📚 IGCSE CCEA Statistics: Formula & Theorem Quick Reference | IGCSE CCEA 统计公式定理速查手册
This handbook is a rapid revision aid for CCEA IGCSE Statistics, compiling the essential formulas, definitions and theorems you need to apply accurately in the exam. Each section pairs English and Chinese explanations, so it works equally well for last-minute review or for checking a method you are unsure about.
本手册是 CCEA IGCSE 统计学的快速复习资料，汇编了考试中需要准确应用的关键公式、定义与定理。每一部分都以中英对照的形式呈现，既适合考前冲刺复习，也适合随时核查某个不熟悉的方法。
The specification rewards precision: marks are lost when a formula is misremembered or applied to the wrong kind of data. Symbols follow the usual CCEA convention - x̄ for a sample mean, μ for a population mean, s for a sample standard deviation, σ for a population standard deviation, and P(A) for the probability of event A.
考纲非常看重准确性：公式记错，或把公式用在不合适的数据类型上，都会直接丢分。符号沿用 CCEA 的惯例：x̄ 表示样本均值，μ 表示总体均值，s 表示样本标准差，σ 表示总体标准差，P(A) 表示事件 A 发生的概率。
1. Measures of Central Tendency | 集中趋势的度量
For ungrouped data, the arithmetic mean (x̄) is the sum of all observations divided by the number of observations. The median is the central value once the data have been arranged in order, and the mode is the most frequently occurring value. For grouped data, class midpoints are used to approximate the mean.
对于未分组数据，算术平均数 (x̄) 是所有观测值之和除以观测值的个数。中位数是把数据排序后位于中间的值，众数是出现频率最高的数值。对于分组数据，则使用组中值来近似计算均值。
x̄ = Σx / n
If n is even, the median is the average of the n/2-th and (n/2 + 1)-th ordered values; if n is odd, it is the ((n + 1)/2)-th ordered value. Always order the data first - taking the middle item of an unordered list gives a meaningless answer.
如果 n 为偶数，中位数是第 n/2 个和第 (n/2 + 1) 个排序值的平均数；如果 n 为奇数，中位数就是第 ((n + 1)/2) 个排序值。务必先排序再看中间项——从未排序的数列中直接取中间项，得出的答案毫无意义。
Mean (grouped) = Σ(f × xₘ) / Σf
In a grouped frequency table, xₘ is the midpoint of each class, found by averaging the class boundaries. The mode becomes the modal class, that is, the class with the highest frequency. When two data sets are combined, the overall mean is the weighted mean (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂).
在分组频数表中，xₘ 是每一组的组中值，由组的上、下边界取平均得到。此时众数变为众数组，即频数最高的那一组。当两组数据合并时，总体均值是加权平均数 (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)。
2. Measures of Dispersion | 离散程度的度量
The simplest measure of spread is the range, obtained by subtracting the smallest value from the largest. The interquartile range (IQR) is more robust because it ignores the extreme 25% at each end, and it is defined as Q₃ − Q₁.
最简单的离散度量是极差，即最大值减去最小值。四分位距 (IQR) 更稳健，因为它忽略了两端各 25% 的极端数据，定义为 Q₃ − Q₁。
Range = max − min IQR = Q₃ − Q₁
For a population with mean μ and size N, the variance is the mean of the squared deviations and the standard deviation is its square root. The sample standard deviation s uses the sample mean x̄ and the divisor (n − 1), which gives an unbiased estimate of the population value.
对于均值为 μ、容量为 N 的总体，方差是偏差平方的平均值，标准差则是方差的平方根。样本标准差 s 使用样本均值 x̄ 并除以 (n − 1)，从而给出总体标准差的无偏估计。
σ² = Σ(x − μ)² / N σ = √[Σ(x − μ)² / N]
s² = Σ(x − x̄)² / (n − 1) s = √[Σ(x − x̄)² / (n − 1)]
ids subtracting the mean from every item: σ² = Σx² / N − μ², and similarly s² = [Σx² − n x̄ viation unchanged, while multiplying every value by k multiplies the standard deviation by |k|.
有一个计算捷径可以避免逐项减去均值：σ² = Σx² / N − μ²，同理 s² = [Σx² − n x̄²] / (n −
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