📚 Year 10 WJEC Statistics Formula and Theorem Quick Reference Handbook | 10年级 WJEC 统计：公式定理速查手册
Welcome to your essential quick-reference handbook for Year 10 WJEC Statistics. This guide covers all key formulae, definitions, and theorems you need to master, from data collection to probability and regression. Keep it handy for revision and exam preparation.
欢迎使用这本为10年级WJEC统计课程准备的核心速查手册。本指南涵盖从数据收集到概率与回归的所有关键公式、定义和定理，方便你随时查阅、复习和备考。
1. Data Collection and Sampling Methods | 数据收集与抽样方法
Accurate statistics start with reliable data. The way data is collected affects its validity, so it is essential to understand different sampling methods and potential biases.
准确的统计始于可靠的数据。数据收集的方式会影响其有效性，因此理解不同的抽样方法和潜在的偏差至关重要。
A population is the entire set of individuals or items of interest. A sample is a subset of the population used to draw conclusions about the whole. A sampling frame is a list of all members of the population from which a sample is drawn.
总体是所有感兴趣的个体或项目的全部集合。样本是总体的一个子集，用于推断总体的特征。抽样框是总体中所有成员的名单，样本从此名单中抽取。
Simple random sampling gives every member of the population an equal chance of selection, often using random number generators. It is free of bias but can be impractical for large populations.
简单随机抽样使总体中的每个成员被选中的概率相等，通常使用随机数生成器。它没有偏差，但对于大总体可能不太实际。
Stratified sampling divides the population into distinct strata (e.g., age groups) and then takes a random sample from each stratum in proportion to its size. This ensures representation of all key subgroups.
分层抽样将总体划分为不同的层（如年龄组），然后按比例从每一层中随机抽取样本。这确保了所有关键子群体都能被代表。
Systematic sampling selects members at regular intervals from an ordered list, starting from a random point. It is simple to use but may introduce bias if there is an underlying pattern in the list.
系统抽样从有序名单中按固定的间隔选取成员，起始点随机。实施简便，但如果名单中存在潜在模式，可能会引入偏差。
Cluster sampling divides the population into clusters, randomly selects a few clusters, and includes every member of those clusters. Convenience sampling uses data that is easiest to reach, but often leads to strong bias.
整群抽样将总体分成群组，随机抽取若干群组，再对选中群组的所有成员进行调查。便利抽样使用最容易获取的数据，但常常导致严重偏差。
2. Measures of Central Tendency | 集中趋势的度量
Measures of central tendency describe the centre of a data set. The three most common are the mean, median, and mode.
集中趋势的度量用来描述数据集的中心位置。最常用的三种是均值、中位数和众数。
The mean (x̄) is the arithmetic average of all values. It uses every data point but is sensitive to outliers. The formula for ungrouped data is:
均值（x̄）是所有数值的算术平均数。它使用了每一个数据点，但对异常值敏感。未分组数据的计算公式为：
x̄ = (∑x) / n
The median is the middle value when data are arranged in order. For an odd number of values it is the central one; for an even number it is the mean of the two central values. To find the position of the median: (n + 1) / 2.
中位数是将数据排序后位于中间的值。当数据个数为奇数时，它就是正中间的值；偶数时则是中间两个值的均值。中位数的位置公式为：(n + 1) / 2。
The mode is the value that occurs most frequently. A data set may have no mode, one mode (unimodal), or more than one mode (bimodal, multimodal). The mode is the only measure that can be used with non‑numerical data.
众数是出现频率最高的值。一个数据集可能没有众数，有一个众数（单峰），或多个众数（双峰、多峰）。众数是唯一能用于非数值数据的集中趋势度量。
For grouped data, the mean is estimated using the midpoints of class intervals: x̄ cy and m is the midpoint. The modal class is the class with the highest frequency density.
对于分组数据，均值用组中值估算：x̄ = ∑(f × m) / ∑f，其中f为频数，m为组中值。众数所在组是频数密度最高的组。
3. Measures of Dispersion | 离散程度的度量
Dispersion tells us how spread out the data are. Important measures include range, interquartile range (IQR), and standard deviation.
离散程度反映数据的分散情况。重要的度量包括极差、四分位距（IQR）和标准差。
The range is the difference between the largest and smallest values: Range = max – min. It is easy to calculate but heavily influenced by outliers.
极差是最大值与最小值之差：极差 = 最大值 – 最小值。计算简单，但受异常值影响很大。
The interquartile range is the range of the middle 50% of the data. It is found by subtracting the lower quartile (Q₁) from the upper quartile (Q₃):
四分位距是数据中间50%的范围。它等于上四分位数（Q₃）减去下四分位数（Q₁）：
IQR = Q₃ – Q₁
The standard deviation measures the average distance of each data value from the mean. For a sample, the formula using n – 1 is:
标准差衡量每个数据值与均值的平均距离。对于样本，使用n – 1的公式为：
s = √[∑(x – x̄)² / (n – 1)]
Variance is simply the square of the standard deviation: s². A large standard deviation indicates data are widely spread; a small one means they are clustered tightly around the mean.
