📚 Year 12 Edexcel Statistics: Formula and Theorem Quick Reference Handbook | Year 12 Edexcel 统计：公式定理速查手册
This quick reference handbook compiles all the essential formulae, theorems and definitions required for the Year 12 Edexcel Statistics course – from measures of central tendency and probability rules right through to the binomial and normal distributions. Use it alongside your revision to check key facts, clarify notation and build confidence before your AS‑Level examination.
本速查手册汇编了 Year 12 Edexcel 统计课程所需的所有核心公式、定理和定义——从集中趋势度量、概率法则直讲到二项分布与正态分布。将其与你的复习结合使用，用以核对关键考点、厘清符号，并在 AS 考试前树立信心。
1. Measures of Central Tendency | 集中趋势度量
The sample mean x̄ (read “x‑bar”) is the arithmetic average of a data set and is the most commonly used measure of central location.
样本均值 x̄（读作“x bar”）是数据集的算术平均值，是最常用的集中位置度量。
x̄ = Σx / n
For grouped data where individual values are not known, the mean is estimated by using class midpoints m and frequencies f.
对于不知道单个数值的分组数据，用组中点 m 和频数 f 估计均值。
x̄ ≈ Σf m / Σf
The median is the middle value when data are ordered; for ungrouped data with n observations, the median is at position (n+1)/2. When data are grouped, linear interpolation is used to locate the median inside the median class.
中位数是排序后处于中间位置的值；对于有 n 个观测值的未分组数据，中位数位置为 (n+1)/2。当数据分组时，使用线性插值在中位数所在组内定位。
Median = L + ( (n/2 − Fbelow) / fmedian ) × w
where L = lower class boundary of the median group, Fbelow = cumulative frequency before the median group, fmedian = frequency of the median group and w = class width.
其中 L = 中位数组的下组界, Fbelow = 中位数组之前的累计频数, fmedian = 中位数组的频数, w = 组距。
The mode is the value that occurs most frequently; for grouped data the modal class is the class with the highest frequency.
众数是出现次数最多的数值；对分组数据而言，众数组是频数最高的组。
2. Measures of Dispersion | 离散程度度量
Range is the simplest measure of spread: maximum − minimum.
极差是最简单的离散度量：最大值 − 最小值。
The interquartile range (IQR) measures the spread of the middle 50% of the data: IQR = Q₃ − Q₁. Quartiles are found in a similar way to the median; for ungrouped data Q₁ is at position (n+1)/4 and Q₃ at position 3(n+1)/4. For grouped data linear interpolation is used.
四分位距（IQR）度量中间 50% 数据的离散程度：IQR = Q₃ − Q₁。四分位数的求法与中位数类似；未分组数据 Q₁ 的位置在 (n+1)/4，Q₃ 在 3(n+1)/4。分组数据使用线性插值。
Variance is the average of the squared deviations from the mean. The sample variance s² uses divisor n−1 for an unbiased estimate.
方差是各数据与均值之差的平方的平均值。样本方差 s² 使用除数 n−1 以获得无偏估计。
s² = Σ(x − x̄)² / (n − 1)
An alternative, often quicker formula is:
另一个通常更快的公式为：
s² = ( Σx² / n − x̄² ) × n/(n−1)
The standard deviation s is the positive square root of the variance and has the same unit as the original data.
标准差 s 是方差的正平方根，具有与原始数据相同的单位。
For grouped data, replace x with the class midpoint m and multiply each squared term by the frequency f.
对于分组数据，用组中点 m 代替 x，并将每个平方项乘以频数 f。
s² = Σf m² / Σf − (Σf m / Σf)²
3. Coding and Its Effects | 编码及其影响
Coding is a linear transformation of the form y = (x − a) / b. It is used to simplify calculations without losing the shape of the distribution.
编码是形如 y = (x − a) / b 的线性变换，用于简化计算且不改变分布形状。
If a constant is subtracted or added to every value, the mean shifts by that same constant, but measures of spread (range, IQR, standard deviation) are unchanged.
若每个值都减去或加上某个常数，均值随之平移相同的常数，但离散度量（极差、IQR、标准差）不变。
If every value is multiplied or divided by a constant, both the mean and the standard deviation (and variance) are multiplied or divided by that constant (variance by the square).
若每个值都乘以或除以某个常数，均值和标准差（及方差）都会乘以或除以该常数（方差则乘以该常数的平方）。
For a coding y = (x − a)/b:
对于编码 y = (x − a)/b：
x̄ = a + b ȳ
sₓ = b s_y
Coding is particularly helpful when working with large or awkward numbers, and you can transform back after calculating summary statistics.
