📚 Year 10 Eduqas Statistics: Formula & Theorem Quick Reference Guide | Year 10 Eduqas 统计学：公式定理速查手册
This quick reference guide compiles the most important formulas and theorems you need for Year 10 Eduqas Statistics. It moves from averages and measures of spread through charts and skewness, then into probability, conditional probability, discrete random variables, the binomial distribution and the normal distribution. Each section states the key concept, the symbolic notation and the exam context in which the formula is normally used. Keep it beside you while revising, and use it as a checklist before you attempt past-paper questions.
本速查手册汇总了 Year 10 Eduqas 统计学所需的最重要公式和定理。内容从平均数与离散度量出发，经过统计图表与偏度，再进入概率、条件概率、离散随机变量、二项分布与正态分布。每节给出核心概念、符号表示法以及该公式在考试中的常见使用场景。复习时请随时查阅，也可在做历年真题前当作自检清单使用。
1. Measures of Central Tendency | 集中趋势的度量
s the arithmetic average. For a data set with values x₁, x₂, ... xₙ, the mean is written x̄ le size. When the data are already organised in a frequency table, the same idea becomes x̄ = Σfx / Σf, where x is each value and f is its frequency.
均值即算术平均值。对于数值 x₁, x₂, ... xₙ，均值写作 x̄ = Σx / n，其中 Σx 为所有数值之和，n 为样本量。当数据已整理成频数表时，同一思想写作 x̄ = Σfx / Σf，其中 x 为各个数值，f 为其频数。
The median is the middle value once the data are sorted into order. If n is odd, the median is the ((n + 1) / 2)-th value. If n is even, it is the mean of the (n / 2)-th and (n / 2 + 1)-th values.
中位数是数据排序后位于中间的值。若 n 为奇数，中位数为第 (n + 1) / 2 个数值；若 n 为偶数，则为第 n / 2 个与第 (n / 2 + 1) 个数值的平均值。
The mode is the value that appears most often. A data set may have one mode, more than one mode (bimodal or multimodal), or no mode at all if every value is unique. The mode is the only average that works for categorical data such as favourite colours or modes of transport.
众数是出现次数最多的值。数据集可以有一个众数、多个众数（双众数或多众数），若所有值均唯一则可没有众数。众数是唯一适用于定类数据（如最喜欢的颜色或出行方式）的平均数。
x̄ = Σx / n x̄ = Σfx / Σf
| Average 平均数 | Best used when 适用场合 | Outlier sensitive? 受离群值影响？ |
| Mean 均值 | Data are roughly symmetric 数据大致对称 | Yes 是 |
| Median 中位数 | Data are skewed or contain extremes 数据偏斜或含极端值 | No 否 |
| Mode 众数 | Data are categorical 数据为定类数据 | No 否 |
2. Measures of Spread | 离散程度的度量
m value. It is quick to calculate but depends on only two values, so a single extreme value can change it dramatically.
极差是最简单的离散度量：极差 = 最大值 − 最小值。它计算快捷，但只依赖两个数值，因此一个极端值就能使它大幅变化。
The population standard deviation σ measures the typical distance of values from the population mean μ:
总体标准差 σ 衡量各数值与总体均值 μ 之间的典型距离：
σ = √( Σ(x − μ)² / N )
The sample standard deviation s uses n − 1 in the denominator so that it gives an unbiased estimate of the population value:
样本标准差 s 的分母使用 n − 1，从而给出总体值的无偏估计：
s = √( Σ(x − x̄)² / (n − 1) )
seful computational form avoids finding every deviation individually: σ² = (Σx² / N) − μ².
方差是标准差的平方，总体记作 σ²，样本记作 s²。一个实用的计算式可以避免逐一求偏差：σ² = (Σx² / N) − μ²。
3. Quartiles, Percentiles and the IQR | 四分位数、百分位数与四分位距
Q₁ is the 25th percentile and Q₃ is the 75th percentile, so the IQR contains the middle 50% of the data.
四分位距是上四分位数与下四分位数之差：IQR = Q₃ − Q₁。Q₁ 是第 25 百分位数，Q₃ 是第 75 百分位数，因此 IQR 包含数据中间 50% 的数值。
For a ranked list of n values, the positions of the quartiles can be found using (n + 1) / 4 and 3(n + 1) / 4; if the position is not a whole number, interpolate between the two neighbouring values.
对于含 n 个数值的排序列表，四分位数的位置可用 (n + 1) / 4 与 3(n + 1) / 4 确定；若位置不是整数，则在相邻两个数值之间插值。
The semi-interquartile range, (Q₃ − Q₁) / 2, is sometimes used because it is roughly half the distance you would expect a typical value to sit from the median in a symmetric distribution.
