IB Mathematics: Statistics Key Points Revision | IB 数学：统计考点精讲

📚 IB Mathematics: Statistics Key Points Revision | IB 数学：统计考点精讲

Statistics in the IB Mathematics course is a powerful tool set for analysing data, quantifying uncertainty, and making evidence‑based decisions. This revision guide distills the core concepts from descriptive measures to hypothesis testing and regression, helping you build a solid understanding for both exam success and practical application.

IB 数学中的统计学是分析数据、量化不确定性并作出基于证据的决策的强大工具。这份精讲将核心概念层层提炼，从描述性度量到假设检验与回归分析，帮助你在考试中脱颖而出，同时建立扎实的实际应用能力。

1. Descriptive Statistics | 描述性统计

Measures of central tendency summarise a data set with a single value. The mean is the arithmetic average of all observations, the median is the middle value when data are ordered, and the mode is the most frequently occurring value.

集中趋势度量用一个数值概括数据集。均值是所有观察值的算术平均，中位数是将数据排序后处于中间位置的值，众数是出现频率最高的值。

For grouped data, the mean is estimated using class midpoints and frequencies. The median class is found by cumulative frequency. Weighted mean is used when different data points carry different importance, computed as Σ(wx) / Σw.

对于分组数据，均值通过组中值与频数来估计，中位数所在组由累积频数确定。当不同数据点重要性不同时，使用加权均值，计算公式为 Σ(wx) / Σw。

Mean formula: x̄ = (Σx) / n; for grouped data: x̄ ≈ Σ(f · midpoint) / n
均值公式：x̄ = (Σx) / n；分组数据：x̄ ≈ Σ(f · 组中值) / n
Median position: (n+1)/2 (ungrouped); locate the median class (grouped)
中位数位置：(n+1)/2（未分组）；定位中位数所在组（分组）

2. Measures of Dispersion | 离散程度度量

Dispersion tells us how spread out the data are. The range is the difference between the maximum and minimum. The interquartile range (IQR = Q₃ − Q₁) measures the spread of the middle 50%.

离散程度表明数据的分散程度。极差是最大值与最小值之差。四分位距（IQR = Q₃ − Q₁）衡量中间50%数据的分布范围。

Variance is the average squared deviation from the mean, and standard deviation is its square root. For a sample, the variance is s² = Σ(x − x̄)² / (n − 1). Population variance is σ² = Σ(x − μ)² / N.

方差是各数据与均值之差的平方的平均值，标准差是它的平方根。对于样本，方差 s² = Σ(x − x̄)² / (n − 1)；总体方差 σ² = Σ(x − μ)² / N。

Measure	Symbol	Formula
Sample variance	s²	Σ(x − x̄)² / (n − 1)
Population variance	σ²	Σ(x − μ)² / N

3. Box Plots and Outliers | 箱线图与异常值

A box plot (box-and-whisker plot) displays the five‑number summary: minimum, Q₁, median (Q₂), Q₃, and maximum. It quickly reveals the centre, spread, and skewness of a distribution.

箱线图展示五数概括：最小值、第一四分位数 Q₁、中位数 Q₂、第三四分位数 Q₃ 和最大值。它能迅速呈现分布的中心、离散程度和偏态。

Outliers are observations that lie far from the main body of data. By the 1.5×IQR rule, a point is an outlier if it is below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR. Whiskers extend to the furthest point within these bounds.

异常值是远离数据主体的观测值。按照1.5×IQR准则，若一点小于 Q₁ − 1.5×IQR 或大于 Q₃ + 1.5×IQR 则为异常值。须线延伸至这些界限内最远的点。

4. Probability Fundamentals | 概率基础

Probability quantifies the likelihood of an event. For a sample space S, P(A) lies between 0 and 1. The addition rule states P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Events A and B are mutually exclusive if they cannot occur together, giving P(A ∩ B) = 0.