在处理大数或不方便的数值时编码非常有用，你可以在计算出概括统计量后再变换回来。
4. Outliers and Box Plots | 离群值与箱形图
An outlier is an observation that lies an abnormal distance from the rest of the data. Boundaries (fences) are usually defined using the IQR.
离群值是与其余数据距离异常的观测值。界限（篱笆）通常使用 IQR 来定义。
Lower fence = Q₁ − 1.5 × IQR
Upper fence = Q₃ + 1.5 × IQR
Any data value below the lower fence or above the upper fence is identified as an outlier.
任何低于下篱笆或高于上篱笆的数据值被识别为离群值。
A box plot (box‑and‑whisker diagram) displays the minimum, Q₁, median, Q₃ and maximum, with outliers plotted individually as crosses. The whiskers extend to the most extreme non‑outlier values.
箱形图（箱须图）显示最小值、Q₁、中位数、Q₃ 和最大值，离群值单独用叉号标出。须线延伸至最极端的非离群值。
Outliers should be examined carefully – they may indicate genuine extreme values, data recording errors or special causes.
对离群值应仔细审查——它们可能代表真实的极端值、数据记录错误或特殊原因。
5. Probability Rules | 概率法则
The probability of an event A, P(A), is a number between 0 and 1. The complement rule states P(not A) = 1 − P(A).
事件 A 的概率 P(A) 是介于 0 和 1 之间的数。互补法则为 P(非 A) = 1 − P(A)。
For any two events A and B, the addition rule is:
对任意两个事件 A 和 B，加法法则为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive (cannot occur together), P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
如果 A 与 B 互斥（不可能同时发生），则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
Conditional probability is the probability of A occurring given that B has occurred:
条件概率是在事件 B 已经发生的条件下，事件 A 发生的概率：
P(A | B) = P(A ∩ B) / P(B)
Two events are independent if the occurrence of one does not affect the probability of the other:
若一个事件的发生不影响另一个事件的概率，则两事件独立：
P(A ∩ B) = P(A) × P(B)
For independent events, conditional probability reduces to the original probability: P(A | B) = P(A).
对于独立事件，条件概率退化为原始概率：P(A | B) = P(A)。
Probability tree diagrams and Venn diagrams are essential tools for organising multi‑stage or overlapping events.
概率树状图和维恩图是组织多阶段或重叠事件的基本工具。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of values, each with an associated probability. The probability distribution must satisfy Σ P(X = x) = 1.
离散随机变量 X 可取可数个值，每个值对应一个概率。概率分布必须满足 Σ P(X = x) = 1。
The expected value E(X) is the long‑run average of the variable, often denoted by μ.
期望值 E(X) 是变量的长期平均值，常用 μ 表示。
E(X) = Σ x P(X = x)
The variance Var(X) measures the spread of the distribution:
方差 Var(X) 度量分布的离散程度：
Var(X) = E(X²) − [E(X)]²
Calculating E(X²) using Σ x² P(X = x) is usually the quickest route.
通常通过 Σ x² P(X = x) 计算 E(X²) 最为快捷。
If a random variable is transformed linearly with Y = aX + b, the expectation and variance follow these rules:
若随机变量经线性变换 Y = aX + b，期望和方差遵循以下法则：
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
The standard deviation of X is σ = √Var(X).
X 的标准差为 σ = √Var(X)。
7. Binomial Distribution | 二项分布
A binomial experiment consists of a fixed number n of independent trials, each with two outcomes (success/failure) and a constant probability of success p. The random variable X = number of successes follows a binomial distribution, written X ~ B(n, p).
二项试验由固定次数 n 次独立试验组成，每次试验有两种结果（成功/失败），且成功概率 p 恒定。随机变量 X = 成功次数服从二项分布，记为 X ~ B(n, p)。
The probability of obtaining exactly r successes is given by the binomial probability function:
恰好获得 r 次成功的概率由二项概率函数给出：
P(X = r) = nCr pr (1−p)n−r
where nCr = n! / (r! (n−r)!) is the binomial coefficient.
其中 nCr = n! / (r! (n−r)!) 是二项式系数。
The mean (expected value) and variance of a binomial distribution are:
二项分布的均值（期望值）和方差为：
E(X) = n p
Var(X) = n p (1−p)
Cumulative probabilities P(X ≤ r) can be found from statistical tables or your calculator’s binomial‑CDF function.
累积概率 P(X ≤ r) 可通过统计表或计算器的二项累积分布函数求得。
8. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution with a symmetric bell‑shaped curve. It is fully described by its mean μ and variance σ², written X ~ N(μ, σ²).
