📚 Quick Reference Handbook of Formulae and Theorems for Statistics | 统计公式定理速查手册
This handbook provides a concise summary of essential formulae, theorems and key statistical measures required for the Year 10 AQA Statistics course. Each section presents the core calculations and concepts in a bilingual, easy-to-refresh format, helping you to master data handling and interpretation.
本手册简要总结了 Year 10 AQA 统计课程所需的基本公式、定理和关键统计量。每个小节以中英双语、便于速查的形式呈现核心计算和概念，帮助你掌握数据处理与解读。
1. Mean, Median and Mode | 平均数、中位数和众数
The mean (x̄) for raw data is the sum of all values divided by the number of values: x̄ = Σx / n.
原始数据的平均数（x̄）是所有数值之和除以数值个数：x̄ = Σx / n。
For a frequency table, the mean is estimated using the midpoint of each class: x̄ ≈ Σ(f × x) / Σf, where x is the data value or class midpoint.
对于频数表，使用各组的组中值来估计平均数：x̄ ≈ Σ(f × x) / Σf，其中 x 是数据值或组中值。
The median is the middle value when data is ordered. If n is odd, median = value at position (n+1)/2. If n is even, median = average of values at positions n/2 and n/2 + 1.
中位数是排序后中间的值。若 n 为奇数，中位数 = 位置 (n+1)/2 处的值；若 n 为偶数，中位数 = 位置 n/2 与 n/2+1 处两值的平均数。
The mode is the value that occurs most frequently. A dataset can have more than one mode (bimodal or multimodal) or no mode at all.
众数是出现次数最多的值。一个数据集可能具有多个众数（双峰或多峰），也可能没有众数。
Estimated mean from grouped data: x̄ ≈ Σ(f × m) / Σf, where m = class midpoint.
分组数据估计平均数：x̄ ≈ Σ(f × m) / Σf，其中 m = 组中值。
2. Range, Quartiles and IQR | 极差、四分位数与四分位距
Range = highest value – lowest value. It is a simple measure of spread but is sensitive to outliers.
极差 = 最大值 – 最小值。它是一种简单的离散度量，但易受异常值影响。
The lower quartile (Q₁) is the median of the lower half of the data; the upper quartile (Q₃) is the median of the upper half. The interquartile range is IQR = Q₃ – Q₁.
下四分位数（Q₁）是数据下半部分的中位数；上四分位数（Q₃）是数据上半部分的中位数。四分位距为 IQR = Q₃ – Q₁。
To find the position of the quartile for n data values, use (n+1)/4 for Q₁ and 3(n+1)/4 for Q₃. If the position is not an integer, interpolate between adjacent values.
对于 n 个数据值，Q₁ 的位置为 (n+1)/4，Q₃ 的位置为 3(n+1)/4。若位置不是整数，则在相邻值之间进行插值。
The IQR gives the range of the middle 50% of the data, making it a robust measure of dispersion.
IQR 给出中间 50% 数据的范围，使其成为一种稳健的离散度量。
3. Standard Deviation and Variance | 标准差与方差
Variance measures the average squared deviation from the mean. The standard deviation is the square root of the variance and has the same units as the data.
方差衡量各数据与平均数之差的平方的平均值。标准差是方差的平方根，与原始数据单位相同。
Sample variance: s² = Σ(x – x̄)² / (n – 1)
样本方差：s² = Σ(x – x̄)² / (n – 1)
Sample standard deviation: s = √[ Σ(x – x̄)² / (n – 1) ]
样本标准差：s = √[ Σ(x – x̄)² / (n – 1) ]
When working with a frequency table, replace x with the class midpoint m and use Σf(m – x̄)² / (Σf – 1). An alternative computing formula is s = √[ (Σfx² / Σf) – x̄² ] for large datasets.
处理频数表时，用组中值 m 替代 x，并使用 Σf(m – x̄)² / (Σf – 1)。对于大数据集，可选用计算式 s = √[ (Σfx² / Σf) – x̄² ]。
A low standard deviation indicates that data points tend to be close to the mean; a high standard deviation indicates that they are spread out.
低标准差表示数据点倾向于集中在均值附近；高标准差则表示数据分布较分散。
4. Box Plots | 箱线图
A box plot (or box-and-whisker diagram) displays the five-number summary: minimum, Q₁, median, Q₃ and maximum.
箱线图（箱形图）展示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。
The box is drawn from Q₁ to Q₃, with a line indicating the median. Whiskers extend to the minimum and maximum values, unless there are outliers.
箱体从 Q₁ 画至 Q₃，中间有一条线标示中位数。胡须延伸至最小值和最大值，除非存在异常值。
Outliers are often defined as values less than Q₁ – 1.5 × IQR or greater than Q₃ + 1.5 × IQR. They may be plotted as individual points beyond the whiskers.
异常值通常定义为小于 Q₁ – 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的值。可将其绘制为胡须之外的单独点。
Box plots are useful for comparing the spread and central tendency of two or more datasets side by side.
