📚 Quick Reference Handbook for Key Formulas and Theorems in Statistics (Year 10 Cambridge) | 公式定理速查手册
This handbook provides a concise summary of the most important formulas and theorems you need for Year 10 Cambridge Statistics. From averages to probability distributions, keep this reference close for quick revision and exam success.
本手册为你提供十年级剑桥统计课程中最重要的公式和定理的简明总结。从平均数到概率分布，随身携带这份参考资料，方便快速复习，迎战考试。
1. Mean, Median, Mode and Range | 平均数、中位数、众数与极差
The arithmetic mean of a list of n numbers is the sum divided by n: x̄ = Σx / n. For a frequency table, the estimated mean is Σ(f × x) / Σf, where x is the midpoint of each class.
一组 n 个数的算术平均数是总和除以 n：x̄ = Σx / n。对于频数表，估计平均数为 Σ(f × x) / Σf，其中 x 是每组的组中值。
The median is the middle value when data are ordered. Its position is at (n + 1)/2. The mode is the value that appears most frequently. The range is the difference between the largest and smallest values: Range = max − min.
中位数是将数据排序后的中间值，位置为 (n + 1)/2。众数是出现次数最多的值。极差是最大值与最小值之差：极差 = 最大值 − 最小值。
2. Quartiles and Interquartile Range | 四分位数与四分位距
Quartiles split ordered data into four equal parts. The lower quartile Q₁ is at position (n + 1)/4, the median Q₂ at (n + 1)/2, and the upper quartile Q₃ at 3(n + 1)/4. The interquartile range is IQR = Q₃ − Q₁.
四分位数将排序后的数据分成四等份。下四分位数 Q₁ 位于 (n + 1)/4，中位数 Q₂ 位于 (n + 1)/2，上四分位数 Q₃ 位于 3(n + 1)/4。四分位距为 IQR = Q₃ − Q₁。
On a cumulative frequency graph, Q₁, Q₂ and Q₃ can be read directly by finding the values corresponding to 25%, 50% and 75% of the total frequency.
在累积频率图上，可直接通过找到总频数的 25%、50% 和 75% 对应的值来读取 Q₁、Q₂ 和 Q₃。
3. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of data around the mean. For a population of N values, σ² = Σ(x − μ)² / N. The standard deviation is σ = √[Σ(x − μ)² / N] or by the computational formula σ = √[(Σx² / N) − μ²].
方差度量数据围绕平均数的离散程度。对于一个拥有 N 个值的总体，σ² = Σ(x − μ)² / N。标准差为 σ = √[Σ(x − μ)² / N]，或使用计算式 σ = √[(Σx² / N) − μ²]。
For a sample of n values, we usually use s² = Σ(x − x̄)² / (n − 1), and the sample standard deviation is s = √[Σ(x − x̄)² / (n − 1)].
对于一个有 n 个值的样本，通常使用 s² = Σ(x − x̄)² / (n − 1)，样本标准差为 s = √[Σ(x − x̄)² / (n − 1)]。
4. Cumulative Frequency | 累积频率
A cumulative frequency table adds each frequency to the sum of the previous ones. The resulting running total can be plotted against the upper class boundary to form a smooth cumulative frequency curve.
累积频率表将每个频数与之前的所有频数累加。得到的累计总数可对着各组的上限绘制成一条光滑的累积频率曲线。
From the curve you can estimate the median, quartiles and percentiles. For instance, the median is the x‑value at half the total cumulative frequency.
从曲线上可以估计中位数、四分位数和百分位数。例如，中位数为累积频率达到总频数一半时对应的 x 值。
5. Histograms and Frequency Density | 直方图与频率密度
In a histogram with unequal class widths, the vertical axis represents frequency density, not the raw frequency. The formula is Frequency density = Frequency / Class width.
在组距不等的直方图中，纵轴表示频率密度而不是原始频数。公式为 频率密度 = 频数 / 组距。
The area of each bar equals the frequency of that class, so the total area represents the total frequency.
