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

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

This article provides a comprehensive summary of the key topics covered in the Year 12 CAIE Statistics course (Paper 1). You will find essential concepts, formulas, and exam tips for mastering data representation, probability, discrete random variables, and the binomial distribution.

本文全面总结 Year 12 CAIE 统计课程（卷1）的核心知识点，涵盖数据表示、概率、离散随机变量及二项分布等重要概念、公式与应考技巧。

---

1. Representing Data | 数据表示

A stem-and-leaf diagram splits each data value into a stem and a leaf, preserving the original numbers while revealing the shape of the distribution. Always provide a key explaining the place value, and use ordered stems for clarity. Back-to-back stem-and-leaf diagrams can compare two related data sets effectively.

茎叶图将每个数据值分为茎和叶，在保留原始数据的同时显示分布形态。务必给出说明位值的图例，并将茎排序。背靠背茎叶图可有效比较两组相关数据。

Box-and-whisker plots display the five-number summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃), and maximum. The box spans the IQR, and whiskers extend to the furthest data point within 1.5 × IQR from the quartiles. Any point beyond these whiskers is treated as an outlier and plotted individually.

箱线图展示五数概括：最小值、下四分位数(Q₁)、中位数(Q₂)、上四分位数(Q₃)和最大值。箱子覆盖四分位距，须线从箱子延伸至距四分位数1.5倍IQR内的最远数据点。任何落在须线之外的点视为异常值并单独标出。

In a histogram, the vertical axis measures frequency density (frequency ÷ class width) so that the area of each bar is proportional to the frequency of that class. When class widths are unequal, frequency density must be used to avoid distorting the visual impression.

直方图中纵轴表示频率密度（频率÷组距），因此每个柱形的面积与对应组的频率成正比。当组距不相等时，必须使用频率密度，以免视觉呈现失真。

A cumulative frequency graph (ogive) plots the running total of frequencies against the upper class boundary of each interval. It allows you to estimate the median (at 50% of the total frequency), quartiles, and any other percentile by reading off the corresponding data values.

累积频率图（肩形图）将累积频率随各组上界描点连线。可利用该图读取总频率50%处估计中位数、下四分位数和上四分位数以及任意百分位数。

---

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

The arithmetic mean (x̄) for a set of n values is x̄ = Σx / n. For grouped data, use the midpoint of each class interval as the representative value.

算术均值（x̄）对n个数据值计算为 x̄ = Σx / n。对于分组数据，使用各组的组中值作为代表值。

The median is the middle value when the data are ordered. If n is odd, the median is the single middle value at position (n+1)/2; if n is even, it is the average of the two middle values. On a cumulative frequency graph, locate the median at the horizontal 50% mark.

中位数是排序后数据的中间值。若n为奇数，中位数是位于 (n+1)/2 位置的单个值；若n为偶数，则是中间两个值的平均。在累积频率图上，从50%的累积频率处读取中位数。

The mode is the most frequently occurring value. A data set may have one mode (unimodal), two modes (bimodal), multiple modes, or no mode at all. For grouped data, the modal class is the class interval with the highest frequency density.

众数是最常出现的值。数据可以是单峰、双峰、多峰或无众数。对于分组数据，众数组是频率密度最高的组区间。

When data are skewed, the median is often a better measure of central tendency than the mean because it is not pulled by extreme values. In a positively skewed distribution, mean > median > mode; in a negatively skewed distribution, mean < median < mode.

当数据偏斜时，中位数通常比均值更能代表集中趋势，因为它不被极端值拉偏。在正偏斜分布中，均值 > 中位数 > 众数；在负偏斜分布中，均值 < 中位数 < 众数。

---

3. Measures of Spread | 离散程度的度量

Range = maximum – minimum. Although simple, it is highly sensitive to outliers and only uses two data points.

极差 = 最大值 – 最小值。尽管计算简单，但它对异常值非常敏感且仅使用两个数据点。

The interquartile range (IQR) = Q₃ − Q₁. It measures the spread of the middle 50% of the data and is a resistant measure because extreme values do not affect the quartiles.

四分位距 (IQR) = Q₃ − Q₁。它衡量中间50%数据的离散程度，属于耐抗性度量，因为异常值不影响四分位数。

Variance quantifies the average squared deviation from the mean. For a set of n values (treated as the population), σ² = Σ(x − μ)² / n. A computationally efficient form is σ² = (Σx² / n) − μ². For grouped data, replace x with the class midpoint.

方差量化数据与均值偏差平方的平均值。对于n个数据（视为总体），σ² = Σ(x − μ)² / n。常用计算式为 σ² = (Σx² / n) − μ²。分组数据使用组中值替换x。

Standard deviation is the positive square root of variance: σ = √[Σ(x − μ)² / n]. It is expressed in the same units as the original data, making it easier to interpret than variance.

标准差是方差的正平方根：σ = √[Σ(x − μ)² / n]。它与原始数据单位相同，因此比方差更易于解释。

When comparing two data sets, quoting both a measure of central tendency and a measure of spread is essential. For symmetric distributions without outliers, use the mean and standard deviation; for skewed data or data with outliers, prefer the median and IQR.

比较两组数据时，必须同时给出集中趋势和离散程度的度量。对无异常值的对称分布，可使用均值和标准差；对偏斜数据或含异常值的数据，优先使用中位数和四分位距。

---

4. Probability Basics | 概率基础

Probability is a numerical measure between 0 and 1 inclusive that quantifies the likelihood of an event. 0 indicates impossibility; 1 indicates certainty.

概率是介于0和1之间的数值度量，用于量化事件发生的可能性。0表示不可能，1表示必然发生。

For a sample space with equally likely outcomes, P(A) = (number of outcomes in A) / (total number of outcomes). This is the basis of classical probability.

对于等可能结果的样本空间，P(A) = 事件A包含的结果数 / 总结果数。这是古典概率的基础。

The addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, they cannot occur simultaneously, so P(A ∩ B) = 0 and the rule simplifies to P(A ∪ B) = P(A) + P(B).

Published by TutorHao | Year 12 统计 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）