📚 Year 11 Edexcel Statistics Formula & Theorem Quick Reference Handbook | 爱德思 Year 11 统计公式定理速查手册
This compact yet comprehensive handbook covers essential formulas, key theorems, and vital statistical concepts required for the Edexcel GCSE Statistics (Year 11) examination. It is designed as a quick revision tool to help you recall definitions, computation steps, and interpretation guidelines at a glance. Use it alongside your class notes and past papers to solidify your understanding.
这本紧凑而全面的手册涵盖了爱德思 GCSE 统计（Year 11）考试所需的核心公式、关键定理和重要统计概念。它是一份快速复习工具，帮助你即时回忆定义、计算步骤和解释指南。请结合课堂笔记与历年真题使用，以巩固理解。
1. Measures of Central Tendency | 集中趋势度量
The mean of a set of n observations x₁, x₂, … se midpoints and frequencies: mean = Σ(f × m) / Σf, where m is the midpoint of each class.
对于 n 个观测值 x₁, x₂, …, xₙ，平均数（均值）计算公式为 Σx / n。对于分组数据，使用组中值与频数：平均数 = Σ(f × m) / Σf，其中 m 是每组的组中值。
The median is the middle value when data are ordered. For ungrouped data with n values, median position is (n+1)/2. If n is even, median is the mean of the two middle values.
中位数是将数据排序后的中间值。对于未分组数据，有 n 个值时，中位数的位置为 (n+1)/2。若 n 为偶数，中位数是中间两个值的平均数。
The mode is the value that occurs most frequently. For grouped data, the modal class is the class interval with the highest frequency.
众数是出现次数最多的值。对于分组数据，众数所在组是频数最高的组距。
- Mean = Σx / n, or Σ(fx) / Σf for a frequency table.
- 平均数 = Σx / n，对于频数表则为 Σ(fx) / Σf。
- Median: the (n+1)/2 th value in ordered list.
- 中位数：有序列表中第 (n+1)/2 个值。
- Mode: most frequent value or class.
- 众数：最常见的值或组。
2. Measures of Dispersion | 离散程度度量
The range is the difference between the largest and smallest values: range = max – min. It gives a very basic measure of spread.
极差是最大值与最小值之差：极差 = 最大值 – 最小值。它提供了一个非常基本的离散度量。
Interquartile range (IQR) = upper quartile (Q3) – ercentile, Q3 the 75th percentile. IQR ignores extreme values and focuses on the middle 50% of data.
四分位距（IQR）= 上四分位数 (Q3) – 下四分位数 (Q1)。Q1 是第 25 百分位数，Q3 是第 75 百分位数。IQR 忽略极端值，聚焦于数据中间 50%。
on measure the average squared distance from the mean. For a population, variance σ² = Σ(x – μ)² / N; standard deviation σ = √σ². For a sample, use n–1 as denominator: s² = Σ(x – x̄)² / (n–1).
方差和标准差度量各数据与均值之间的平均平方距离。对于总体，方差 σ² = Σ(x – μ)² / N；标准差 σ = √σ²。对于样本，使用 n–1 作分母：s² = Σ(x – x̄)² / (n–1)。
σ² = Σ(x – μ)² / N, s² = Σ(x – x̄)² / (n – 1)
In Edexcel GCSE Statistics, the sample standard deviation formula with divisor (n–1) is often required for sets of data, while the population formula may be used for a given distribution.
在爱德思 GCSE 统计中，通常要求使用分母为 (n–1) 的样本标准差公式计算数据集，而总体公式可能用于给定分布。
3. Frequency Density and Histograms | 频数密度与直方图
The class width is the difference between the upper and lower boundaries of the interval.
对于组距不等的直方图，每个柱形的面积与频数成正比。频数密度 = 频数 ÷ 组距宽度。组距宽度是区间上界与下界之差。
You use frequency density on the vertical axis, not frequency, to ensure the area of the bar represents frequency correctly. To find a frequency from a histogram, multiply the frequency density by the class width.