方差就是标准差的平方：s²。标准差大表明数据分散得广；标准差小则意味着数据紧密地聚集在均值周围。
4. Frequency Distributions and Histograms | 频数分布与直方图
Frequency distributions organise raw data into classes. Histograms are used to display grouped continuous data. In a histogram, the area of each bar represents frequency, not the height.
频数分布将原始数据归入不同的组。直方图用于展示分组的连续数据。在直方图中，每个条形的面积（而非高度）代表频数。
When class widths are unequal, you must use frequency density. The formula is:
当组距不相等时，必须使用频数密度。公式为：
Frequency density = frequency / class width
Frequency density is plotted on the vertical axis, and the class boundaries on the horizontal axis. The total area of all bars equals the total frequency.
纵轴表示频数密度，横轴表示组界。所有条形的总面积等于总频数。
To draw a histogram, first calculate class boundaries to avoid gaps, then compute the frequency density for each class.
绘制直方图时，首先计算组界以避免间隙，然后为每个组计算频数密度。
5. Representing Data: Charts and Diagrams | 数据表示：图表与示意图
Data can be represented visually using many types of charts. Bar charts are for categorical data, while pie charts show proportions. Scatter graphs show relationships between two variables.
数据可以通过多种图表直观地表示。条形图适用于分类数据，饼图显示比例。散点图展示两个变量之间的关系。
A stem‑and‑leaf diagram retains the original data values while showing their distribution. The 'stem' typically represents the leading digits and the 'leaf' the trailing digits. A key, e.g. '2|3 means 23', is essential.
茎叶图既能保留原始数据值，又能展示其分布。“茎”通常表示前导数字，“叶”表示尾随数字。图例是必不可少的，例如“2|3表示23”。
A box‑and‑whisker plot (box plot) summarises data using the five‑ inside. Whiskers extend to the minimum and maximum, or to 1.5 × IQR to highlight outliers.
箱线图（盒须图）利用五数概括法总结数据：最小值、Q₁、中位数、Q₃和最大值。箱体从Q₁延伸到Q₃，中位数标在箱内。须线延伸到最小值和最大值，或延伸到1.5 × IQR以标示异常值。
6. Cumulative Frequency and Box Plots | 累积频数与箱线图
A cumulative frequency table adds up frequencies as you move through the classes. The cumulative frequency graph, or ogive, plots the running total against the upper class boundaries.
累积频数表是在依次经过各个组时累加频数。累积频数图（折线图）将累计总数相对于上组界作图。
From a cumulative frequency curve you can estimate the median, quartiles, and percentiles. The median corresponds to the 50th percentile (½ of the total frequency), Q₁ to the 25th percentile, and Q₃ to the 75th percentile.
通过累积频数曲线可以估算中位数、四分位数和百分位数。中位数对应第50百分位数（总频数的一半），Q₁对应第25百分位数，Q₃对应第75百分位数。
To construct a box plot from the ogive, read off the five‑number summary directly: minimum and maximum from the ends of the curve, and the quartiles from the 25th, 50th, and 75th percentile points.
利用累积频数图构建箱线图时，直接读出五数概括：最小值和最大值从曲线两端读取，四分位数从第25、50和75百分位点读取。
7. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph displays paired numerical data. Correlation describes the strength and direction of a linear relationship between two variables.
散点图展示成对的数值数据。相关性描述两个变量之间线性关系的强度和方向。
Positive correlation means that as one variable increases, the other tends to increase. Negative correlation means that as one increases, the other tends to decrease. No correlation implies no discernible linear pattern.
正相关意味着一个变量增大时，另一个也倾向于增大。负相关意味着一个增大时，另一个倾向于减小。无相关则意味着没有可辨识的线性模式。
Correlation is not causation. A strong correlation between two variables does not prove that one causes the other; there may be a lurking third variable.
相关性不代表因果性。两个变量之间很强的相关性并不能证明其中一个导致了另一个；可能存在隐藏的第三个变量。
The strength can be assessed visually or using the product‑moment correlation coefficient r, which ranges from –1 to +1. The closer to ±1, the stronger the linear relation.
相关强度可以通过目测判断，或使用积差相关系数r来衡量，其取值范围从–1到+1。越接近±1，线性关系越强。
8. Linear Regression and Line of Best Fit | 线性回归与最佳拟合线
When a scatter graph shows a linear trend, a straight line can be fitted to model the relationship. The equation of the regression line of y on x is:
当散点图显示线性趋势时，可以用一条直线来模拟关系。y关于x的回归直线方程为：
y = a + bx
Here b is the slope (gradient) and a is the y‑intercept. The slope b is calculated using:
其中b是斜率（梯度），a是y轴截距。斜率b的计算公式为：
b = Sxy / Sxx
Sxy = ∑(x – x̄)(y – ȳ) and Sxx = ∑(x – x̄)². Once b is found, the intercept a is obtained from:
Sxy = ∑(x – x̄)(y – ȳ)，Sxx = ∑(x – x̄)²。求出b后，截距a由下式得到：
a = ȳ – b x̄
The line of best fit passes through the mean point (x̄, ȳ). It can be used for interpolation (predicting within the range of data) but extrapolation (predicting outside the range) is less reliable.
