Formula and Theorem Quick Reference Handbook | 公式定理速查手册

📚 Formula and Theorem Quick Reference Handbook | 公式定理速查手册

This handbook provides a concise summary of the key formulas, definitions and theorems you need for Year 11 CIE Statistics. It covers measures of centre and spread, probability, counting techniques, binomial and normal distributions, correlation and regression. Use it as a quick revision aid before tests and exams.

本手册为你总结了 Year 11 CIE 统计课程中最核心的公式、定义和定理，涵盖中心趋势、离散程度、概率、计数原理、二项分布与正态分布、相关与回归等内容，适合考试前快速查阅和复习。

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

The arithmetic mean (or average) is the sum of all data values divided by the number of values.

算术平均数（均值）是所有数据值之和除以数据个数。

Mean, x̄ = Σx / n

The median is the middle value when the data are arranged in order. For n values, if n is odd, the median is the (n+1)/2 th value; if n is even, it is the average of the n/2 th and (n/2 + 1) th values.

中位数是将数据排序后位于中间位置的值。若数据个数 n 为奇数，中位数为第 (n+1)/2 个值；若 n 为偶数，则为第 n/2 和第 (n/2+1) 个值的平均数。

The mode is the value that occurs most frequently. A data set can have one mode, more than one mode (multimodal) or no mode at all.

众数是出现次数最多的值。一组数据可以有一个众数、多个众数（多峰）或没有众数。

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2. Measures of Dispersion | 离散程度的度量

The range is the difference between the largest value and the smallest value.

极差是最大值与最小值之差。

Range = max(x) – min(x)

The interquartile range (IQR) measures the spread of the middle 50% of data. It is the difference between the upper quartile Q₃ and the lower quartile Q₁.

四分位距（IQR）衡量中间 50% 数据的离散程度，它是上四分位数 Q₃ 与下四分位数 Q₁ 之差。

IQR = Q₃ – Q₁

Variance measures the average squared deviation from the mean. For a population of size N with mean μ:

方差衡量数据与均值之差的平方的平均值。对于容量为 N、均值为 μ 的总体：

σ² = Σ(x – μ)² / N

For a sample of size n with mean x̄, the unbiased estimate of population variance uses (n – 1); however, in CIE IGCSE Statistics the formula often used is:

对于容量为 n、均值为 x̄ 的样本，总体方差的无偏估计使用 (n – 1)；但 CIE IGCSE 统计考试中常用以下公式：

s² = Σ(x – x̄)² / n

Standard deviation is the square root of variance. It has the same units as the original data.

标准差是方差的平方根，其单位与原数据相同。

s = √[Σ(x – x̄)² / n]

The five-number summary consists of minimum, Q₁, median, Q₃ and maximum. It is used to construct box‑and‑whisker diagrams.

五数综合概括包括最小值、Q₁、中位数、Q₃ 和最大值，用于绘制箱线图。

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3. Weighted and Combined Means | 加权平均与合并均值

When data values carry different importance, a weighted mean is calculated using assigned weights w.

当数据值具有不同重要性时，需使用指定的权重 w 计算加权平均数。

Weighted mean = Σ(wx) / Σw

To find the combined mean of two or more sets, multiply each group's mean by its size, sum these products and divide by the total number of items.

求两个或多个数据集的合并均值，需将各组均值乘以该组样本量，求和后再除以总样本量。

Combined mean = (n₁x̄₁ + n₂x̄₂ + ...) / (n₁ + n₂ + ...)

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4. Basic Probability Rules | 基本概率法则

The probability of an event A is a number between 0 and 1, where 0 means impossible and 1 means certain.

事件 A 的概率是 0 到 1 之间的一个数，0 表示不可能，1 表示必然发生。

The complement rule states that the probability of 'not A' is 1 minus P(A).

互补法则：事件 '非 A' 的概率等于 1 减去 P(A)。

P(A') = 1 – P(A)

For any two events A and B, the addition rule is:

对于任意两个事件 A 和 B，加法法则为：

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

If A and B are mutually exclusive, they cannot occur together, so P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).

若 A 与 B 互斥（不能同时发生），则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。

If A and B are independent, the occurrence of one does not affect the probability of the other, and the multiplication rule becomes:

若 A 与 B 独立，一个事件的发生不影响另一个事件的概率，此时乘法法则为：

P(A ∩ B) = P(A) × P(B)

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

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred.

条件概率 P(A|B) 是指在事件 B 已经发生的条件下，事件 A 发生的概率。

P(A|B) = P(A ∩ B) / P(B)

The multiplication rule for any two events uses conditional probability:

任意两个事件的乘法法则使用条件概率：

P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)

Tree diagrams help visualise multi-stage experiments. Probabilities along each branch multiply, and probabilities of final outcomes add.

