📚 Year 11 CCEA Statistics: Formula & Theorem Quick Reference Handbook | 11年级CCEA统计：公式定理速查手册
This handbook provides a concise compilation of all the essential formulas, theorems, and key concepts required for the CCEA GCSE Statistics (Year 11) specification. It is designed to be a quick revision aid for students preparing for their examinations. Use it to check definitions, recall calculation steps, and reinforce your understanding of statistical foundations.
本手册简明汇编了CCEA GCSE统计学（11年级）大纲要求的所有重要公式、定理和关键概念，旨在为学生备考提供快速复习帮助。你可以随时查阅定义、回顾计算步骤，并巩固统计学基础的掌握。
1. Measures of Central Tendency | 集中趋势测量
The arithmetic mean (average) is the sum of all data values divided by the total number of values. It is the most common measure of centre.
算术平均值（均值）是所有数据值的总和除以数据总个数，是最常用的集中趋势度量。
Mean (x̄) = Σx / n
For ungrouped data, the median is the middle value when the data are arranged in order. If n is odd, it is the ((n+1)/2)th value; if n is even, it is the average of the (n/2)th and (n/2 + 1)th values.
对于未分组数据，中位数是将数据排序后位于中间位置的数值。若n为奇数，则取第((n+1)/2)个值；若n为偶数，则取第(n/2)个与第(n/2+1)个值的平均数。
The mode is the value that appears most frequently. A dataset may have one mode, more than one mode, or no mode at all.
众数是出现次数最多的数据值。一组数据可以有一个众数、多个众数或无众数。
2. Measures of Dispersion | 离散程度测量
The range is the difference between the maximum and minimum values. It shows the spread but can be affected by outliers.
极差是最大值与最小值之差，反映数据跨度但易受异常值影响。
Range = Maximum value − Minimum value
The interquartile range (IQR) is the difference between the upper quartile (Q3) and the lower quartile (Q1). It measures the spread of the middle 50% of data and is resistant to outliers.
四分位距（IQR）是上四分位数（Q3）与下四分位数（Q1）之差，衡量中间50%数据的离散程度，且不受异常值影响。
IQR = Q3 − Q1
Variance and standard deviation measure the average squared deviation from the mean. For a population:
方差与标准差衡量各数据与均值的平均平方偏差。对于总体：
σ² = Σ(x − μ)² / N σ = √[Σ(x − μ)² / N]
For a sample, we divide by (n − 1) to obtain an unbiased estimate:
对于样本，我们除以(n − 1)以获得无偏估计：
s² = Σ(x − x̄)² / (n − 1) s = √[Σ(x − x̄)² / (n − 1)]
3. Grouped Data: Mean and Interpolation | 分组数据：平均值与插值
When data are presented in a frequency table with class intervals, the estimated mean uses the midpoint (x) of each interval and the frequency (f).
当数据以组距频数表呈现时，估算平均值使用各组的组中点（x）与频数（f）。
x̄ = Σ(f × x) / Σf
The median for grouped data is found by linear interpolation within the class that contains the (n/2)th value.
分组数据的中位数通过在包含第(n/2)个值的组内进行线性插值求得。
Median = L + ( (n/2 − F) / f ) × w
where L = lower boundary of the median class, F = cumulative frequency before the median class, f = frequency of the median class, w = class width.
其中 L = 中位数所在组的下限，F = 中位数组之前的累积频数，f = 中位数组的频数，w = 组距。
The same formula can be applied to estimate quartiles, deciles, and percentiles by replacing n/2 with the appropriate position (e.g. n/4 for Q1, 3n/4 for Q3).
同样公式可用于估算四分位数、十分位数和百分位数，只需将n/2替换为相应位置（如Q1用n/4，Q3用3n/4）。
4. Frequency Density and Histograms | 频数密度与直方图
In a histogram, the area of each bar is proportional to the frequency it represents. When class widths are unequal, frequency density must be used to construct the vertical scale.
在直方图中，每个柱形的面积与其代表的频数成比例。当组距不相等时，必须使用频数密度作为纵轴来绘图。
Frequency Density = Frequency / Class Width
You can recover the frequency of a class by multiplying the frequency density by the class width: Frequency = Frequency Density × Class Width.
你可以通过频数密度乘以组距恢复该组的频数：频数 = 频数密度 × 组距。
Always label the vertical axis as ‘Frequency density’ and ensure the horizontal axis is continuous and scaled correctly.
务必标记纵轴为“频数密度”，并确保横轴连续且刻度正确。
5. Cumulative Frequency and Percentiles | 累积频率与百分位数
A cumulative frequency curve (or ogive) is plotted using the upper class boundary and the running total of frequencies. It allows you to estimate medians, quartiles, and percentiles graphically.
累积频率曲线（折线图）以组上限和累积频数描点连线，可用于图形法估算中位数、四分位数和百分位数。
The median corresponds to the value at 50% of the total frequency; the lower quartile at 25%; the upper quartile at 75%.