箱线图非常适合并排比较两个或多个数据集的离散程度和集中趋势。
5. Probability Basics | 概率基础
The probability of an event A is given by P(A) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
若所有结果等可能，事件 A 的概率为 P(A) = 有利结果数 / 可能结果总数。
Probabilities always lie between 0 and 1 inclusive. A probability of 0 means impossible; a probability of 1 means certain.
概率值始终介于 0 和 1 之间（含）。概率为 0 表示不可能事件；概率为 1 表示必然事件。
For any event A, P(not A) = 1 – P(A). The sum of probabilities of all mutually exclusive outcomes of an experiment is 1.
对于任意事件 A，P(非 A) = 1 – P(A)。一个实验所有互斥结果的概率之和为 1。
Expected frequency of an event is calculated as probability × number of trials. This is used to predict how many times an event should occur.
事件的期望频数 = 概率 × 试验次数。用于预测某事件应发生的次数。
Relative frequency from an experiment can be used as an estimate of probability: Relative frequency = frequency of event / total number of trials.
实验获得的相对频率可作为概率的估计值：相对频率 = 事件发生频数 / 总试验次数。
6. Conditional Probability and Tree Diagrams | 条件概率与树形图
Conditional probability P(A|B) is the probability that event A occurs given that event B has already occurred.
条件概率 P(A|B) 表示在事件 B 已发生的条件下事件 A 发生的概率。
P(A|B) = P(A ∩ B) / P(B) , provided P(B) > 0.
P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。
Tree diagrams help to visualise sequences of events and multiply probabilities along branches. The sum of probabilities on branches from the same point equals 1.
树形图有助于可视化事件序列，并沿分支相乘概率。从同一点出发的分支概率之和为 1。
For independent events, P(A ∩ B) = P(A) × P(B). For dependent events, use the general multiplication rule: P(A ∩ B) = P(A) × P(B|A).
对于独立事件，P(A ∩ B) = P(A) × P(B)。对于非独立事件，使用一般乘法法则：P(A ∩ B) = P(A) × P(B|A)。
When completing a tree diagram, label each branch with its probability and write the outcome at the end. The final probability of each combined outcome is found by multiplying along the path.
完成树形图时，在每条分支上标注概率，并在末端写出结果。每个组合结果的最终概率由路径上的概率相乘得出。
7. Histograms and Frequency Density | 直方图与频率密度
A histogram displays continuous grouped data. Unlike a bar chart, the area of each bar is proportional to frequency, not just its height.
直方图显示连续的分组数据。与条形图不同，每个直条的面积（而非仅高度）与频数成正比。
When class widths are unequal, you must use frequency density on the vertical axis. Frequency density = Frequency ÷ Class width.
当组距不等时，纵轴必须使用频率密度。频率密度 = 频数 ÷ 组距。
Frequency Density = Frequency / Class Width
频率密度 = 频数 / 组距
To find the frequency represented by a bar, multiply the frequency density by the class width: Frequency = Frequency density × Class width.
求某一直条所代表的频数时，将频率密度乘以组距：频数 = 频率密度 × 组距。
The total area of all bars in a histogram equals the total frequency. Histograms can also be used to estimate the modal class and proportions.
直方图中所有直条的总面积等于总频数。直方图还可用于估计众数组和比例。
8. Cumulative Frequency | 累积频率
Cumulative frequency is the running total of frequencies up to a given upper boundary. It is used to construct cumulative frequency graphs (ogives).
累积频率是截至某一给定上限的频数累计值。它用于绘制累积频率图（折线图）。
Plot the upper boundary of each class against its cumulative frequency and join the points with a smooth curve. The cumulative frequency starts at 0 and ends at the total frequency.
将每个组的上限与对应的累积频率在坐标图上描点，并用光滑曲线连接。累积频率从 0 开始，到总频数结束。
From a cumulative frequency graph, you can estimate the median (at 50% of total frequency), quartiles (at 25% and 75%) and interpercentile ranges.
通过累积频率图可以估计中位数（总频数的 50% 处）、四分位数（25% 和 75% 处）以及百分位距。
To find any given percentile, locate the corresponding cumulative frequency on the vertical axis, draw a horizontal line to the curve and then a vertical line down to read the value.
要找出任一百分位数，可在纵轴上定位相应的累积频率，画水平线交于曲线，再向下画垂直线轴读取数值。
9. Scatter Graphs and Correlation | 散布图与相关
A scatter graph shows the relationship between two variables. Each point represents a pair of values (x, y).
散布图显示两个变量之间的关系。每个点代表一对 (x, y) 数值。
Correlation describes the strength and direction of the linear relationship. Positive correlation means that as x increases, y tends to increase; negative correlation means that as x increases, y tends to decrease.