半四分位距 (Q₃ − Q₁) / 2 有时也会被使用，因为对称分布中它大约等于典型数值到中位数距离的一半。
IQR = Q₃ − Q₁ Pₖ position = k(n + 1) / 100
4. Grouped Data and Frequency Tables | 分组数据与频数表
individual values are unknown, so the mean can only be estimated using class midpoints: x̄ ≈ Σfx / Σf, where x is the midpoint of each class and f is its frequency.
当数据被分成组时，原始数值未知，因此只能用组中值估计均值：x̄ ≈ Σfx / Σf，其中 x 为每组的组中值，f 为该组频数。
The modal class is the class with the highest frequency, and the median class is the class in which the (n / 2)-th value falls when cumulative frequencies are used.
众数所在组是频数最高的那一组；中位数所在组则是累积频数达到第 n / 2 个数值时所在的那一组。
The median of grouped data can be estimated by linear interpolation, where L is the lower class boundary of the median class, F is the cumulative frequency before that class, f is the frequency of the median class and w is the class width:
分组数据的中位数可用线性插值估计，其中 L 为中位数所在组的下组界，F 为该组之前的累积频数，f 为该组频数，w 为组距：
Median ≈ L + ( (n / 2 − F) / f ) × w
Remember that a grouped estimate is only an approximation; the true mean and median depend on the original ungrouped values.
请记住，分组估计值只是近似值；真实的均值与中位数取决于未分组的原始数据。
5. Presenting Data: Charts, Cumulative Frequency and Box Plots | 数据展示：图表、累积频数与箱线图
A bar chart is used for discrete or categorical data, while a histogram is used for continuous grouped data. In a histogram the vertical axis shows frequency density rather than frequency, so the area of each bar represents the frequency of that class.
条形图用于离散或定类数据，直方图用于连续的分组数据。直方图的纵轴表示频数密度而非频数，因此每个矩形的面积代表该组的频数。
Frequency density is calculated as frequency divided by class width. If a histogram question gives you area and class width, you can recover the frequency by multiplying them together.
频数密度等于频数除以组距。若题目给出面积与组距，将二者相乘即可还原频数。
Frequency density = frequency / class width
A cumulative frequency curve is plotted against upper class boundaries and rises to n. Reading across from n / 2, n / 4 and 3n / 4 on the vertical axis gives the median, the lower quartile and the upper quartile respectively.
累积频数曲线以各组上组界为横坐标绘制，最终上升到 n。在纵轴上分别从 n / 2、n / 4 和 3n / 4 横读，即可得到中位数、下四分位数和上四分位数。
A box plot (box-and-whisker diagram) shows the five-number summary: minimum, Q₁, median, Q₃ and maximum. Two box plots drawn on the same scale are the quickest way to compare two distributions for location and spread.
箱线图（箱须图）展示五数概括：最小值、Q₁、中位数、Q₃ 与最大值。在同一坐标尺度上绘制两个箱线图，是比较两个分布位置与离散程度最快的方法。
6. Outliers and Skewness | 离群值与偏度
rule marks any value as an outlier if it is below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
离群值是远离其余数据的数值。常用的判定规则是：若数值低于 Q₁ − 1.5 × IQR 或高于 Q₃ + 1.5 × IQR，则视为离群值。
Lower fence = Q₁ − 1.5 × IQR Upper fence = Q₃ + 1.5 × IQR
Skewness describes which way a distribution leans. In a positively skewed distribution the mean is greater than the median and the right whisker of the box plot is longer; in a negatively skewed distribution the mean is less than the median and the left whisker is longer.
偏度描述分布偏向哪一侧。正偏（右偏）分布中均值大于中位数，箱线图右侧须更长；负偏（左偏）分布中均值小于中位数，左侧须更长。
A quick check uses quartiles as well: if (Q₃ − median) is greater than (median − Q₁), the distribution is positively skewed, and if it is smaller, the distribution is negatively skewed.
也可以用量距快速判断：若 (Q₃ − 中位数) 大于 (中位数 − Q₁)，则分布正偏；若小于，则分布负偏。
When you compare two distributions in an exam, comment on both an average and a measure of spread, and mention outliers or skew if they affect which average is the better summary.
考试中比较两个分布时，既要评论平均数，也要评论离散度量；若离群值或偏度影响平均数选择，也应予以说明。
7. Basic Probability Rules | 基本概率法则
0 ≤ P(A) ≤ 1. A probability of 0 means the event is impossible and 1 means it is certain.
事件概率等于有利结果数除以等可能结果总数，且始终满足 0 ≤ P(A) ≤ 1。概率为 0 表示事件不可能发生，为 1 表示必然发生。
The complement rule states that P(A') = 1 − P(A), where A' that A does not happen. This is often the fastest route to an answer when a question says 'at least one'.
补集法则为 P(A') = 1 − P(A)，其中 A' 表示 A 不发生。当题目出现"至少一个"时，这通常是最快的解法。
For two events, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If the events are mutually exclusive they cannot happen together, so P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
对于两个事件，加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若两事件互斥，则它们不可能同时发生，故 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
The multiplication rule states that P(A ∩ d B are independent. Independence can be tested by checking whether P(A) × P(B) equals P(A ∩ B).
