📚 Year 12 CIE Statistics: Formula & Theorem Quick Reference | Year 12 CIE 统计公式定理速查手册
This quick reference handbook covers every essential formula and theorem you will encounter in the Year 12 CIE Statistics (Probability & Statistics 1) course. It is designed for rapid revision, helping you recall definitions, computational shortcuts and key properties for your AS‑Level exam.
这本速查手册涵盖了 Year 12 CIE 统计（概率与统计 1）课程中你将遇到的所有核心公式与定理。它专为快速复习而设计，帮助你回顾定义、计算技巧和关键性质，为 AS 阶段考试做好充分准备。
1. Data Presentation and Summary Statistics | 数据呈现与汇总统计量
Data can be displayed using stem‑and‑leaf diagrams, box‑and‑whisker plots and histograms. From any list of ungrouped data we can compute the five‑number summary: minimum, lower quartile (Q₁), median (Q₂), upper quartile (Q₃) and maximum.
数据可以通过茎叶图、箱线图和直方图进行展示。对任何未分组数据列表，我们可以计算五数概括：最小值、下四分位数 (Q₁)、中位数 (Q₂)、上四分位数 (Q₃) 和最大值。
2. Measures of Central Tendency | 集中趋势度量
The three most common averages are the mean, median and mode.
最常用的三种平均数分别是均值、中位数和众数。
Mean (ungrouped data): x̄ = Σx / n, where Σx is the sum of all n observations.
均值（未分组数据）：x̄ = Σx / n，这里 Σx 是所有 n 个观测值的总和。
The median is the middle value when the data are ordered. If n is odd, median = x(n+1)/2. If n is even, median = ½[xn/2 + xn/2+1].
中位数是将数据排序之后的中间值。如果 n 是奇数，中位数 = x(n+1)/2。如果 n 是偶数，中位数 = ½[xn/2 + xn/2+1]。
The mode is the value that occurs most frequently.
众数是出现频率最高的数值。
3. Measures of Dispersion | 离散度量
Dispersion tells us how spread out the data are. Three fundamental measures are the range, interquartile range (IQR) and standard deviation.
离散度表示数据的分散程度。三个基本度量是极差、四分位距 (IQR) 和标准差。
Range = maximum − minimum.
极差 = 最大值 − 最小值。
IQR = Q₃ − Q₁. It is the width of the middle 50 % of the data and is not affected by outliers.
IQR = Q₃ − Q₁。它是中间 50 % 数据的宽度，不受异常值影响。
4. Variance and Standard Deviation Formulas | 方差与标准差公式
For a set of n values x₁, x₂, ..., xₙ with mean x̄, the variance (using the divisor n) is:
对于具有均值 x̄ 的 n 个数值 x₁, x₂, ..., xₙ，方差（使用除数 n）为：
Variance = σ² = Σ(x − x̄)² / n
方差 = σ² = Σ(x − x̄)² / n
The computationally efficient form, which avoids calculating every deviation, is:
避免计算每一个离差的计算高效形式是：
σ² = Σx² / n − ( Σx / n )²
σ² = Σx² / n − ( Σx / n )²
The standard deviation is the square root of the variance: σ = √σ². In many CIE S1 problems you are expected to use the divisor n (rather than n−1).
标准差是方差的平方根：σ = √σ²。在众多 CIE S1 题目中，你应使用的除数是 n（而非 n−1）。
5. Grouped Data: Estimating Mean, Median, Quartiles | 分组数据：估计均值、中位数、四分位数
When data are presented in a frequency table, the mean is estimated using midpoints:
当数据以频数表的形式给出时，均值通过组中值来估计：
x̄ ≈ Σ(f × mid‑point) / Σf
x̄ ≈ Σ(f × 组中值) / Σf
For the median and quartiles we use linear interpolation:
对于中位数和四分位数，我们使用线性插值：
Estimated median = L + [ (n/2 − F) / f ] × c
估计中位数 = L + [ (n/2 − F) / f ] × c
where L is the lower boundary of the median class, F is the cumulative frequency before that class, f is the frequency of the class, and c is the class width. Replace n/2 with n/4 or 3n/4 for the lower and upper quartiles respectively.
这里 L 是中位数所在组的下限，F 是该组之前的累积频数，f 是该组的频数，c 是组距。将 n/2 替换为 n/4 或 3n/4 即可得到下四分位数和上四分位数的对应公式。
For variance of grouped data use Σf × x² / Σf − ( Σf × x / Σf )², where x is the mid‑point.
分组数据的方差使用 Σf × x² / Σf − ( Σf × x / Σf )²，其中 x 是组中值。
6. Probability Basics | 概率基础
is measured on a scale from 0 (impossible) to 1 (certain). For any event A, 0 ≤ P(A) ≤ 1.
概率的测量范围是从 0（不可能）到 1（确定）。对于任一事件 A，0 ≤ P(A) ≤ 1。
Complement rule: P(A') = 1 − P(A)
互补法则：P(A') = 1 − P(A)
Addition rule for mutually exclusive events: P(A ∪ B) = P(A) + P(B) when A and B cannot occur together.
互斥事件的加法法则：当 A 和 B 不能同时发生时，P(A ∪ B) = P(A) + P(B)。
General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
一般加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
7. Conditional Probability and Independence | 条件概率与独立性
The conditional probability of A given B is
在 B 发生的条件下 A 发生的概率是
P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.