相关描述线性关系的强度和方向。正相关意味着当 x 增加时，y 趋于增加；负相关意味着当 x 增加时，y 趋于减少。
If points lie close to a straight line, the correlation is strong. If they are widely scattered, the correlation is weak. No correlation means there is no discernible linear pattern.
若点紧密分布在一条直线附近，则相关性较强；若点较为分散，则相关性较弱。无相关意味着不存在明显的线性模式。
A line of best fit can be drawn by eye through the points to show the trend. It should have roughly equal numbers of points above and below the line.
可以通过目测画一条最佳拟合线来显示趋势，该线应使线上方和线下方的点数大致相等。
The equation of the line of best fit (y = mx + c) can be used to make predictions. Extrapolating beyond the range of the data is unreliable.
最佳拟合线的方程（y = mx + c）可用于进行预测。超出数据范围的外推是不可靠的。
10. Spearman’s Rank Correlation | 斯皮尔曼等级相关系数
Spearman’s rank correlation coefficient (rₛ) measures the strength and direction of the monotonic relationship between two sets of ranked data.
斯皮尔曼等级相关系数（rₛ）衡量两组排序数据之间单调关系的强度和方向。
rₛ = 1 – (6Σd²) / [n(n² – 1)]
rₛ = 1 – (6Σd²) / [n(n² – 1)]
To calculate rₛ: rank the data for each variable separately, find the difference d between the ranks for each pair, square them and sum to get Σd². Plug the values into the formula.
计算 rₛ 的步骤：分别对每组变量排序，找出每对数据的秩次差 d，求其平方并累加得 Σd²，再代入公式。
rₛ always lies between -1 and +1. A value of +1 indicates perfect positive rank correlation, -1 indicates perfect negative rank correlation, and 0 indicates no rank correlation.
rₛ 值始终介于 -1 和 +1 之间。+1 表示完全正等级相关，-1 表示完全负等级相关，0 表示无等级相关。
When tied ranks occur, assign the average rank to each tied value. Spearman’s rank is useful when data is not normally distributed or when only ordinal data is available.
当出现相同等级时，给每个相同值分配平均秩次。当数据不服从正态分布或只有顺序数据时，斯皮尔曼等级相关系数非常有用。
11. Sampling Methods | 采样方法
A sample is a subset of a population used to draw conclusions about the whole population. A good sample should be representative and unbiased.
样本是总体的一个子集，用于推断总体的特征。好的样本应具有代表性且无偏。
Simple random sampling: every member of the population has an equal chance of being selected. This can be achieved using random number generators or lottery methods.
简单随机采样：总体中每个成员被选中的机会均等。可通过随机数生成器或抽签法实现。
Stratified sampling: the population is divided into groups (strata) based on a characteristic, and a random sample is taken from each stratum in proportion to its size.
分层采样：根据某一特征将总体分成层级，从每个层级按比例随机抽取样本。
Systematic sampling: select every kth member from a list after a random start. The sampling interval k = population size ÷ sample size.
系统采样：在列表中随机起点后，每隔 k 个成员选取一个。采样间隔 k = 总体大小 ÷ 样本大小。
Cluster sampling: divide the population into clusters and randomly select entire clusters. Quota sampling involves selecting a fixed number of individuals with specific characteristics, but it is non-random and can introduce bias.
整群采样：将总体分成群组，随机选择整个群组。配额采样则是选取固定数量的具有特定特征的个体，但它是非随机的，可能引入偏差。
The choice of method affects the reliability of conclusions. Random methods are generally preferred for inferential statistics.
采样方法的选择会影响结论的可靠性。在推断统计中通常优先使用随机方法。
12. Time Series and Moving Averages | 时间序列与移动平均数
A time series is a sequence of data recorded at regular time intervals. It can show trend, seasonal variation and irregular fluctuations.
时间序列是按固定时间间隔记录的数据序列，可显示趋势、季节变动和不规则波动。
A moving average smooths out short-term fluctuations to reveal the underlying trend. For a given point, it is the average of a fixed number of consecutive values centred on that point.
移动平均数可平滑短期波动以揭示潜在趋势。对于某一点，它是以此点为中心固定项数连续数值的平均值。
4-point moving average: (y₁ + y₂ + y₃ + y₄)/4, then (y₂ + y₃ + y₄ + y₅)/4, and so on.
四点移动平均：(y₁ + y₂ + y₃ + y₄)/4，然后是 (y₂ + y₃ + y₄ + y₅)/4，以此类推。
When an even number of points is used, the moving average falls between time periods. Centring is then achieved by taking a two-point moving average of the moving averages.
当使用偶数个点时，移动平均会落在时间周期之间。此时可通过对移动平均数再取两项移动平均来实现中心化。
Seasonal variation = actual value – trend value. By averaging seasonal variations over several cycles, you can obtain typical seasonal effects for forecasting.
季节变动 = 实际值 – 趋势值。通过求多个周期的季节变动的平均值，可以得到用于预测的典型季节效应。
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