IB Core Statistics: Data, Probability, and Distributions | IB 核心统计：数据、概率与分布

📚 IB Core Statistics: Data, Probability, and Distributions | IB 核心统计：数据、概率与分布

Welcome to the IB Core Statistics revision guide. This article gathers essential statistical concepts that every IB Mathematics student must master, covering data handling, probability theory, discrete and continuous distributions, and bivariate analysis. Whether you are following the Analysis and Approaches or Applications and Interpretation pathway, a firm grasp of these topics will strengthen both your internal assessment and final examination performance.

欢迎来到 IB 核心统计复习指南。本文汇集了每位 IB 数学学生必须掌握的核心统计概念，涵盖数据处理、概率论、离散与连续分布以及双变量分析。无论你修读的是分析与方法还是应用与解释课程，扎实掌握这些主题都会增强你的内部评估和最终考试成绩。

1. Types of Data and Sampling | 数据类型与抽样

Data can be classified as qualitative (categorical) or quantitative (numerical). Quantitative data is further split into discrete (countable, e.g., number of students) and continuous (measurable, e.g., height in cm). Understanding the data type determines which statistical techniques are appropriate.

数据可分为定性（分类）和定量（数值）两类。定量数据又分为离散（可计数，如学生人数）和连续（可测量，如身高 cm）。理解数据类型决定了适用的统计方法。

Reliable conclusions require representative samples. Simple random sampling gives each member of the population an equal chance. Stratified sampling divides the population into subgroups and samples proportionally, preserving key characteristics. Convenience sampling is easier but often biased.

可靠的结论需要代表性样本。简单随机抽样让总体中每个成员有同等机会被抽中。分层抽样先将总体分成子群再按比例抽样，以保留关键特征。便利抽样虽然容易操作，但常带有偏差。

A sample must be large enough to reduce variability and avoid undercoverage. IB exam questions frequently ask you to identify sampling methods and discuss potential sources of bias.

样本必须足够大以减少变异并避免覆盖不足。IB 试题经常要求你识别抽样方法并讨论潜在的偏差来源。

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

Central tendency describes the typical value in a data set. The mean, median, and mode each give a different measure of 'centre'. The sample mean is the sum of all observations divided by the count:

集中趋势描述数据集的典型值。平均数、中位数和众数分别提供不同的“中心”度量。样本平均数是所有观察值之和除以个数：

x̄ = (Σxᵢ) / n

The median is the middle value when data are ordered. For an even number of values, it is the average of the two middle numbers. The median resists extreme outliers, while the mean is pulled towards them.

中位数是数据排序后位于中间的值。若数据个数为偶数，则为中间两个数的平均值。中位数能抵抗极端异常值，而平均数会被异常值拉向它们。

The mode is the most frequent value and is particularly useful for categorical data. In a symmetric distribution the mean, median and mode coincide; in a skewed distribution they diverge.

众数是最常出现的值，对分类数据尤其有用。在对称分布中，平均数、中位数和众数重合；在偏态分布中它们则分离。

3. Measures of Dispersion | 离散程度的度量

Dispersion measures how spread out the data are. The range (maximum – minimum) is simple but sensitive to outliers. The interquartile range (IQR = Q₃ – Q₁) covers the middle 50% of data and is robust to outliers.

离散程度衡量数据的分散情况。极差（最大值 – 最小值）简单但易受异常值影响。四分位距（IQR = Q₃ – Q₁）覆盖中间 50% 的数据，对异常值稳健。

The variance and standard deviation provide numerical summaries that use all data points. For a sample, the variance s² is calculated with divisor (n–1):

方差和标准差提供利用所有数据点的数值概括。样本方差 s² 使用除数 (n–1) 计算：

s² = Σ(xᵢ – x̄)² / (n – 1)

The standard deviation s is the square root of the variance. A small standard deviation indicates data clustered around the mean; a large one signals high variability.

标准差 s 是方差的平方根。标准差小表示数据聚集在均值周围；大则表明变异性高。

Outliers can be identified using the rule: any point below Q₁ – e Q₃ + 1.5×IQR is a suspected outlier. You must know how to apply this rule on a box plot.