P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。
Two events are independent if the occurrence of one does not affect the probability of the other. Independence can be tested with either of these equivalent conditions:
如果某一事件的发生不影响另一事件发生的概率，则称这两个事件独立。独立性可以用下列任一等价条件来检验：
P(A ∩ B) = P(A) × P(B)
P(A | B) = P(A) (and P(B | A) = P(B))
P(A ∩ B) = P(A) × P(B)
P(A | B) = P(A)（且有 P(B | A) = P(B)）
ually exclusive events cannot be independent (unless one probability is zero), because P(A ∩ B) = 0, which would contradict the product rule for non‑zero probabilities.
互斥事件不可能是独立的（除非其中一个概率为零），因为 P(A ∩ B) = 0，这与非零概率下的乘积法则相矛盾。
8. Permutations and Combinations | 排列与组合
When we arrange or select items, we use factorials, permutations and combinations.
当我们需要排列或选择项目时，会用到阶乘、排列与组合。
Factorial: n! = n × (n−1) × (n−2) × ... × 2 × 1, with 0! = 1.
阶乘：n! = n × (n−1) × (n−2) × ... × 2 × 1，且 0! = 1。
Permutations (order matters): the number of ways to arrange r objects chosen from n distinct objects is
排列（顺序重要）：从 n 个不同物品中选出 r 个进行排列的方式数为
ⁿPᵣ = n! / (n−r)!
Combinations (order does not matter): the number of ways to select r objects from n is
组合（顺序不重要）：从 n 个物品中选出 r 个的方式数为
ⁿCᵣ = n! / (r! (n−r)!)
The binomial coefficient ⁿCᵣ is also written as C(n, r) or (n r).
二项式系数 ⁿCᵣ 也写作 C(n, r) 或 (n r)。
9. Discrete Random Variables | 离散随机变量
each with an assigned probability P(X = x). All probabilities must satisfy Σ P(X = x) = 1.
离散随机变量 X 取可数的数值集合，每个数值都对应一个概率 P(X = x)。所有概率必须满足 Σ P(X = x) = 1。
The expected value (mean) of X is
X 的期望值（均值）为
E(X) = μ = Σ x · P(X = x)
The variance of X can be computed using
X 的方差可以用下式计算
Var(X) = E(X²) − [E(X)]² = Σ x² P(X = x) − μ²
The standard deviation is σ = √Var(X).
标准差为 σ = √Var(X)。
10. Expectation and Variance of Linear Functions | 线性函数的期望与方差
If a and b are constants and X is a random variable, the following rules hold:
如果 a 和 b 是常数，X 是随机变量，则下列规则成立：
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
Adding a constant shifts the mean but does not affect the variance or standard deviation. Multiplying by a constant scales the mean by that constant and scales the variance by its square.
加上一个常数会平移均值，但不影响方差或标准差。乘以一个常数将使均值按该常数缩放，并使方差按该常数的平方缩放。
These rules are especially useful when combining independent random variables, but note that CIE S2 covers combinations; in S1 you normally apply them to a single variable.
这些规则在组合独立随机变量时特别有用，但请注意 CIE S2 才涉及变量组合；在 S1 中通常仅对一个变量进行应用。
11. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability of success p.
二项分布适用于在固定次数 n 的独立试验中，每次试验的成功概率 p 相同时，成功次数的分布。
We write X ~ B(n, p). The probability of exactly r successes is
记作 X ~ B(n, p)。恰好有 r 次成功的概率为
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, for r = 0, 1, 2, ..., n
P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，其中 r = 0, 1, 2, ..., n
Key properties for X ~ B(n, p):
X ~ B(n, p) 的关键性质：
| Mean (E(X)) | np |
| Variance (Var(X)) | np(1 − p) |
| Shape | Symmetric if p = 0.5; skewed otherwise |
| 均值 (E(X)) | np |
| 方差 (Var(X)) | np(1 − p) |
| 形状 | p = 0.5 时对称；否则有偏 |
lative probabilities P(X ≤ k) are often found using tables or by summing individual terms.
累积概率 P(X ≤ k) 通常通过查表或对单项求和得到。
12. Normal Distribution and Standardization | 正态分布与标准化
The normal distribution is a continuous distribution with a bell‑shaped curve. It is fully described by its mean μ and standard deviation σ. We denote it by X ~ N(μ, σ²).
正态分布是一种具有钟形曲线的连续分布。它由均值 μ 和标准差 σ 完全描述，记作 X ~ N(μ, σ²)。
To find probabilities, we convert to the standard normal distribution Z ~ N(0, 1) using
为求概率，我们通过下式将其转换为标准正态分布 Z ~ N(0, 1)：
Z = (X − μ) / σ
rd normal table gives Φ(z) = P(Z < z). Symmetry is often used: P(Z < −z) = 1 − Φ(z).
标准正态表给出了 Φ(z) = P(Z < z)。常利用对称性：P(Z < −z) = 1 − Φ(z)。
When using the normal distribution to approximate a binomial distribution (continuity correction is required), apply
在用正态分布近似二项分布（需要连续性校正）时，使用：
μ = np, σ² = np(1 − p) and Z = (X ± 0.5 − μ)/σ
For the CIE S1 exam, you must be able to find an unknown mean or standard deviation by setting up a Z‑equation and using the inverse normal table.
在 CIE S1 考试中，你必须能够通过建立 Z 方程并使用逆正态表来求出未知的均值或标准差。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图