乘法法则指出，当 A 与 B 相互独立时 P(A ∩ B) = P(A) × P(B)。判断独立性即检验 P(A) × P(B) 是否等于 P(A ∩ B)。
Experimental (relative) probability is calculated from data as the number of successful trials divided by the total number of trials. As the number of trials increases, this value tends to move closer to the theoretical probability.
实验（相对）概率由数据计算，等于成功试验次数除以试验总次数。随着试验次数增加，该值会逐渐接近理论概率。
8. Venn Diagrams and Set Notation | 韦恩图与集合记法
A Venn diagram shows the sample space ξ as a rectangle and events as circles inside it. The notation A ∪ B means 'A or B' and A ∩ B means 'A and B', while A' means 'not A'.
韦恩图用一个矩形表示样本空间 ξ，用圆表示事件。记法 A ∪ B 表示"A 或 B"，A ∩ B 表示"A 且 B"，A' 表示"非 A"。
When filling in a Venn diagram, always start with the intersection of all the events and work outwards, and place any outcomes belonging to no event outside the circles but inside the rectangle.
填写韦恩图时，应先填所有事件的交集，再向外推进；不属于任何事件的结果要放在圆外、矩形内的区域。
The relationship n(A ∪ B) = n(A) + n(B) − n(A ∩ is very useful for two-way tables, where the totals along the rows and columns must agree.
关系式 n(A ∪ B) = n(A) + n(B) − n(A ∩ B) 与加法法则相对应，在处理行、列合计必须一致的二联表时非常有用。
Remember that P(A ∪ B) + P(A' ∩ B') does not necessarily equal 1; only P(A) + P(A') = 1 always holds for a single event.
请记住 P(A ∪ B) + P(A' ∩ B') 不一定等于 1；只有对单个事件，P(A) + P(A') = 1 始终成立。
9. Conditional Probability and Tree Diagrams | 条件概率与树形图
Conditional probability is the probability of one event given that another has already occurred, written P(A | B) and calculated as P(A ∩ B) / P(B), provided P(B) is not zero.
条件概率是在另一事件已经发生的前提下某事件发生的概率，写作 P(A | B)，计算式为 P(A ∩ B) / P(B)，前提是 P(B) 不为零。
P(A | B) = P(A ∩ B) / P(B)
A tree diagram multiplies along the branches and adds across the outcomes: the probability of a complete path is the product of the probabilities on its branches, and the probability of an event with several paths is the sum of those path probabilities.
树形图沿分支相乘、按结果相加：一条完整路径的概率等于各分支概率之积，而由多条路径组成的事件的概率等于这些路径概率之和。
When items are chosen without replacement, the denominator on the second set of branches decreases, so the probabilities change. With replacement, the probabilities stay the same on every branch.
当抽取是不放回时，第二层分支的分母会减小，概率随之改变；若是有放回抽取，每条分支的概率保持不变。
ependence and conditional probability are linked: A and B are independent exactly when P(A | B) = P(A), that is, when knowing B gives no information about A.
独立性与条件概率相互关联：A 与 B 独立的充要条件是 P(A | B) = P(A)，即知道 B 发生对 A 毫无信息。
10. Discrete Random Variables and Expectation | 离散随机变量与期望值
A discrete random variable X takes numerical values each with a probability, and these probabilities must sum to 1 across the whole distribution.
离散随机变量 X 取数值时各有对应概率，整个分布中所有概率之和必须等于 1。
x P(X = x). It represents the long-run average if the experiment were repeated many times.
X 的期望值（均值）等于每个取值与其概率乘积之和：E(X) = Σ x P(X = x)。它表示在大量重复试验下的长期平均值。
E(X) = Σ x P(X = x) E(X²) = Σ x² P(X = x)
the standard deviation is the square root of that value. This shortcut form is usually faster than working from the definition.
离散随机变量的方差为 Var(X) = E(X²) − [E(X)]²，标准差即其平方根。这一简捷式通常比用定义计算更快。
Expected value is also used to judge games and decisions as fair or unfair. A game is fair when the expected gain is zero; if E(X) is positive for the player, the game favours the player.
期望值也用于判断游戏或决策是否公平。当期望收益为零时游戏公平；若玩家的 E(X) 为正，则游戏对玩家有利。
11. The Binomial Distribution | 二项分布
A binomial distribution models the number of successes in n repeated trials. It applies only when there is a fixed number of trials, each trial has exactly two outcomes, the trials are independent, and the probability of success p stays constant.
二项分布用于描述 n 次重复试验中成功次数的分布。它只在满足以下条件时适用：试验次数固定；每次试验只有两种结果；各次试验相互独立；成功概率 p
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