你需要在纵轴上使用频数密度而非频数，以确保柱形面积正确表示频数。要从直方图求频数，将频数密度乘以组距宽度。
Frequency Density = Frequency ÷ Class Width
Remember that if a bar is missing, the frequency can be found by calculating the area of the bar using the scale of frequency density and class width.
请记住，如果某个柱形缺失，可以通过频数密度的刻度和组距宽度计算柱形面积来求得频数。
4. Cumulative Frequency and Percentiles | 累积频数与百分位数
A cumulative frequency table adds frequencies consecutively down the classes. The cumulative frequency curve (ogive) plots cumulative frequency against the upper class boundary.
累积频数表将各组的频数按顺序累加。累积频数曲线（肩形图）以累积频数对组距上界作图。
Use the ogive to estimate medians, quartiles, and percentiles. Draw a horizontal line from the required cumulative frequency (e.g. n/2 for median) to the curve, then drop vertically to read the value on the x‑axis.
可利用累积频数曲线估计中位数、四分位数和百分位数。从所需的累积频数（如求中位数时用 n/2）处画水平线交曲线，然后向下画垂直线在 x 轴上读数。
The interpercentile range, such as the 10th to 90th percentile range, can also be read from the ogive. It indicates the spread of the middle 80% of data.
百分位数间距，例如第 10 到第 90 百分位数间距，也可从累积频数曲线读取。它表明中间 80% 数据的分布范围。
- Cumulative frequency adds up the frequencies.
- 累积频数是频数的累加。
- Median from ogive: read value at n/2.
- 从累积频数曲线求中位数：在 n/2 处读取的值。
- IQR = Q3 – Q1; Q1 at n/4, Q3 at 3n/4.
- IQR = Q3 – Q1；Q1 在 n/4 处，Q3 在 3n/4 处。
5. Probability Rules and Notation | 概率规则与记号
r of favourable outcomes / total number of equally likely outcomes. Probabilities range from 0 (impossible) to 1 (certain).
基本概率：P(事件) = 有利结果数 / 等可能结果总数。概率范围从 0（不可能）到 1（确定）。
P(A or B) = P(A) + P(B). If events are not mutually exclusive, use P(A or B) = P(A) + P(B) – P(A and B).
互斥事件的加法法则：P(A 或 B) = P(A) + P(B)。若事件不互斥，使用 P(A 或 B) = P(A) + P(B) – P(A 且 B)。
pendent events, use conditional probability: P(A and B) = P(A) × P(B given A) = P(A) × P(B|A).
独立事件的乘法法则：P(A 且 B) = P(A) × P(B)。对于不独立事件，使用条件概率：P(A 且 B) = P(A) × P(B 给定 A) = P(A) × P(B|A)。
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
The ‘given that’ notation P(B|A) means the probability of B occurring given that A has occurred. It is calculated by P(B|A) = P(A ∩ B) / P(A).
‘给定’记号 P(B|A) 表示在 A 已发生的情况下 B 发生的概率。计算公式为 P(B|A) = P(A ∩ B) / P(A)。
6. Expectation and Venn Diagrams | 期望与韦恩图
× probability of the event. This is used in goodness-of-fit tests and simple predictions.
n 次试验中某事件的期望频数为 n × 该事件的概率。这用于拟合优度检验和简单预测。
Venn diagrams display sets and their overlaps. Numbers inside each region represent frequencies or probabilities. The rectangle is the sample space, with total probability 1.
韦恩图展示集合及其交集。每块区域内的数字表示频数或概率。矩形框是样本空间，总概率为 1。
To find missing probabilities from a Venn diagram, remember that the sum of all disjoint region probabilities equals 1. Use the union and intersection formulas to solve for unknowns.
要从韦恩图找出缺失的概率，请记住所有互斥区域概率之和等于 1。使用并集与交集公式解出未知数。
| P(A ∪ B) | P(A) + P(B) – P(A ∩ B) |
| P(A') = 1 – P(A) | Complement rule |
| Expected frequency | n × P(event) |
Always label regions clearly and check that probabilities add up to 1. For conditional probability using a Venn diagram, restrict the sample space to the given event.
