📚 Year 10 SQA Statistics: Formula & Theorem Quick Reference Handbook | SQA 10年级统计：公式定理速查手册
This concise reference guide collects every key formula, definition and theorem required for the SQA Year 10 Statistics course. Designed for rapid revision, it puts essential quantitative facts at your fingertips so you can check them whenever you need.
这本简明速查手册汇集了 SQA 10 年级统计学课程所需的每一个关键公式、定义和定理。专为快速复习设计，让你随时查阅核心的量化要点。
1. Measures of Central Tendency | 集中趋势度量
Measures of central tendency summarise the centre of a data set. The three most common measures are the mean, median and mode.
集中趋势度量概括数据集的中心。最常见的三种度量是均值、中位数和众数。
The sample mean (often written as x̄) is the arithmetic average. For a data set of n values, it is given by: x̄ = Σx / n, where Σx means “sum of all x-values”.
样本均值（常写作 x̄）是算术平均数。对于包含 n 个值的数据集，公式为：x̄ = Σx / n，其中 Σx 表示“所有 x 值的总和”。
The median is the middle value when the data are listed in order. If n is odd, the median is the (n+1)/2th value; if n is even, it is the average of the n/2th and (n/2 + 1)th values.
中位数是将数据排序后处于中间位置的数值。若 n 为奇数，中位数是第 (n+1)/2 个值；若 n 为偶数，则是第 n/2 个与第 (n/2 + 1) 个值的平均数。
The mode is the value that appears most frequently. A data set may have no mode, one mode (unimodal) or more than one mode (bimodal / multimodal).
众数是出现次数最多的值。一个数据集可能没有众数、有一个众数（单峰）或多个众数（双峰/多峰）。
2. Measures of Dispersion | 离散程度度量
Dispersion tells us how spread out the data are. Three key measures are the range, interquartile range (IQR) and standard deviation.
离散程度告诉我们数据的分散情况。三个关键度量是极差、四分位距（IQR）和标准差。
The range is the simplest measure: Range = maximum – minimum. It shows the total spread but is sensitive to outliers.
极差是最简单的度量：极差 = 最大值 – 最小值。它显示了总跨度，但对异常值敏感。
The interquartile range is the difference between the upper (Q₃) and lower (Q₁) quartiles: IQR = Q₃ – Q₁. It covers the middle 50% of data and is more robust against outliers.
四分位距是上四分位数（Q₃）与下四分位数（Q₁）之差：IQR = Q₃ – Q₁。它覆盖中间 50% 的数据，对异常值更稳健。
The sample standard deviation s measures the average distance of data points from the mean. The defining formula is:
样本标准差 s 衡量数据点与均值的平均距离。其定义公式为：
s = √[ Σ(x – x̄)² / (n – 1) ]
s = √[ Σ(x – x̄)² / (n – 1) ]
The numerator Σ(x – x̄)² is the sum of squared deviations from the mean; dividing by n–1 gives an unbiased estimate of the population variance. The square of the standard deviation, s², is called the variance.
分子 Σ(x – x̄)² 是每个值与均值之差的平方和；除以 n–1 可得到总体方差的无偏估计。标准差的平方 s² 称为方差。
3. Quartiles and Box Plots | 四分位数与箱线图
Quartiles divide an ordered data set into four equal parts. The lower quartile Q₁ is the median of the lower half, the middle quartile Q₂ is the overall median, and the upper quartile Q₃ is the median of the upper half.
四分位数把排序后的数据集分成四等份。下四分位数 Q₁ 是下半部分的中位数，中四分位数 Q₂ 是总中位数，上四分位数 Q₃ 是上半部分的中位数。
A five‑number summary consists of: minimum, Q₁, median (Q₂), Q₃, maximum. This summary is the basis for drawing a box plot.
五数概括包括：最小值、Q₁、中位数（Q₂）、Q₃、最大值。该概括是绘制箱线图的基础。
inimum and maximum within 1.5×IQR from the quartiles; data beyond are plotted as outliers.