异常值可用以下规则识别：任何低于 Q₁ – 1.5×IQR 或高于 Q₃ + 1.5×IQR 的数据点被视为可疑异常值。你必须知道如何在箱线图上应用此规则。

4. Graphical Representations | 图形表示

Graphs summarise data visually. A histogram displays the frequency distribution of a continuous variable; the area of each bar is proportional to the frequency. Frequency density is used when class widths are unequal.

图形以视觉方式概括数据。直方图显示连续变量的频率分布；每个条形的面积与频率成比例。当组距不相等时需使用频率密度。

A box-and-whisker plot (box plot) shows the minimum, Q₁, median, Q₃ and maximum. It highlights the central tendency, spread, and possible outliers at a glance. The cumulative frequency curve visualises cumulative totals and helps estimate medians and percentiles.

箱线图（盒须图）显示最小值、Q₁、中位数、Q₃ 和最大值。它能一览无余地突出集中趋势、离散程度和可能的异常值。累积频率曲线则可视化累积总数，并帮助估计中位数和百分位数。

IB exams regularly ask you to interpret these graphs, sketch them from data, or compare distributions using shape, centre, and spread. Always label axes and comment on skewness.

IB 考试经常要求你解释这些图形、根据数据绘制图形，或从形状、中心和离散程度比较分布。务必标注坐标轴并评论偏态。

5. Introduction to Probability | 概率入门

Probability is the measure of how likely an event is to occur, expressed as a number between 0 and 1. The sample space (S) lists all possible outcomes. An event A is a subset of the sample space, and its complement A' is everything not in A.

概率是对事件发生可能性的度量，用 0 到 1 之间的数字表示。样本空间（S）列出所有可能的结果。事件 A 是样本空间的子集，其补集 A′ 是 A 之外的所有结果。

number of favourable outcomes) / (total number of outcomes). The addition rule handles unions:

对于等可能结果，P(A) = （有利结果数）/（总结果数）。加法法则处理并集：

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive (cannot happen together), P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

若 A 与 B 互斥（不能同时发生），P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

Venn diagrams and tables are powerful tools for organising outcomes and calculating probabilities. Practice translating word problems into these visual structures.

维恩图和表格是组织结果和计算概率的强大工具。练习将文字题转化为这些可视化结构。

6. Conditional Probability and Tree Diagrams | 条件概率与树状图

Conditional probability P(A | B) is the probability that A occurs given that B has occurred. It is defined as:

条件概率 P(A | B) 是在 B 已发生的条件下 A 发生的概率。其定义为：

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0

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This leads to the multiplication rule: P(A ∩ B) = P(A) × P(B | h complete branch, the joint probability is the product of the probabilities on that path.

由此引出乘法法则：P(A ∩ B) = P(A) × P(B | A)。树状图能展示顺序事件，其中条件概率沿分支相乘。在每个完整分支末端，联合概率是该路径上概率的乘积。

Two events are independent if P(A | B) = P(A), or equivalently P(A ∩ B) = P(A) × P(B). Always check independence using these conditions.

若满足 P(A | B) = P(A) 或等价条件 P(A ∩ B) = P(A) × P(B)，则两个事件独立。务必使用这些条件检验独立性。

IB problems often combine conditional probability with 'given that' wording. Highlight keywords and draw the tree or table to avoid confusion.

IB 问题常将条件概率与“已知……”的措辞结合。突出关键词并画出树状图或表格以避免混淆。

7. Discrete Random Variables | 离散随机变量

probability. The probability distribution must satisfy 0 ≤ P(X = x) ≤ 1 and ΣP(X = x) = 1.

离散随机变量 X 可取一组可计数的值，每个值有指定的概率。概率分布必须满足 0 ≤ P(X = x) ≤ 1 且 ΣP(X = x) = 1。

The expected value E(X) is the theoretical mean of X. It is calculated as:

期望值 E(X) 是 X 的理论均值，其计算公式为：

E(X) = Σ x · P(X = x)

Variance measures the spread of the distribution: Var(X) = E(X²) – where E(X²) = Σ x² · P(X = x). The standard deviation is the square root of the variance.