始终清晰标注各区域，并检查概率相加是否等于 1。对于使用韦恩图的条件概率，将样本空间限制在所给的事件内。
7. Scatter Diagrams and Correlation | 散点图与相关
A scatter diagram graphs bivariate data. Plot the independent variable on the x‑axis and the dependent variable on the y‑axis. Describe the correlation as positive, negative, or none.
散点图用于绘制双变量数据。将自变量标在 x 轴，因变量标在 y 轴上。将相关性描述为正相关、负相关或无相关。
Correlation strength can be strong, moderate, or weak. Beware of outliers that can affect the line of best fit and correlation coefficient.
相关强度可以是强、中等或弱。注意可能影响最佳拟合线和相关系数的异常值。
The line of best fit is drawn ‘by eye’ (or using mean point) and should have roughly equal numbers of points above and below. It should pass through the mean point (x̄, ȳ).
最佳拟合线凭目测（或利用均值点）画出，应使线上方和下方的点数大致相等，并应通过均值点 (x̄, ȳ)。
Use the line to estimate values: interpolation (within the data range) is reliable; extrapolation (outside the range) is unreliable because the trend may not continue.
利用该直线估计取值时：内插（数据范围内）可靠；外推（数据范围外）不可靠，因为趋势可能不再延续。
8. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rank correlation coefficient, rₛ, measures the strength of monotonic association between two ranked variables. It is computed using the differences dᵢ between the ranks of each pair.
斯皮尔曼等级相关系数 rₛ 度量两个排序变量之间的单调关联强度。它利用每对数据的秩次之差 dᵢ 进行计算。
rₛ = 1 – (6 Σdᵢ²) / [n(n² – 1)]
Here, n is the number of paired observations, and dᵢ = rank of x – rank of y for the i‑th pair. If tied ranks occur, assign the average of the ranks they would have occupied.
此处 n 是成对观测的数量，dᵢ = 第 i 对数据中 x 秩次 – y 秩次。如果出现并列，则分配它们本应占据的秩次的平均值。
The value of rₛ ranges from –1 (perfect negative correlation) to +1 (perfect positive correlation). A value near 0 suggests no monotonic relationship. Interpret it in context: strong correlation does not imply causation.
rₛ 的取值范围从 –1（完全负相关）到 +1（完全正相关）。接近 0 的值表明不存在单调关系。在具体语境中解释：强相关并不意味着因果关系。
Always state the significance level if a hypothesis test is required. Edexcel often provides a critical value table for Spearman’s rank test.
如果需要进行假设检验，需说明显著性水平。爱德思通常会提供斯皮尔曼等级检验的临界值表。
9. Pearson’s Product Moment Correlation Coefficient (PMCC) | 皮尔逊积矩相关系数
Pearson’s r measures the strength of a linear relationship between two numerical variables. Its formula can be given in terms of sums of squares and cross-products.
皮尔逊 r 度量两个数值变量之间线性关系的强度。其公式可用平方和与交叉积和表示。
r = Sₓᵧ / √Sₓₓ × Sᵧᵧ
Where Sₓₓ = Σ(x – x̄)², Sᵧᵧ = Σ(y – ȳ)², and Sₓᵧ = Σ(x – x̄)(y – ȳ). Alternatively, use the computational formula with Σx², Σy², Σxy, and means.
其中 Sₓₓ = Σ(x – x̄)²，Sᵧᵧ = Σ(y – ȳ)²，Sₓᵧ = Σ(x – x̄)(y – ȳ)。也可以使用含有 Σx²、Σy²、Σxy 及均值的计算式。
r also ranges from –1 to +1. r = 1 indicates a perfect positive linear relationship, r = – egative linear, and r = 0 no linear correlation. However, a zero r does not rule out a non‑linear relationship.
r 同样在 –1 到 +1 之间。r = 1 表示完全正线性关系，r = –1 表示完全负线性关系，r = 0 表示无线性相关。但 r 为零不能排除存在非线性关系。
In Year 11, you may use calculator functions to find r. Interpret the result alongside a scatter diagram, as a single outlier can distort the correlation coefficient dramatically.
