GCSE CAIE Statistics: Formula & Theorem Quick Reference | GCSE CAIE 统计：公式定理速查手册

📚 GCSE CAIE Statistics: Formula & Theorem Quick Reference | GCSE CAIE 统计：公式定理速查手册

This revision handbook collects all essential formulas, definitions and theorems you need for the CAIE GCSE Statistics examination. Each section is presented as a concise bilingual reference to help you revise efficiently.

这本复习手册汇集了 CAIE GCSE 统计学考试所需的所有基本公式、定义和定理。每个部分均以简洁的双语对照形式呈现，帮助你高效复习。

1. Measures of Central Tendency | 集中趋势量度

The mean for a set of n observations x₁, x₂, …, xₙ is the sum of the values divided by n. For grouped data, the estimated mean uses midpoints multiplied by frequencies.

对于 n 个观测值 x₁, x₂, …, xₙ，平均值等于数值之和除以 n。对于分组数据，估计平均值使用组中值乘以频数。

Mean: x̄ = Σx / n Grouped mean: x̄ = Σfx / Σf

The median is the middle value when data are ordered. For n observations, the position of the median is (n + 1)/2. If n is even, the median is the mean of the two middle values.

中位数是将数据排序后的中间值。对于 n 个观测值，中位数的位置为 (n + 1)/2。若 n 为偶数，中位数是中间两个数值的平均值。

The mode is the value that occurs most frequently. A data set may have no mode, one mode (unimodal), two modes (bimodal) or more (multimodal).

众数是出现频率最高的数值。一组数据可能没有众数、有一个众数（单峰）、两个众数（双峰）或多个众数（多峰）。

2. Measures of Spread | 离散程度量度

The range is the difference between the largest and smallest values. The interquartile range (IQR) is Q₃ – Q₁, the spread of the middle 50% of the data.

极差是最大值与最小值之差。四分位距 (IQR) 为 Q₃ – Q₁，即中间 50% 数据的分布范围。

Range = max – min IQR = Q₃ – Q₁

Variance measures the average squared deviation from the mean. For a population, divide by n; for a sample, divide by n-1. Standard deviation is the square root of the variance.

方差衡量各数据与平均值之差的平方的平均值。对于总体，除以 n；对于样本，除以 n-1。标准差是方差的平方根。

σ² = Σ(x – μ)² / N s² = Σ(x – x̄)² / (n – 1)

An alternative formula for variance is Σx²/n – (Σx/n)², which is often easier for calculation without having to find each deviation.

方差的另一种计算公式为 Σx²/n – (Σx/n)²，这样无需先计算每个偏差，常使计算更简便。

3. Quartiles and Percentiles | 四分位数与百分位数

The lower quartile Q₁ is the median of the lower half of the data; the upper quartile Q₃ is the median of the upper half. When using cumulative frequency graphs, Q₁ is the value at 25% of the total frequency.

下四分位数 Q₁ 是数据下半部分的中位数；上四分位数 Q₃ 是数据上半部分的中位数。使用累积频数图时，Q₁ 对应于总频数的 25% 处的数值。

The kth percentile Pₖ is the value below which k% of the observations fall. For instance, the 90th percentile is found at 90% of cumulative frequency.

第 k 百分位数 Pₖ 表示有 k% 的观测值低于该值。例如，第 90 百分位数位于累积频数的 90% 处。

To find quartiles from a stem-and-leaf diagram or ordered list, use positions: Q₁ position = (n + 1)/4, Q₃ position = 3(n + 1)/4. If the position is not an integer, interpolate.

要从茎叶图或排序列表中求四分位数，可使用位置公式：Q₁ 位置 = (n + 1)/4，Q₃ 位置 = 3(n + 1)/4。若位置不是整数，则进行插值。

4. Frequency Distributions | 频数分布

In a frequency table with class intervals, the class midpoint is (lower bound + upper bound) / 2. For open-ended classes, use reasonable assumptions or the given bound.

在含组距的频数表中，组中值 = (下限 + 上限) / 2。对于开放式组距，需使用合理假设或题目给出的边界。

Cumulative frequency is the running total of frequencies. To estimate the median from a grouped table, plot cumulative frequency against upper class boundaries and read at half the total frequency.

累积频数是频数的累加求和。要从分组数据表估计中位数，需将累积频数相对于组上限作图，并在总频数的一半处读取数值。

5. Probability Fundamentals | 概率基础

The probability of an event A is P(A) = number of favourable outcomes / total number of equally likely outcomes. Probability always lies between 0 and 1 inclusive.

事件 A 的概率 P(A) = 有利结果数 / 所有等可能结果总数。概率始终介于 0 和 1 之间（含 0 和 1）。

P(A) = n(A) / n(S) 0 ≤ P(A) ≤ 1

The complement rule states that P(not A) = 1 – P(A). Events A and B are mutually exclusive if they cannot occur together, so P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).

互补规则：P(非 A) = 1 – P(A)。若事件 A 和 B 互斥（不能同时发生），则 P(A ∩ B) = 0，且 P(A ∪ B) = P(A) + P(B)。

For independent events, P(A ∩ B) = P(A) × P(B). For dependent events, use conditional probability: P(A ∩ B) = P(A) × P(B|A).