方差衡量分布的离散程度：Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ x² · P(X = x)。标准差为方差的平方根。

You should be able to construct a probability distribution table from a word problem and use the expectation to make decisions in contexts such as games of chance.

你应能根据文字题构建概率分布表，并利用期望值在诸如机会游戏的情境中做出决策。

8. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. If X follows a binomial distribution, we write X ~ B(n, p).

二项分布用于描述在固定次数的独立试验中成功的次数，每次试验的成功概率 p 相同。若 X 服从二项分布，我们记作 X ~ B(n, p)。

The probability of obtaining exactly k successes in n trials is:

在 n 次试验中获得恰好 k 次成功的概率为：

P(X = k) = ⁿCₖ · pᵏ · (1 – p)ⁿ⁻ᵏ

where ⁿCₖ is the binomial coefficient 'n choose k'. The mean and variance of a binomial distribution are E(X) = np and Var(X) = np(1 – p).

其中 ⁿCₖ 是二项式系数“n 选 k”。二项分布的均值与方差为 E(X) = np 和 Var(X) = np(1 – p)。

You will use your GDC to calculate binomial probabilities, cumulative probabilities, and inverse binomial values. Recognise the binomial setting: fixed n, constant p, independence, and two outcomes (success/failure).

你将使用 GDC 计算二项概率、累积概率和逆二项值。要能识别二项分布的前提：n 固定、p 恒定、独立性和两种结果（成功/失败）。

9. The Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution with a symmetric bell-shaped curve. It is fully characterised by its mean μ and standard deviation σ: X ~ N(μ, σ²). The total area under the curve equals 1.

正态分布是一种连续型概率分布，具有对称的钟形曲线。它完全由均值 μ 和标准差 σ 描述：X ~ N(μ, σ²)。曲线下的总面积为 1。

The empirical rule states that approximately 68% of observations lie within μ ± σ, 95% within μ ± 2σ, and 99.7% should be able to sketch the curve and shade regions corresponding to given probabilities.

经验法则指出：约 68% 的观测值落在 μ ± σ 内，95% 在 μ ± 2σ 内，99.7% 在 μ ± 3σ 内。你应能画出曲线并涂色标示给定概率对应的区域。

To compare values from different normal distributions, standardise using the z-score:

要比较来自不同正态分布的值，需用 z 分数进行标准化：

z = (x – μ) / σ

The standard normal distribution Z ~ N(0, 1) allows you to find probabilities and inverse normal values with a GDC. Always draw a diagram, label the mean, and clearly state the probability you are finding.

标准正态分布 Z ~ N(0, 1) 可让你用 GDC 求出概率和逆正态值。始终画出图示，标注均值，并清晰说明你正在求的概率。

10. Bivariate Data and Correlation | 双变量数据与相关性

Bivariate data involve two variables measured on the same individuals. A scatter plot is the first step to visualise the relationship. Look for direction (positive or negative), form (linear or non‑linear), and strength.

双变量数据涉及对同一个体测量的两个变量。散点图是可视化关系的第一步。观察方向（正或负）、形状（线性或非线性）和强度。

tion coefficient r measures the strength and direction of a linear relationship. r ranges from – ate strong linear correlation, while values near 0 suggest weak or no linear relationship.

皮尔逊积矩相关系数 r 衡量线性关系的强度和方向。r 的取值范围是 –1 到 +1；接近 ±1 的值表示强线性相关，接近 0 的值表示弱或无线性关系。

It is essential to note that correlation does not imply causation. A high correlation between two variables could be due to a lurking third variable or coincidence. IB examiners often ask you to interpret r in context and comment on reliability.

必须注意，相关性并不意味着因果关系。两个变量之间的高度相关可能源于潜藏的第三变量或巧合。IB 考官经常要求你在上下文中解释 r，并评论可靠性。

+ bx. Use it only within the range of the observed data to make predictions, and beware of extrapolation.

当线性模型合适时，你的 GDC 可计算出最小二乘回归线方程 y = a + bx。只可在观测数据范围内使用它进行预测，并谨防外推。

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