Year 12 CAIE Statistics: Core Knowledge Summary | Year 12 CAIE 统计：核心知识点梳理

📚 Year 12 CAIE Statistics: Core Knowledge Summary | Year 12 CAIE 统计：核心知识点梳理

This comprehensive guide covers all the essential topics in the Year 12 CAIE Statistics syllabus (S1), including data representation, measures of central tendency and variation, probability, permutations and combinations, discrete random variables, binomial and geometric distributions, and the normal distribution. Mastering these concepts is the key to success in your AS Level examination.

这份全面的指南涵盖了 Year 12 CAIE 统计学（S1）教学大纲中的所有核心主题，包括数据表示、集中趋势和离散程度的度量、概率、排列与组合、离散随机变量、二项分布与几何分布，以及正态分布。掌握这些概念是你在 AS 阶段考试中取得成功的关键。

1. Data Representation | 数据表示

Stem-and-leaf diagrams allow raw data to be displayed and sorted by stems and leaves, making it easy to find medians and quartiles. A key must always be provided to show how to read the plot.

茎叶图通过茎和叶展示原始数据并进行排序，便于找到中位数和四分位数。必须提供图例以说明如何读图。

Box-and-whisker plots, or box plots, illustrate the minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃) and maximum. They clearly show the spread and skewness; an outlier is typically defined as any value more than 1.5 × IQR below Q₁ or above Q₃.

箱线图（或盒须图）展示了最小值、下四分位数(Q₁)、中位数(Q₂)、上四分位数(Q₃)和最大值。它们清晰地显示了数据的散布和偏度；异常值通常定义为低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR 的任何值。

Histograms are used for continuous grouped data. The area of each bar is proportional to the frequency; therefore, the vertical axis must be frequency density, calculated as frequency ÷ class width. For unequal class widths, this is essential.

直方图用于连续分组数据。每个矩形的面积与频数成正比；因此，纵轴必须是频率密度，计算公式为 频数 ÷ 组距。对于不等组距的数据，这一点至关重要。

Cumulative frequency graphs, or ogives, can be drawn by plotting cumulative frequency against the upper class boundary. They are used to estimate medians, quartiles and percentiles, and to find inter-percentile ranges.

累积频率图（或折线图）可以通过绘制累积频率与组上限的对应关系得到。它们用于估计中位数、四分位数和百分位数，并求百分位数间距。

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

The mean (x̄) of a set of data is the sum of all values divided by the number of values. For grouped data, linear interpolation uses the midpoints of classes as an approximation. Alternatively, a coding method can simplify calculations: if y = (x – a)/b, then x̄ = a + b ȳ.

一组数据的均值 (x̄) 是所有数值之和除以数值个数。对于分组数据，使用各组的组中值进行线性插值可以给出近似值。此外，编码法可以使计算简化：若 y = (x – a)/b，则 x̄ = a + b ȳ。

The median is the middle value when data are ordered. For ungrouped data with an even number of values, it is the mean of the two central values. For grouped data, linear interpolation is used to estimate the median from the cumulative frequency table.

中位数是数据排序后位于中间的值。对于偶数值的未分组数据，它是两个中间值的平均值。对于分组数据，使用线性插值法从累积频率表中估计中位数。

The mode is the value that occurs most frequently. In a histogram, the modal class is the class with the highest frequency density, not necessarily the highest frequency.

众数是出现最频繁的值。在直方图中，众数所在组是频率密度最高的组，而不一定是频数最高的组。

3. Measures of Variation | 离散程度的度量

The range is the difference between the maximum and minimum values. The interquartile range (IQR) = Q₃ − Q₁ and describes the spread of the middle 50 % of the data, making it resistant to outliers.

极差是最大值与最小值的差。四分位距 (IQR) = Q₃ − Q₁，它描述了中间 50% 数据的离散程度，因此不受异常值影响。

Variance and standard deviation measure the average squared deviation from the mean. For ungrouped data, the variance s² = Σ(x − x̄)²/(n−1) for a sample; the CAIE syllabus commonly uses the formula Σx²/n − (x̄)² for a population or when all data are known. For grouped data, replace x with the class midpoint and weight by frequency.

方差和标准差衡量的是每个数据与均值的偏差平方的平均值。对于未分组数据，样本方差 s² = Σ(x − x̄)²/(n−1)；CAIE 大纲中当数据为总体或全部已知时通常使用公式 Σx²/n − (x̄)²。对于分组数据，用组中值代替 x，并乘以频数加权。

Standard deviation (σ) is the positive square root of variance. It is useful because it has the same units as the original data. Comparing coefficients of variation (standard deviation ÷ mean) can help assess relative spread.

标准差 (σ) 是方差的正平方根。它之所以常用，是因为它的单位与原数据相同。比较变异系数（标准差 ÷ 均值）有助于评估相对散布程度。

4. Probability Basics | 概率基础

The probability of an event A, P(A), satisfies 0 ≤ P(A) ≤ 1. The complement rule states P(A’) = 1 − P(A). For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B), since they cannot occur together.

事件 A 的概率 P(A) 满足 0 ≤ P(A) ≤ 1。互补规则为 P(A’) = 1 − P(A

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