对于独立事件，P(A ∩ B) = P(A) × P(B)。对于不独立事件，使用条件概率：P(A ∩ B) = P(A) × P(B|A)。

6. Conditional Probability & Tree Diagrams | 条件概率与树形图

Conditional probability P(A|B) is the probability of A given that B has occurred. The formula is P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率 P(A|B) 表示在事件 B 发生的条件下事件 A 的概率。公式为 P(A|B) = P(A ∩ B) / P(B)，且 P(B) > 0。

Tree diagrams help organise multi-stage probabilities. Multiply along branches for combined events; add probabilities of mutually exclusive paths. Probabilities on branches after the first level are conditional.

树形图有助于梳理多阶段概率。沿分支相乘得到联合事件的概率；将互斥路径的概率相加。第一级后的分支概率均为条件概率。

7. Expectation and Fair Games | 期望值与公平游戏

The expectation or expected value E(X) of a discrete random variable X is the sum of each value multiplied by its probability: E(X) = Σ [x · P(X = x)].

离散随机变量 X 的期望值（或期望）E(X) 为每个值乘以其概率的总和：E(X) = Σ [x · P(X = x)]。

A game is fair if the expected gain (or net gain) is zero. Otherwise, the expected gain is the average profit or loss per game in the long run.

若期望收益（或净收益）为零，则游戏是公平的。否则，期望收益表示长期平均每次游戏的利润或损失。

8. Binomial Distribution | 二项分布

A binomial distribution arises from n independent trials, each with two outcomes (success/failure) and constant probability of success p. The random variable X is the number of successes.

二项分布产生于 n 次独立试验，每次试验只有两种结果（成功/失败），且成功的概率 p 恒定。随机变量 X 表示成功的次数。

X ~ B(n, p)

The probability of exactly r successes is given by the binomial probability formula. Here nCr (or C(n, r)) is the binomial coefficient.

恰好获得 r 次成功的概率由二项概率公式给出。其中 nCr（或 C(n, r)）为二项系数。

P(X = r) = nCr × pʳ × (1 – p)ⁿ⁻ʳ

The mean and variance of a binomial distribution are μ = np and σ² = np(1 – p). The standard deviation is √[np(1 – p)].

二项分布的均值与方差分别为 μ = np 和 σ² = np(1 – p)。标准差为 √[np(1 – p)]。

9. Scatter Diagrams & Correlation | 散点图与相关性

A scatter diagram plots one variable against another to show the relationship. Positive correlation means both variables increase together; negative correlation means one increases as the other decreases.

散点图将一个变量相对于另一个变量绘图，以显示关系。正相关表示两个变量共同增加；负相关表示一个增加而另一个减少。

Correlation is not causation. The strength of linear correlation can be judged visually or by calculating the product-moment correlation coefficient r.

相关性不代表因果性。线性相关性的强度可通过目测或计算积矩相关系数 r 加以判断。

10. Product-Moment Correlation Coefficient | 积矩相关系数

The Pearson correlation coefficient r measures the strength and direction of a linear relationship between two variables x and y. Its value is always between -1 and 1.

皮尔逊相关系数 r 衡量两个变量 x 和 y 之间线性关系的强度和方向。其值始终介于 -1 和 1 之间。

r = [ nΣxy – Σx Σy ] / √[ (nΣx² – (Σx)²) (nΣy² – (Σy)²) ]

A value close to 1 indicates strong positive correlation, close to -1 strong negative correlation, and close to 0 weak or no linear correlation. Always interpret r in the context of the data.

接近 1 表示强正相关，接近 -1 表示强负相关，接近 0 表示弱线性相关或无线性相关。应始终结合数据背景解释 r 值。

11. Equation of Regression Line | 回归直线方程

The regression line of y on x (used to predict y from x) is of the form y = a + bx, where b is the gradient and a is the y-intercept.

y 对 x 的回归直线（用于由 x 预测 y）形式为 y = a + bx，其中 b 为斜率，a 为 y 轴截距。

b = [ nΣxy – Σx Σy ] / [ nΣx² – (Σx)² ] a = (Σy – b Σx) / n

The regression line always passes through the mean point (x̄, ȳ). Use it only for interpolation within the range of the given data; extrapolation can be unreliable.

回归直线始终经过均值点 (x̄, ȳ)。回归线只能用于给定数据范围内的内插；外推可能不可靠。

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

Spearman’s rank correlation coefficient rₛ is used for monotonic relationships, especially with ordinal data or non-linear trends. Rank each data set separately and find the differences d between the ranks of each pair.

斯皮尔曼等级相关系数 rₛ 用于单调关系，尤其适用于顺序数据或非线性趋势。分别对每组数据排序，并计算每对排名之差 d。

rₛ = 1 – [ 6 Σd² / n(n² – 1) ]

If there are tied ranks, the formula is only an approximation. rₛ also ranges from -1 to +1, with similar interpretations of strength and direction as Pearson’s r.

若存在并列排名，该公式仅为近似值。rₛ 的取值范围同样为 -1 到 +1，其强度和方向的解释与皮尔逊 r 类似。

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GCSE CAIE Statistics: Formula & Theorem Quick Reference | GCSE CAIE 统计：公式定理速查手册