中位数对应总频数的50%处；下四分位数在25%处；上四分位数在75%处。
For a given percentile Pk, find the k% of the total frequency on the cumulative frequency axis and read off the corresponding data value from the horizontal axis.
对于给定的第k百分位数Pk，在累积频率轴上找到总频数的k%，然后从横轴上读取对应的数据值。
Linear interpolation (as shown in Section 3) can also be used for a more precise estimate when the grouped frequency table is available.
当有分组频数表时，也可使用线性插值（见第3节）以获得更精确的估计值。
6. Probability Fundamentals | 概率基础
The probability of an event A is the number of favourable outcomes divided by the total number of equally likely outcomes. It always lies between 0 and 1 inclusive.
事件A的概率等于有利结果数除以所有等可能结果总数，其值总在0到1之间（含0和1）。
P(A) = Number of favourable outcomes / Total number of outcomes
The complement of A, denoted by A’, is the event that A does not occur. The sum of their probabilities is 1.
A的补事件，记作A’，表示A不发生的事件。它们的概率之和为1。
P(A’) = 1 − P(A)
Expected frequency is the number of times an event is predicted to occur over a fixed number of trials. It is found by multiplying the probability by the number of trials.
期望频数是预估事件在固定试验次数中发生的次数，由概率乘以试验次数得到。
Expected frequency = P(event) × Number of trials
7. Mutually Exclusive and Independent Events | 互斥事件与独立事件
Mutually exclusive events cannot occur at the same time. For such events A and B, P(A ∩ B) = 0 and the addition rule simplifies.
互斥事件不可能同时发生。对于互斥事件A和B，有P(A ∩ B) = 0，且加法法则简化。
P(A ∪ B) = P(A) + P(B)
For any two events, the general addition rule accounts for overlap:
对于任意两个事件，通用加法法则需考虑重叠部分：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent events are those where the occurrence of one does not affect the probability of the other. The multiplication rule for independent events is:
独立事件指一个事件的发生不影响另一个事件的概率。独立事件的乘法法则为：
P(A ∩ B) = P(A) × P(B)
When events are not independent, tree diagrams and conditional probabilities must be used; the multiplication rule then becomes P(A ∩ B) = P(A) × P(B|A).
当事件不独立时，必须使用树图和条件概率，此时乘法法则变为P(A ∩ B) = P(A) × P(B|A)。
8. Conditional Probability | 条件概率
Conditional probability is the probability of event A occurring given that event B has already occurred. It is fundamental for handling dependent events.
条件概率是指在事件B已经发生的条件下事件A发生的概率，是处理相依事件的基础。
P(A|B) = P(A ∩ B) / P(B) provided P(B) ≠ 0
You can rearrange this to find the probability of the intersection: P(A ∩ B) = P(B) × P(A|B). This is frequently used in probability tree diagrams.
你可以将此式变形求交事件的概率：P(A ∩ B) = P(B) × P(A|B)，这常用于概率树图。
In a tree diagram, multiply probabilities along branches to find combined probabilities; add relevant branch outcomes for ‘or’ scenarios. Always check if events are independent to decide whether conditional probabilities change on successive branches.
在树图中，沿分支相乘得到联合概率；对于“或”的情形，将相关分支结果相加。始终检查事件是否独立以确定后续分支的条件概率是否改变。
9. Scatter Graphs and Correlation | 散点图与相关分析
A scatter graph displays the relationship between two variables. Correlation measures the strength and direction of a linear association.
散点图显示两个变量间的关系。相关性衡量线性关联的强度和方向。
Pearson’s product-moment correlation coefficient (r) quantifies linear correlation. Its formula, based on sums of squares, is:
皮尔逊积矩相关系数（r）量化线性相关性，其基于平方和的公式为：
r = Sxy / √(Sxx × Syy)
where the sums of squares are:
其中各平方和的定义为：
Sxx = Σx² − (Σx)² / n
Syy = Σy² − (Σy)² / n
Sxy = Σxy − (Σx)(Σy) / n
The value of r ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation). r = 0 indicates no linear correlation.
r值范围为−1（完全负相关）到+1（完全正相关）。r = 0表示无线性相关。
10. Linear Regression | 线性回归
The least squares regression line of y on x is the straight line that best fits the data by minimising the sum of squared vertical residuals. It can be used to predict y from x.
y对x的最小二乘回归线是通过最小化竖直残差平方和得到的最佳拟合直线，可用于由x预测y。
y = a + b x
The slope (b) and intercept (a) are calculated as:
斜率（b）和截距（a）计算如下：
b = Sxy / Sxx
a = ȳ − b x̄
The line always passes through the mean point (x̄, ȳ). Interpolation is prediction within the range of the data; extrapolation outside the range may be unreliable.
该直线恒过均值点（x̄, ȳ）。内插是在数据范围内预测；外推至范围外可能不可靠。
11. Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. It is written
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