最佳拟合线经过均值点(x̄, ȳ)。它可用于内插（在数据范围内进行预测），但外推（超出范围预测）的可靠性较低。
9. Basic Probability Rules | 基本概率法则
Probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain). For equally likely outcomes:
概率衡量一个事件发生的可能性大小，取值范围从0（不可能）到1（必然）。对于等可能结果：
P(A) = number of favourable outcomes / total number of outcomes
robability of an event not occurring is 1 minus the probability that it does: P(not A) = 1 – P(A).
互补法则指出，一个事件不发生的概率等于1减去它发生的概率：P(非A) = 1 – P(A)。
For mutually exclusive events (they cannot happen at the same time), the addition rule is:
对于互斥事件（它们不能同时发生），加法法则为：
P(A or B) = P(A) + P(B)
f events are not mutually exclusive, we must subtract the overlap: P(A or B) = P(A) + P(B) – P(A and B).
如果事件不是互斥的，我们必须减去重叠部分：P(A或B) = P(A) + P(B) – P(A和B)。
Independent events have no influence on each other. The multiplication rule for independent events is:
独立事件互不影响。独立事件的乘法法则为：
P(A and B) = P(A) × P(B)
10. Tree Diagrams and Conditional Probability | 树状图与条件概率
Tree diagrams help visualise sequences of events. Probabilities are written on branches, and outcomes at the ends. The sum of probabilities on branches from a single point is 1.
树状图有助于将一系列事件可视化。概率写在枝干上，结果写在末尾。从同一个点出发的各枝干上的概率之和为1。
To find the probability of a combined sequence along a path, multiply the probabilities along that path. For multiple paths leading to the same outcome, add their probabilities.
要计算某条路径上的组合事件概率，将沿途的概率相乘。如果有多条路径通向同一结果，则将这些路径的概率相加。
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted P(A|B) and read as 'probability of A given B'. The formula is:
条件概率是指在另一个事件已经发生的情况下，某个事件发生的概率。记作P(A|B)，读作“在B发生的条件下A发生的概率”。公式为：
P(A|B) = P(A ∩ B) / P(B)
From this we obtain the general multiplication rule: P(A ∩ B) = P(A| bility of another. Tree diagrams naturally incorporate conditional probabilities on second‑stage branches.
由此可得一般乘法法则：P(A ∩ B) = P(A|B) × P(B)。它适用于因一个事件的结果影响另一个事件概率的相依事件。树状图在第二阶段枝干上自然地体现了条件概率。
11. Index Numbers | 指数
Index numbers measure changes over time in variables such as prices, quantities, or economic indicators. They always involve a base period, given the value 100.
指数用来衡量价格、数量或经济指标等变量随时间的变化。它们总会设定一个基期，赋值为100。
The simple price relative for a single commodity comparing the current price (Pₙ) with the base price (P₀) is:
单个商品的简单价比，将现期价格（Pₙ）与基期价格（P₀）比较，其计算公式为：
Price relative = (Pₙ / P₀) × 100
A weighted aggregate index uses quantities or expenditure as weights to reflect the importance of different items. The base‑weighted (Laspeyres) price index formula is:
加权总指数使用数量或支出作为权重，以反映不同商品的重要程度。基期加权（拉氏）价格指数公式为：
∑(Pₙ × Q₀) / ∑(P₀ × Q₀) × 100
where Q₀ represents base‑period quantities. An index above 100 indicates an increase relative to the base period; below 100 indicates a decrease.
其中Q₀表示基期数量。指数大于100表示相对于基期有所增长；小于100则表示下降。
12. Time Series and Moving Averages | 时间序列与移动平均
A time series records data at regular intervals over time. It may show a trend (long‑term movement), seasonal variation (regular fluctuations), and random (irregular) components.
时间序列是按固定时间间隔记录的数据。它可能表现出趋势（长期动向）、季节性变化（规律性波动）和随机（不规则）成分。
Moving averages smooth out short‑term fluctuations to reveal the underlying trend. A 3‑point moving average replaces each value with the mean of itself and its immediate neighbours:
移动平均通过平滑短期波动来揭示潜在趋势。三点移动平均将每个值替换为它自身及其相邻点的均值：
MA(3) = (xₙ₋₁ + xₙ + xₙ₊₁) / 3
For quarterly data, a 4‑point centred moving average is often used to remove seasonal effects. It involves averaging four consecutive values and then centring by averaging two successive moving averages.
对于季度数据，常使用四点中心移动平均来消除季节性影响。它先对四个连续值求平均，再对连续的两个移动平均值求平均以实现中心化。
Once the trend is identified, seasonal variation can be estimated by subtracting the trend from the original data (additive model) or by division (multiplicative model). These estimates help make adjusted forecasts.
一旦识别出趋势，就可以通过用原始数据减去趋势（加法模型）或除以趋势（乘法模型）来估算季节性变化。这些估算值有助于进行经调整的预测。
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