正态分布是一种具有对称钟形曲线的连续概率分布，由其均值 μ 和方差 σ² 完全确定，记作 X ~ N(μ, σ²)。
Standardising transforms any normal variable into the standard normal Z ~ N(0, 1):
标准化将任意正态变量转化为标准正态 Z ~ N(0, 1)：
Z = (X − μ) / σ
Probabilities for X can then be obtained from the standard normal table: P(X < a) = Φ((a−μ)/σ), where Φ is the cumulative distribution function of Z.
然后可通过标准正态表求得 X 的概率：P(X < a) = Φ((a−μ)/σ)，其中 Φ 是 Z 的累积分布函数。
Because the normal curve is symmetric, Φ(−z) = 1 − Φ(z). Common interval probabilities are:
由于正态曲线对称，Φ(−z) = 1 − Φ(z)。常见的区间概率为：
P(|Z| < 1) ≈ 0.6827, P(|Z| < 2) ≈ 0.9545, P(|Z| < 3) ≈ 0.9973
Inverse normal calculations find the value a such that P(X < a) = p; use the inverse Φ⁻¹ function on your calculator or the percentage points table.
逆正态计算找出满足 P(X < a) = p 的 a 值；请使用计算器的逆 Φ⁻¹ 函数或百分位数表。
Approximating a binomial with a normal distribution is possible when n is large and np > 5 and n(1−p) > 5. A continuity correction is then required.
当 n 较大且 np > 5、n(1−p) > 5 时，可用正态分布近似二项分布，此时需进行连续性校正。
9. Correlation | 相关性
The product‑moment correlation coefficient (PMCC), denoted by r, measures the strength and direction of a linear relationship between two variables.
积矩相关系数（PMCC），记作 r，度量两个变量间线性关系的强度和方向。
r = Σ((x − x̄)(y − ȳ)) / √( Σ(x − x̄)² Σ(y − ȳ)² )
The formula for r is symmetric and always satisfies −1 ≤ r ≤ 1. Values close to +1 indicate strong positive correlation, values close to −1 strong negative correlation, and values near 0 weak or no linear correlation.
r 的公式是对称的，且始终满足 −1 ≤ r ≤ 1。接近 +1 的数值表明强正相关，接近 −1 表明强负相关，接近 0 表明弱或无线性相关。
Correlation does not imply causation. Outliers can heavily distort r, so always plot a scatter diagram first.
相关不代表因果关系。离群值会严重扭曲 r，因此务必先绘制散点图。
You can test whether the population correlation coefficient ρ is zero using a hypothesis test with the test statistic r and critical values from the Spearman/Pearson table, but this is often covered in Year 13.
你可以用检验统计量 r 和 Pearson/Spearman 表的临界值来检验总体相关系数 ρ 是否为 0，但这通常在 Year 13 深入涉及。
10. Hypothesis Testing for the Binomial | 二项分布的假设检验
A hypothesis test provides a formal framework for deciding whether observed data support a particular claim about a population parameter. For the binomial distribution, the parameter tested is the success probability p.
假设检验提供了一套形式化框架，用于判断观测数据是否支持关于总体参数的某种主张。对于二项分布，被检验的参数是成功概率 p。
The null hypothesis H₀ states a specific value for p, usually p = p₀, while the alternative hypothesis H₁ is that p is less than, greater than, or simply not equal to p₀.
原假设 H₀ 陈述 p 的一个具体值，通常 p = p₀，而备择假设 H₁ 则是 p 小于、大于或仅仅不等于 p₀。
The test statistic is the observed number of successes X.
检验统计量是观测到的成功次数 X。
The p‑value is the probability of obtaining a result at least as extreme as the observed X, assuming H₀ is true. If the p‑value is less than the significance level α (commonly 0.05), the result is statistically significant and H₀ is rejected.
p 值是在 H₀ 为真的前提下，获得至少如观测值 X 那样极端的结果的概率。若 p 值小于显著性水平 α（常用 0.05），则结果具有统计显著性，拒绝 H₀。
The critical region (rejection region) consists of those values of X for which the cumulative probability under H₀ is ≤ α (for a one‑tailed test) or ≤ α/2 in each tail (for a two‑tailed test).
临界域（拒绝域）由在 H₀ 下累积概率 ≤ α（单尾检验）或在每尾 ≤ α/2（双尾检验）的 X 值构成。
Always state your conclusion in the context of the problem, linking back to the original claim.
请始终在问题背景中给出结论，并联系原始主张。
Published by TutorHao | Statistics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导