📚 Year 10 Cambridge Statistics: Formula & Theorem Quick Reference Guide | Year 10 Cambridge 统计：公式定理速查手册
This article provides a concise yet comprehensive collection of key formulas and theorems for Year 10 Cambridge Statistics. It is designed as a quick revision aid to help you recall essential concepts, from measures of central tendency and dispersion to probability distributions, regression and index numbers. Each formula is presented with a brief explanation and its counterpart in Chinese.
本文为 Year 10 Cambridge 统计课程提炼了一份简明而全面的公式定理速查手册。内容涵盖集中趋势、离散程度、概率分布、回归分析及指数等核心知识，并以中英对照的方式逐条呈现，方便快速复习与记忆。
1. Measures of Central Tendency | 集中趋势的度量
The arithmetic mean x̄ of a set of n observations x₁, x₂, …, xₙ is the sum divided by the number of items.
x̄ = (x₁ + x₂ + … + xₙ) / n = ∑x / n
对于包含 n 个观测值 x₁, x₂, …, xₙ 的数据集，算术平均数 x̄ 等于总和除以项数。
The median is the middle value when data are arranged in order. If n is odd, it 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.
Median position: (n + 1) / 2
中位数是将数据排序后位于中间的数值。若 n 为奇数，即第 (n+1)/2 个值；若 n 为偶数，则是第 n/2 与第 (n/2 + 1) 个值的平均数。
The mode is the value that occurs most frequently in a data set. A set may have one mode, more than one mode, or no mode at all.
Mode = most frequent value
众数是数据集中出现次数最多的值。一组数据可能有一个众数、多个众数或无众数。
2. Measures of Dispersion | 离散程度的度量
The range is the simplest measure of spread, calculated as the difference between the largest and the smallest observations.
Range = xₘₐₓ – xₘᵢₙ
极差是最简单的离散量数，由最大值减去最小值得到。
The interquartile range (IQR) measures the spread of the middle 50% of the data. It is defined as the difference between the upper quartile Q₃ and the lower quartile Q₁.
IQR = Q₃ – Q₁
四分位距 (IQR) 衡量中间 50% 数据的分散程度，定义为上四分位数 Q₃ 与下四分位数 Q₁ 之差.
For a sample, the variance is the mean of the squared deviations from the sample mean, using n-1 as divisor. The standard deviation is its square root.
s² = ∑(x – x̄)² / (n – 1)
s = √[∑(x – x̄)² / (n – 1)]
对于样本，方差是各数据与样本均值离差平方的平均数，分母为 n-1。标准差则是方差的平方根。
If the data represent a whole population, divide by N instead of n-1 and denote the variance by σ² and the standard deviation by σ.
σ² = ∑(x – μ)² / N, σ = √σ²
若数据代表整个总体，则分母用 N 而不是 n-1，方差记作 σ²，标准差记作 σ。
3. Estimation with Grouped Data | 分组数据的估计
When data are grouped into classes, the mean is estimated by multiplying each class midpoint m by its frequency f, summing, and dividing by total frequency.
x̄ ≈ ∑(f × m) / ∑f
当数据以组距形式呈现时，均值的估计方法是将每组的组中值 m 乘以相应的频数 f，求和后除以总频数。
The median for grouped data is estimated by linear interpolation within the median class.
Median ≈ L + ( (n/2 – CF) / fₘ ) × 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，fₘ 为中位数所在组的频数，w 为组距。
分组数据的中位数通过在中位数所在组内进行线性插值来估计。
The mode for grouped data is estimated using the boundaries and frequencies of the modal class and its neighbours.
Mode ≈ L + ( (fₘ – f₁) / (2fₘ – f₁ – f₂) ) × w
其中 fₘ 为众数所在组的频数，f₁ 为前一组的频数，f₂ 为后一组的频数。
分组数据的众数通过利用众数组及其前后组的频数与组界来估计。
4. Basic Probability Rules | 概率基本法则
For any two events A and B, the addition rule gives the probability that either event occurs.
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
对于任意两个事件 A 和 B，加法法则给出了至少一个事件发生的概率。
If A and B are mutually exclusive (they cannot occur together), then P(A ∩ B) = 0 and the rule simplifies.
P(A ∪ B) = P(A) + P(B)
若 A 与 B 互斥（不能同时发生），则 P(A ∩ B) = 0，加法法则简化如上。
For independent events, the probability of both occurring is the product of their individual probabilities.
P(A ∩ B) = P(A) × P(B)
对于独立事件，两者同时发生的概率等于各自概率的乘积。
The conditional probability of A given B is the probability that A occurs when B is known to have occurred.
P(A | B) = P(A ∩ B) / P(B), given P(B) > 0
条件概率 P(A | B) 表示在已知事件 B 发生的前提下事件 A 发生的概率。
5. Permutations and Combinations | 排列与组合
The factorial of a positive integer n is the product of all positive integers from 1 to n.
n! = n × (n-1) × (n-2) × … × 2 × 1, 0! = 1
正整数 n 的阶乘是从 1 到 n 的所有正整数的乘积，并规定 0! = 1。
A permutation is an ordered arrangement of r objects taken from n distinct objects.
P(n, r) = n! / (n – r)!
排列是指从 n 个不同元素中取出 r 个元素的有序安排。
A combination is a selection of r objects from n distinct objects where order does not matter.
C(n, r) = n! / [r!(n – r)!]
组合是从 n 个不同元素中选取 r 个元素的无序选择。
6. Binomial Distribution | 二项分布
If a discrete random variable X follows a binomial distribution with parameters n (number of trials) and p (probability of success), the probability of exactly r successes is given by the binomial formula.
P(X = r) = C(n, r) × p^r × (1 – p)^(n – r), r = 0, 1, 2, …, n
若离散随机变量 X 服从参数为 n（试验次数）和 p（成功概率）的二项分布，则取得恰好 r 次成功的概率由二项式公式给出。
The mean and variancePublished by TutorHao | Year 10 统计 Revision Series | aleveler.com
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