correlation describes the strength and direction of a linear relationship. Positive correlation: as x increases, y tends to increase. Negative correlation: as x increases, y tends to decrease. No correlation: no clear pattern.
相关性描述线性关系的强度和方向。正相关：x 增大时，y 倾向于增大。负相关：x 增大时，y 倾向于减小。无相关：没有明显模式。
Correlation does not imply causation. Just because two variables move together does not mean one causes the other.
相关性不代表因果性。两个变量一起变化，并不意味着其中一个导致了另一个。
A line of best fit (trend line) can be drawn by eye through the points. It should have roughly equal numbers of points above and below the line. Use it to estimate values within the given data range (interpolation) but avoid extending beyond (extrapolation) as this can be unreliable.
最佳拟合线（趋势线）可以通过观察画出，使线上下的点数大致相等。可利用该线在已知数据范围内进行估计（插值），但应避免超出范围（外推），因为外推可能不可靠。
8. Basics of Probability | 概率基础
Probability (P) is a measure of how likely an event is to happen. It is a number between 0 (impossible) and 1 (certain). The probability of an event A is: P(A) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
概率（P）是对事件发生可能性的度量，其值介于 0（不可能）与 1（必然）之间。若所有可能结果等可能，事件 A 的概率为：P(A) = 有利结果数 / 所有可能结果总数。
The sample space is the set of all possible outcomes. An event is a subset of the sample space.
样本空间是所有可能结果的集合。事件是样本空间的一个子集。
The complement of event A, written A’, means ‘A does not happen’. P(A’) = 1 − P(A).
事件 A 的补集，写作 A’，意思是“A 不发生”。P(A’) = 1 − P(A)。
For two events, the addition rule: P(A or B) = P(A) + P(B) − P(A and B). This prevents double-counting outcomes that belong to both events.
对于两个事件，加法法则为：P(A 或 B) = P(A) + P(B) − P(A 和 B)。这能避免重复计算同时属于两个事件的结果。
9. Independent and Mutually Exclusive Events | 独立事件与互斥事件
Events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, getting a 3 and getting a 5 are mutually exclusive. Then P(A and B) = 0, so the addition rule simplifies to P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则称为互斥事件。例如掷骰子时，出现 3 点和出现 5 点互斥。此时 P(A 和 B) = 0，加法法则简化为 P(A 或 B) = P(A) + P(B)。
Events are independent if the occurrence of one does not affect the probability of the other. For independent events A and B, P(A and B) = P(A) × P(B).
如果一个事件的发生不影响另一个事件发生的概率，则称这两个事件独立。对于独立事件 A 和 B，P(A 和 B) = P(A) × P(B)。
A common exam question involves selecting items with or without replacement. ‘With replacement’ keeps events independent; ‘without replacement’ means probabilities change, so events are dependent.
常考的一种题型涉及有放回或无放回抽取。“有放回”时事件保持独立；“无放回”意味着概率会变化，事件因此是相关的。
Remember: ‘mutually exclusive’ deals with ‘OR’ (cannot both happen), while ‘independence’ deals with ‘AND’ (probabilities multiply).
记忆窍门：互斥性关系到“或”（不能同时发生），独立性关系到“且”（概率相乘）。
10. Sampling Methods | 抽样方法
Random sampling: Every member of the population has an equal chance of being selected. This can be achieved using random number generators or drawing names from a hat. It minimizes bias.
随机抽样：总体中的每个成员都有相等的机会被选中。可通过随机数生成器或抽签来实现，能最大程度减少偏差。
Stratified sampling: The population is divided into groups (strata) based on a characteristic (e.g., age, gender), and a random sample is taken from each stratum in proportion to its size. This ensures the sample reflects the population structure.
分层抽样：先将总体按某种特征（如年龄、性别）划分为不同层，然后从每一层中按其规模比例随机抽取样本。这样能保证样本反映总体结构。
Systematic sampling: Items are selected at regular intervals from an ordered list. For example, choosing every 10th person on a register. Quick and easy, but can introduce bias if there is a hidden pattern.
系统抽样：从有序列表中按固定间隔选取样本。例如从名册上每隔 9 人选取 1 人。方法简便快捷，但若存在隐藏规律，可能产生偏差。
Opportunity (convenience) sampling: Choosing individuals who are easily available. It is fast but often produces biased results and should be avoided in rigorous studies.
便利抽样：选取最容易接触到的个体。方法快速，但常常产生偏差，在严谨研究中应避免使用。
For your exam, be ready to describe a method, state its advantages and disadvantages, and identify potential sources of bias.
考试中要准备好描述一种抽样方法，说明其优缺点，并识别潜在的偏差来源。
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