在箱线图中，箱子从 Q₁ 延伸到 Q₃，中间用一条线标出中位数。触须延伸到距四分位数 1.5×IQR 范围内的最小值和最大值；超出该范围的数据点被标记为异常值。
4. Probability Fundamentals | 概率基础
Probability is a number between 0 and 1 (inclusive) that measures the chance of an event occurring. A probability of 0 means the event is impossible; a probability of 1 means it is certain.
概率是介于 0 和 1（含）之间的数，用于度量事件发生的可能性。概率为 0 表示事件不可能发生；概率为 1 表示事件必然发生。
The theoretical probability of an event A is: P(A) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
事件 A 的理论概率为：P(A) = 有利结果数 / 所有可能结果总数，假设所有结果等可能发生。
The sample space is the set of all possible outcomes. For example, when rolling a fair six‑sided die, the sample space is {1, 2, 3, 4, 5, 6}.
样本空间是所有可能结果的集合。例如，投掷一枚均匀六面骰子时，样本空间为 {1, 2, 3, 4, 5, 6}。
Experimental probability (or relative frequency) is calculated as: P(A) = frequency of event A / total number of trials. It approaches theoretical probability as the number of trials increases.
实验概率（或相对频率）计算为：P(A) = 事件 A 发生的次数 / 总试验次数。随着试验次数增加，它会趋近理论概率。
5. Probability Rules | 概率运算规则
Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则它们互斥。对于互斥事件 A 和 B：P(A 或 B) = P(A) + P(B)。
If events can occur together, the general addition rule applies: P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where A ∩ B means “A and B”.
如果事件可以同时发生，则适用一般加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)，其中 A ∩ B 表示“A 与 B”。
Two events are independent if the occurrence of one does not affect the probability of the other. For independent events A and B: P(A and B) = P(A) × P(B), written as P(A ∩ B) = P(A)P(B).
两个事件相互独立是指一个事件的发生不影响另一个事件的发生概率。对于独立事件 A 和 B：P(A 且 B) = P(A) × P(B)，记作 P(A ∩ B) = P(A)P(B)。
When events are not independent, the multiplication rule becomes P(A ∩ B) = P(A) × P(B|A), introducing conditional probability.
当事件不独立时，乘法规则变为 P(A ∩ B) = P(A) × P(B|A)，引入了条件概率。
The probability that A does not happen (complement) is: P(not A) = 1 – P(A).
事件 A 不发生的概率（补集）为：P(非 A) = 1 – P(A)。
6. Conditional Probability and Tree Diagrams | 条件概率与树状图
Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred. The formula is:
条件概率 P(A|B) 是在事件 B 已经发生的前提下，事件 A 发生的概率。公式为：
P(A|B) = P(A ∩ B) / P(B)
P(A|B) = P(A ∩ B) / P(B)
r analysing sequential experiments. Rearranging gives the general multiplication rule: P(A ∩ B) = P(B) × P(A|B).
这一关系是分析连续试验的基础。移项可得一般乘法规则：P(A ∩ B) = P(B) × P(A|B)。
Tree diagrams help visualise multi‑stage probability problems. Branches show probabilities; end‑of‑branch probabilities are multiplied along the path, and probabilities of different outcomes are added.
树状图有助于可视化多阶段概率问题。分支显示概率；沿路径将概率相乘得到最终结果的概率，不同结果的概率则相加。
When drawing a tree diagram, the probabilities on the first set of branches are the marginal probabilities, and later branches carry conditional probabilities. The sum of probabilities on branches from the same point must be 1.
绘制树状图时，第一组分支上的概率是边缘概率，后续分支上的是条件概率。从同一点伸出的分支概率之和必为 1。
7. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (scatter plot) displays the relationship between two numerical variables. Each point represents a pair of values (x, y).
散点图展示两个数值变量
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