每个长方形的面积等于该组的频数，因此总面积代表总频数。
6. Probability Rules and Notation | 概率规则与记号
The probability of an event A is written P(A), a number between 0 and 1. The complement is P(A’) = 1 − P(A). For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
事件 A 的概率记作 P(A)，是介于 0 和 1 之间的一个数。对立事件为 P(A’) = 1 − P(A)。对于互斥事件 A 和 B，P(A 或 B) = P(A) + P(B)。
When events are not mutually exclusive, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For independent events, the multiplication rule is P(A ∩ B) = P(A) × P(B).
当事件并非互斥时，加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。对于独立事件，乘法法则为 P(A ∩ B) = P(A) × P(B)。
7. Conditional Probability and Tree Diagrams | 条件概率与树形图
Conditional probability P(A|B) means the probability of A given that B has occurred. It is calculated as P(A|B) = P(A ∩ B) / P(B).
条件概率 P(A|B) 表示在事件 B 已发生的条件下事件 A 发生的概率，计算公式为 P(A|B) = P(A ∩ B) / P(B)。
Tree diagrams help visualise successive events. Multiply probabilities along branches to find the probability of a combined outcome. All probabilities on branches from a single point must sum to 1.
树形图有助于可视化相继发生的事件。沿着分支将概率相乘即可得到组合结果的概率。从同一点出发的各分支概率之和必须等于 1。
8. Expected Value and Discrete Random Variables | 期望值与离散随机变量
For a discrete random variable X with probability function P(X = x), the expected value (mean) is E(X) = Σ[x · P(X = x)]. The variance can be found by Var(X) = Σ[x² · P(X = x)] − [E(X)]².
对于具有概率函数 P(X = x) 的离散随机变量 X，期望值（均值）为 E(X) = Σ[x · P(X = x)]。方差可通过 Var(X) = Σ[x² · P(X = x)] − [E(X)]² 求得。
In a binomial distribution with n trials and success probability p, the probability of exactly r successes is P(X = r) = C(n, r) pʳ qⁿ⁻ʳ, where q = 1 − p. The expected value is E(X) = np.
在试验次数为 n、成功概率为 p 的二项分布中，恰好成功 r 次的概率为 P(X = r) = C(n, r) pʳ qⁿ⁻ʳ，其中 q = 1 − p。期望值为 E(X) = np。
9. Bivariate Data and Correlation | 双变量数据与相关性
A scatter graph shows the relationship between two variables. Positive correlation means that as one variable increases, the other tends to increase; negative correlation means that as one increases, the other tends to decrease.
散点图显示两个变量之间的关系。正相关意味着当一个变量增加时，另一个变量倾向于增加；负相关意味着当一个变量增加时，另一个变量倾向于减少。
A line of best fit can be drawn by eye, passing as close as possible to all points. Its equation takes the form y = mx + c, where m is the gradient and c is the y‑intercept.
最佳拟合线可以通过目测画出，并尽可能靠近所有点。其方程形式为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
10. Formula Summary Table | 公式汇总表
Here is a quick‑reference table of the essential formulas for Year 10 Cambridge Statistics.
以下是十年级剑桥统计课程核心公式的速查表。
| Concept | Formula | Notes |
|---|---|---|
| Mean (ungrouped) | x̄ = Σx / n | Sum of values divided by count |
| Mean (grouped) | x̄ = Σ(f × x) / Σf | x is the class midpoint |
| Range | max − min | Difference between extremes |
| Interquartile range | IQR = Q₃ − Q₁ | Middle 50% spread |
| Population variance | σ² = Σ(x − μ)² / N | N population size |
| Sample variance | s² = Σ(x − x̄)² / (n − 1) | n sample size |
| Standard deviation (pop.) | σ = √[Σ(x − μ)² / N] | Square root of variance |
| Frequency density | fd = frequency / class width | Used in histograms |
| Probability complement | P(A’) = 1 − P(A) | A’ is ‘not A’ |
| Addition rule | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | General case |
| Conditional probability | P(A|B) = P(A ∩ B) / P(B) | Given B has occurred |
| Expected value | E(X) = Σ[x · P(X = x)] | Discrete random variable |
| Binomial probability | P(X = r) = C(n, r) pʳ qⁿ⁻ʳ | q = 1 − p |
| Binomial mean | E(X) = np | n trials, success prob. p |
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