概率量化事件发生的可能性。对于样本空间 S，P(A) 介于 0 和 1 之间。加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若事件互斥，则它们不能同时发生，P(A ∩ B) = 0。

Conditional probability P(A|B) is the probability of A given B, defined as P(A ∩ B) / P(B). Events are independent if P(A ∩ B) = P(A)P(B), or equivalently P(A|B) = P(A).

条件概率 P(A|B) 表示在 B 发生的条件下 A 的概率，定义为 P(A ∩ B) / P(B)。若事件独立，则满足 P(A ∩ B) = P(A)P(B)，或等价于 P(A|B) = P(A)。

5. Discrete Random Variables | 离散随机变量

A discrete random variable X takes a countable set of values. Its probability distribution lists each value with its associated probability P(X = x). All probabilities sum to 1.

离散随机变量 X 取可数个值。它的概率分布列出每个值及其相应概率 P(X = x)。所有概率之和为 1。

The expected value (mean) is E(X) = Σ x·P(X = x). The variance is Var(X) = E(X²) − [E(X)]², or equivalently Σ (x − μ)² P(X = x). Standard deviation is √Var(X).

期望值（均值）E(X) = Σ x·P(X = x)。方差 Var(X) = E(X²) − [E(X)]²，或等价于 Σ (x − μ)² P(X = x)。标准差为 √Var(X)。

E(aX + b) = a E(X) + b Var(aX + b) = a² Var(X)

6. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in n independent trials, each with constant probability of success p. Conditions: fixed n, each trial independent, two possible outcomes, constant p.

二项分布模型描述了在 n 次独立试验中成功的次数，每次成功概率 p 恒定。条件：n 固定、每次试验独立、两种可能结果、p 恒定。

The probability of exactly k successes is P(X = k) = nCk · pᵏ · (1−p)ⁿ⁻ᵏ. The mean is E(X) = np, and the variance is Var(X) = np(1−p).

恰好得到 k 次成功的概率为 P(X = k) = nCk · pᵏ · (1−p)ⁿ⁻ᵏ。均值 E(X) = np，方差 Var(X) = np(1−p)。

X ~ B(n, p) P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ

7. Normal Distribution | 正态分布

The normal distribution is a continuous, bell‑shaped curve fully described by its mean μ and standard deviation σ. Notation: X ~ N(μ, σ²). Approximately 68% of data lie within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

正态分布是连续、钟形的曲线，完全由均值 μ 和标准差 σ 决定。记作 X ~ N(μ, σ²)。约 68% 的数据落在 μ ± σ 内，95% 落在 μ ± 2σ 内，99.7% 落在 μ ± 3σ 内。

To calculate probabilities, standardise using Z = (X − μ) / σ, giving Z ~ N(0, 1). Use the standard normal table or GDC to find Φ(z). For inverse normal, find the value x such that P(X < x) = given probability.

计算概率时，使用 Z = (X − μ) / σ 进行标准化，得到 Z ~ N(0, 1)。利用标准正态分布表或计算器求 Φ(z)。对于逆正态，求 x 使得 P(X < x) = 给定概率。

8. Sampling and the Central Limit Theorem | 抽样与中心极限定理

When drawing samples of size n from a population with mean μ and standard deviation σ, the sample mean x̄ has a sampling distribution with mean μ and standard error σ/√n.

从均值为 μ、标准差为 σ 的总体中抽取样本容量为 n 的样本时，样本均值 x̄ 的抽样分布具有均值 μ 和标准误 σ/√n。

The Central Limit Theorem (CLT) states that for sufficiently large sample sizes (usually n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population distribution.

中心极限定理指出，当样本容量足够大时（通常 n ≥ 30），样本均值的抽样分布近似正态，无论总体分布形态如何。

This allows the use of normal probabilities for confidence intervals and hypothesis tests about the mean when n is large or the population is normal.

这使得在 n 较大或总体正态时，可用正态概率来构建均值的置信区间和进行假设检验。

9. Confidence Intervals | 置信区间

A confidence interval gives a range of plausible values for a population parameter. For the population mean μ with known population standard deviation σ, a (1−α)100% confidence interval is x̄ ± z* × (σ/√n).

