IGCSE CIE Statistics: Quick Reference Handbook of Formulas and Theorems | IGCSE CIE 统计：公式定理速查手册

📚 IGCSE CIE Statistics: Quick Reference Handbook of Formulas and Theorems | IGCSE CIE 统计：公式定理速查手册

This quick reference guide brings together all the essential formulas, notation and key theorems required for the Cambridge IGCSE Statistics syllabus (0479). Use it to reinforce your understanding of data handling, probability, distributions and bivariate analysis. Each section presents the core rule in English first, followed by its Chinese translation, so you can revise confidently in either language.

这本速查手册汇集了剑桥 IGCSE 统计学（0479）所有必备公式、符号与核心定理，涵盖数据处理、概率、分布与双变量分析。每一条规则均先呈现英文解释，再给出对应的中文译文，方便你同时巩固双语概念，自信备考。

1. Types of Data and Charts | 数据类型与图表

Data can be qualitative (categorical) or quantitative (numerical). Quantitative data may be discrete (countable) or continuous (measurable). When constructing histograms for grouped continuous data, the vertical axis must represent frequency density.

数据可分为定性（分类）数据与定量（数值）数据。定量数据又分为离散型（可数）与连续型（可度量）。绘制分组连续数据的直方图时，纵轴必须表示频率密度。

Frequency density = Frequency / Class width

频率密度 = 频数 ÷ 组距

In cumulative frequency graphs, the plotted points are joined by a smooth curve, and the median, quartiles and percentiles can be estimated from the graph using interpolation.

在累积频数图中，各点用光滑曲线相连，中位数、四分位数与百分位数可通过插值从图中估算。

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

For ungrouped data, the mean is the sum of values divided by the number of items. The median is the middle value when data are ordered; the mode is the most frequent value.

对于未分组数据，平均数等于所有数值之和除以数据个数。中位数是排序后中间的值，众数是出现次数最多的值。

Mean x̄ = Σx / n

平均数 x̄ = Σx / n

For grouped frequency tables, an estimate of the mean uses the midpoints of classes:

对于分组频数表，可使用组中点估算平均数：

Estimated mean = Σfx / Σf

估算平均数 = Σfx / Σf

The median for grouped data is found using linear interpolation: Median = L + [ (n/2 – CF) / f ] * w, where L is the lower boundary of the median class, CF is the cumulative frequency before the class, f is the class frequency, and w is the class width.

分组数据的中位数通过线性插值求得：中位数 = L + [ (n/2 – CF) / f ] * w。其中 L 为中位数所在组的下限，CF 为该组之前的累积频数，f 为组频数，w 为组距。

3. Measures of Dispersion | 离散程度的度量

Range = maximum value – minimum value. The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁), resistant to extreme values.

极差 = 最大值 – 最小值。四分位距 (IQR) 是上四分位数 (Q₃) 与下四分位数 (Q₁) 之差，不易受极端值影响。

IQR = Q₃ – Q₁

四分位距 = Q₃ – Q₁

Variance and standard deviation measure the spread around the mean. For a population, use σ²; for a sample, use s². The standard deviation is the square root of the variance.

方差与标准差衡量数据在平均数周围的分散程度。总体方差用 σ²，样本方差用 s²。标准差为方差的平方根。

Population variance σ² = Σ(x – μ)² / N

总体方差 σ² = Σ(x – μ)² / N

Sample variance s² = Σ(x – x̄)² / (n – 1)

样本方差 s² = Σ(x – x̄)² / (n – 1)

For grouped data, the same formulas apply using midpoints x and frequencies f: σ² = Σf(x – μ)² / Σf.

对于分组数据，用组中点 x 与频数 f 代入同样公式：σ² = Σf(x – μ)² / Σf。

4. Basic Probability | 概率基础

Probability of an event A is a number between 0 and 1 inclusive. The complement rule states that the probability of A not occurring is 1 – P(A).

事件 A 的概率是介于 0 到 1（含）的数。互补规则表明 A 不发生的概率为 1 – P(A)。

P(A’) = 1 – P(A)

For mutually exclusive events, addition rule: P(A or B) = P(A) + P(B). For independent events, multiplication rule: P(A and B) = P(A) * P(B).

对于互斥事件，使用加法法则：P(A 或 B) = P(A) + P(B)。对于独立事件，使用乘法法则：P(A 与 B) = P(A) * P(B)。

Conditional probability is the probability of A given B: P(A|B) = P(A and B) / P(B). This formula rearranges to give the general multiplication rule.

条件概率是在 B 发生的前提下 A 发生的概率：P(A|B) = P(A 与 B) / P(B)。该公式变形可得一般乘法法则。

5. Permutations and Combinations | 排列与组合

The factorial n! is the product of all positive integers up to n: n! = n * (n-1) * … * 1. By definition, 0! = 1.

阶乘 n! 表示从 1 到 n 所有正整数的乘积：n! = n × (n-1) × … × 1。规定 0! = 1。

Permutations count arrangements where order matters. The number of ways to arrange r objects from n is:

排列用于顺序重要的情形。从 n 个中取 r 个排列的方法数为：

ⁿPᵣ = n! / (n – r)!

Combinations count selections where order does not matter. The number of ways to choose r objects from n is:

组合用于顺序无关的选择。从 n 个中选 r 个的组合数为：

ⁿCᵣ = n! / [r! (n – r)!]

6. Discrete Random Variables and Binomial Distribution | 离散随机变量与二项分布

For a discrete random variable X, the expected value E(X) and variance Var(X) are:

对于离散随机变量 X，期望值 E(X) 与方差 Var(X) 为：

E(X) = Σ x P(X = x)

Var(X) = E(X²) – [E(X)]² = Σ x² P(X = x) – μ²

A binomial distribution arises from a fixed number n of independent trials, each with the same probability of success p. Notation: X ~ B(n, p). The probability of exactly r successes is:

二项分布源自固定次数 n 的独立试验，每次成功概率为 p。记为 X ~ B(n, p)。恰好 r 次成功的概率为：

P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ

For the binomial distribution, mean = np and variance = np(1 – p).