树状图有助于可视化多阶段试验。沿各分支的概率相乘，最终结果的概率相加。

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6. Counting Principles: Permutations and Combinations | 计数原理：排列与组合

The factorial of a positive integer n is the product of all positive integers from 1 to n.

正整数 n 的阶乘是所有从 1 到 n 的正整数的乘积。

n! = n × (n – 1) × (n – 2) × ... × 2 × 1

A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is:

排列是指按特定顺序排列对象。从 n 个不同对象中选取 r 个的排列数为：

nPr = n! / (n – r)!

A combination is a selection of objects without regard to order. The number of combinations of n distinct objects taken r at a time is:

组合是不考虑顺序的对象选择。从 n 个不同对象中选取 r 个的组合数为：

nCr = n! / [r! (n – r)!]

Use permutations when the order matters (e.g. race positions); use combinations when only the set matters (e.g. selecting a committee).

当顺序重要时使用排列（如比赛名次）；当仅与集合有关时使用组合（如选举委员会）。

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7. The Binomial Distribution | 二项分布

sts of n independent trials, each with the same probability of success p and failure q = 1 – p.

二项试验由 n 次独立试验组成，每次试验的成功概率 p 相同，失败概率 q = 1 – p。

If a random variable X follows a binomial distribution, we write X ~ B(n, p). The probability of obtaining exactly r successes is:

若随机变量 X 服从二项分布，记为 X ~ B(n, p)。得到恰好 r 次成功的概率为：

P(X = r) = nCr × p^r × q^{n – r}

The mean (expected value) and variance of a binomial distribution are:

二项分布的均值（期望）和方差为：

Mean, E(X) = np

Variance, Var(X) = npq

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8. The Normal Distribution | 正态分布

The normal distribution is a continuous, bell‑shaped distribution described by its mean μ and standard deviation σ.

正态分布是一种连续的钟形分布，由均值 μ 和标准差 σ 描述。

To find probabilities for any normal variable X ~ N(μ, σ²), standardise it to the standard normal variable Z using:

为了求任意正态变量 X ~ N(μ, σ²) 的概率，需将其标准化为标准正态变量 Z：

Z = (X – μ) / σ

Φ(z) = P(Z < z). Symmetry properties can help obtain probabilities for negative z-values.

然后使用标准正态分布表求 Φ(z) = P(Z < z)。对称性质可帮助获得负 z 值的概率。

Approximately 68% of values lie within 1σ of the mean, 95% within 2σ and 99.7% within 3σ (the empirical rule).

大约 68% 的数值落在均值的一个标准差范围内，95% 落在两个标准差范围内，99.7% 落在三个标准差范围内（经验法则）。

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9. Scatter Diagrams and Correlation | 散点图与相关

A scatter diagram displays the relationship between two variables. The pattern reveals the direction (positive/negative), form (linear/non‑linear) and strength of correlation.

散点图展示两个变量之间的关系。图形模式能反映相关的方向（正/负）、形状（线性/非线性）和强度。

The product‑moment correlation coefficient r (Pearson's r) measures the strength of linear correlation:

积差相关系数 r（皮尔逊相关系数）衡量线性相关的强度：

r = Σ(x – x̄)(y – ȳ) / √[Σ(x – x̄)² Σ(y – ȳ)²]

r always lies between –1 and 1. r = 1 indicates perfect positive correlation, r = –1 perfect negative correlation, and r = 0 no linear correlation.

r 始终介于 –1 与 1 之间。r = 1 表示完全正相关，r = –1 表示完全负相关，r = 0 表示没有线性相关。

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10. Spearman's Rank Correlation Coefficient | 斯皮尔曼等级相关系数

Spearman's rank correlation coefficient r_s assesses the strength of a monotonic relationship using ranked data. It does not require the relationship to be linear.

斯皮尔曼等级相关系数 r_s 使用排序数据评估单调关系的强度，不要求关系是线性的。

r_s = 1 – 6Σd² / [n(n² – 1)]

Here, d is the difference between the ranks of each pair, and n is the number of paired observations. r_s also ranges from –1 to +1.

式中 d 为每对数据的等级之差，n 为成对观测值的数量。r_s 的取值范围也在 –1 到 +1 之间。

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11. Linear Regression (Least Squares Regression Line) | 线性回归（最小二乘回归线）

. The slope b is found by minimising the sum of squared vertical deviations from the line.

y 对 x 的回归直线方程为 y = a + bx。斜率 b 通过最小化各点到直线的纵向偏差平方和求得。

b = Σ(x – x̄)(y – ȳ) / Σ(x – x̄)²

The intercept a is determined from the fact that the regression line passes through the mean point (x̄, ȳ):

截距 a 根据回归直线经过均值点 (x̄, ȳ) 这一条件确定：

a = ȳ – b x̄

This line can be used to predict values of y for given x, but care should be taken with extrapolation beyond the data range.

该直线可用来根据 x 值预测 y 值，但对

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