在 Year 11，你可能会使用计算器函数求 r。需要结合散点图解释结果，因为单个异常值可能极大地扭曲相关系数。
10. Equation of Regression Line | 回归直线方程
used to predict y from x. It has the form y = a + bx, where b is the slope and a is the y‑intercept.
若存在线性关系，可使用 y 对 x 的回归直线通过 x 预测 y。其形式为 y = a + bx，其中 b 是斜率，a 是截距。
b = Sₓᵧ / Sₓₓ, a = ȳ – b x̄
Here, b is the gradient (change in y per unit increase in x). The line always passes through the mean point (x̄, ȳ).
此处 b 是梯度（x 每增加一个单位 y 的变化量）。该直线总是通过均值点 (x̄, ȳ)。
Once the equation is obtained, substitute an x‑value to predict the corresponding y. Remember to comment on reliability based on whether the value lies within the original data range.
得到方程后，代入 x 值预测对应的 y。请记住根据该值是否位于原始数据范围内来评述其可靠性。
are different unless r = ±1. In Edexcel GCSE, you are typically asked for the y on x line.
不要混淆 y 对 x 的回归线与 x 对 y 的回归线；除非 r = ±1，否则它们是不同的。在爱德思 GCSE 中，通常要求给出 y 对 x 的回归线。
11. The Normal Distribution | 正态分布
The normal distribution is a continuous, symmetric, bell‑shaped curve defined by its mean μ and standard deviation σ. About 68% of data lie within 1σ of μ, 95% within 2σ, and 99.7% within 3σ.
正态分布是一条连续的、对称的钟形曲线，由均值 μ 和标准差 σ 决定。大约 68% 的数据落在 μ 的 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。
Standardised scores (z‑scores) tell you how many standard deviations a value is above or below the mean: z = (x – μ) / σ. z‑scores allow comparison across different normal distributions.
标准化分数（z 分数）告诉你某个值距离均值有多少个标准差：z = (x – μ) / σ。z 分数使得跨不同正态分布的比较成为可能。
z = (x – μ) / σ
To find probabilities, you may use given normal distribution tables. The total area under the curve is 1. The area to the left of a z‑value gives the cumulative probability P(Z < z).
要计算概率，你可以使用给定的正态分布表。曲线下的总面积为 1。z 值左侧的面积给出累积概率 P(Z < z)。
If you are given a probability and asked to find the x‑value, rearrange the z‑ + zσ, where z is the corresponding value from the inverse table or symmetrical reasoning.
如果给定概率并要求找出 x 值，可重排 z 公式：x = μ + zσ，其中 z 是对应的逆表值或根据对称性得到。
12. Key Theorems and Checking Methods | 关键定理与校验方法
The law of large numbers: as the number of trials increases, the experimental probability tends to the theoretical probability. This underpins why a larger sample gives a more reliable estimate.
大数定律：随着试验次数增加，实验概率趋近于理论概率。这解释了为什么更大的样本会给出更可靠的估计。
ly exclusive vs independent: mutually exclusive events cannot occur simultaneously, so P(A∩B)=0. Independent events have no influence on each other, P(B|A)=P(B).
互斥与独立：互斥事件不能同时发生，因此 P(A∩B)=0。独立事件互不影响，P(B|A)=P(B)。
Always check reasonableness of answers: probabilities must lie between 0 and 1, standard deviation cannot be negative, correlation coefficients must be in [−1, 1]. Common mistake: confusing sum of squares with square of sums in correlation formulas.
始终检查答案的合理性：概率必须在 0 到 1 之间，标准差不能为负，相关系数必须在 [−1, 1] 内。常见错误：在相关公式中将平方与和平方混淆。
- Use tree diagrams for conditional probability problems.
- 使用树状图解决条件概率问题。
- Check the gradient of a regression line for real‑world plausibility.
- 检查回归直线的梯度在现实世界中是否合理。
- If data are skewed, median and IQR are better than mean and range.
- 如果数据偏斜，中位数和 IQR 优于均值和极差。
These fundamental principles will help you not only solve problems but also critically evaluate statistical claims in your examination.
这些基本原则不仅帮助你解题，还能帮你在考试中批判性地评价统计论断。
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