置信区间给出了总体参数一个合理的数值范围。对于总体标准差 σ 已知的总体均值 μ，(1−α)100% 置信区间为 x̄ ± z* × (σ/√n)。

When σ is unknown, use the sample standard deviation s and the t‑distribution: x̄ ± t* × (s/√n) with degrees of freedom n−1. For a population proportion p, the confidence interval is p̂ ± z* × √[p̂(1−p̂)/n].

当 σ 未知时，使用样本标准差 s 和 t 分布：x̄ ± t* × (s/√n)，自由度 n−1。对于总体比例 p，置信区间为 p̂ ± z* × √[p̂(1−p̂)/n]。

CI for μ (σ known): x̄ ± z* · (σ/√n)

10. Hypothesis Testing | 假设检验

A hypothesis test assesses evidence against a null hypothesis H₀. The alternative hypothesis H₁ can be one‑sided or two‑sided. The test statistic measures how far the sample result is from the null value, standardised by its standard error.

假设检验评估反对零假设 H₀ 的证据。备择假设 H₁ 可以是单侧或双侧。检验统计量衡量样本结果与零假设值的差异，并以标准误标准化。

The p‑value is the probability of obtaining a result at least as extreme as the observed one, assuming H₀ is true. If p‑value < significance level α, reject H₀. Common test statistics include z, t, and χ².

p 值是在 H₀ 为真的条件下，得到至少与实际观测一样极端结果的概率。若 p 值 < 显著性水平 α，则拒绝 H₀。常用的检验统计量包括 z、t 和 χ²。

For a test about one mean with unknown σ, use t = (x̄ − μ₀) / (s/√n) with df = n−1. For two‑sample t‑tests, be careful about equal or unequal variances.

关于单一均值且 σ 未知的检验，使用 t = (x̄ − μ₀) / (s/√n)，自由度 df = n−1。对于双样本 t 检验，注意方差相等或不等的情形。

11. Bivariate Data and Linear Regression | 双变量数据与线性回归

A scatter plot displays the relationship between two quantitative variables. Pearson’s correlation coefficient r measures the strength and direction of a linear relationship; it ranges from −1 to 1.

散点图展示两个定量变量之间的关系。皮尔逊相关系数 r 衡量线性关系的强度和方向，取值范围为 −1 到 1。

The least‑squares regression line y = a + bx minimises the sum of squared residuals. Slope b = r · (s_y / s_x), and intercept a = ȳ − b x̄. The coefficient of determination r² indicates the proportion of variation in y explained by x.

最小二乘回归线 y = a + bx 使得残差平方和最小。斜率 b = r · (s_y / s_x)，截距 a = ȳ − b x̄。决定系数 r² 表示 y 的变异中可由 x 解释的比例。

y = a + bx b = r · (s_y / s_x) a = ȳ − b x̄

12. Chi‑squared Tests (HL) | 卡方检验 (HL)

The chi‑squared (χ²) test is used for categorical data. The goodness‑of‑fit test checks if observed frequencies match an expected distribution. The test statistic is χ² = Σ (O − E)² / E.

卡方 (χ²) 检验用于分类数据。拟合优度检验检查观测频数是否与期望分布相符。检验统计量为 χ² = Σ (O − E)² / E。

The test for independence (contingency table) examines whether two categorical variables are associated. The expected frequency for each cell is (row total × column total) / grand total. Degrees of freedom are (rows−1)(cols−1).

独立性检验（列联表）考察两个分类变量是否有关联。每个单元格的期望频数为 (行合计 × 列合计) / 总计。自由度为 (行数−1)×(列数−1)。

If the p‑value is less than the significance level, we conclude there is evidence of an association or that the model does not fit well.

若 p 值小于显著性水平，我们得出结论：有证据表明存在关联或模型拟合不佳。

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IB Mathematics: Statistics Key Points Revision | IB 数学：统计 考点精讲