二项分布的均值 = np，方差 = np(1 – p)。

7. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution symmetric about the mean μ, with spread determined by the standard deviation σ. The total area under the curve equals 1.

正态分布是一种连续型概率分布，关于均值 μ 对称，分散程度由标准差 σ 决定，曲线下总面积为 1。

To find probabilities, any normal variable X ~ N(μ, σ²) is standardised to Z ~ N(0, 1) using:

求概率时，将任意正态变量 X ~ N(μ, σ²) 标准化为标准正态变量 Z ~ N(0, 1)：

Z = (X – μ) / σ

Standard normal tables then give P(Z < z). For inverse problems, use the tables backwards to find the z-value corresponding to a given left-tail probability.

标准正态表提供 P(Z < z) 的值。对于逆向问题，可利用表格反查给定左侧概率对应的 z 值。

8. Bivariate Data: Correlation | 双变量数据：相关

Correlation measures the strength and direction of a linear relationship between two variables. Spearman’s rank correlation coefficient is a non-parametric measure based on ranks.

相关性衡量两个变量之间线性关系的强度和方向。Spearman 等级相关系数是一种基于排位的非参数度量。

rₛ = 1 – [6 Σd²] / [n(n² – 1)]

Here d is the difference between the ranks of each pair, and n is the number of paired observations. The value lies between -1 (perfect negative) and +1 (perfect positive).

其中 d 是每对数据的等级差值，n 是成对观测值的个数。系数值介于 -1（完全负相关）与 +1（完全正相关）之间。

9. Bivariate Data: Least Squares Regression | 双变量数据：最小二乘回归

The least squares regression line of y on x has the form y = a + b x, where b is the slope and a is the intercept. The slope is calculated as:

y 对 x 的最小二乘回归直线方程为 y = a + b x，其中 b 为斜率，a 为截距。斜率计算如下：

b = Sxy / Sxx

where Sxy = Σ(x – x̄)(y – ȳ) = Σxy – (Σx Σy)/n, and Sxx = Σ(x – x̄)² = Σx² – (Σx)²/n. The intercept a = ȳ – b x̄.

其中 Sxy = Σ(x – x̄)(y – ȳ) = Σxy – (Σx Σy)/n，Sxx = Σ(x – x̄)² = Σx² – (Σx)²/n。截距 a = ȳ – b x̄。

The line passes through the mean point (x̄, ȳ). Use the equation to estimate y for a given x (interpolation), but only within the range of the data.

回归直线经过均值点 (x̄, ȳ)。可用该方程对给定的 x 值估计 y（内插），但仅限于数据范围之内。

10. Time Series and Moving Averages | 时间序列与移动平均

A time series shows data collected at regular intervals. Four components are typically identified: trend, seasonal variation, cyclical variation and random (irregular) component.

时间序列显示按固定间隔采集的数据，通常可识别出四种成分：长期趋势、季节变动、循环变动与随机（不规则）成分。

Moving averages smooth out short-term fluctuations to reveal the trend. For odd period k, a k-point centred moving average is the mean of k consecutive values placed at the middle time point.

移动平均可平滑短期波动以反映趋势。对于奇数期 k，k 点中心移动平均是 k 个连续值的平均数，置于中间时点。

The seasonal effect for each period can be estimated by subtracting the trend from the actual value (additive model) or dividing by the trend (multiplicative model).

各时期的季节效应可通过从实际值中减去趋势（加法模型）或除以趋势（乘法模型）来估算。

11. Index Numbers | 指数

An index number measures the relative change in a variable compared to a base period. A simple price relative for a single commodity is:

指数衡量变量相对于基期的相对变化。单一商品的简单价比为：

Price relative = (Pₙ / P₀) * 100

价比 = (Pₙ / P₀) × 100

Weighted aggregate indices combine several items. The Laspeyres index uses base-period quantities as weights, while the Paasche index uses current-period quantities. For the Laspeyres price index:

加权综合指数合并多种商品。Laspeyres 指数以基期数量为权重，Paasche 指数以现期数量为权重。Laspeyres 价格指数公式为：

Laspeyres Price Index = (Σ Pₙ Q₀ / Σ P₀ Q₀) * 100

Laspeyres 价格指数 = (Σ Pₙ Q₀ / Σ P₀ Q₀) × 100

Changing the base or splicing indices allows comparison across different time periods.

更换基期或拼接指数可在不同时间段之间进行比较。

12. Sampling and Estimation Concepts | 抽样与估计概念

A statistic calculated from a sample is an estimator of the corresponding population parameter. The sample mean x̄ is an unbiased estimator of the population mean μ.

由样本计算出的统计量是对应总体参数的估计量。样本均值 x̄ 是总体均值 μ 的无偏估计量。

The distribution of sample means tends to be normal for large sample sizes (Central Limit Theorem), with standard error SE = σ / √n. This underpins confidence intervals and hypothesis testing.

当样本容量较大时，样本均值的分布趋向正态（中心极限定理），其标准误差 SE = σ / √n。这是置信区间与假设检验的基础。

For a population proportion p, the sample proportion p̂ = x / n has mean p and standard error √[p(1 – p) / n].

对于总体比例 p，样本比例 p̂ = x / n 的均值为 p，标准误差为 √[p(1 – p) / n]。

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IGCSE CIE Statistics: Quick Reference Handbook of Formulas and Theorems | IGCSE CIE 统计：公式定理速查手册