📚 Year 10 Edexcel Statistics: Quick Reference Formulas & Theorems | Year 10 Edexcel 统计：公式定理速查手册
This quick‑reference handbook compiles the essential formulas and theorems you need for the Year 10 Edexcel Statistics course. Use it as a revision checklist, a moment‑before‑exam refresher, or a foundation for your GCSE Statistics studies. Every formula is stated in its simplest mathematical form, with clear notation and a reminder of when and how to apply it.
这份速查手册整理了 Year 10 Edexcel 统计课程所需的所有核心公式与定理。你可以把它当作复习清单、考前速览，或 GCSE 统计学习的基石。每个公式都以最简洁的数学形式给出，并配有清晰的符号说明与应用情境，方便你随时查阅。
1. Types of Data | 数据类型
Data is categorised as qualitative or quantitative. Quantitative data splits into discrete and continuous types.
数据分为定性数据和定量数据。定量数据又分为离散数据和连续数据。
Qualitative data describes qualities or categories (e.g. eye colour, type of car). Quantitative data describes numerical values.
定性数据描述品质或类别（例如眼睛颜色、汽车型号）。定量数据描述数值。
Discrete data can only take specific, separate values, often whole numbers (e.g. shoe size, number of pets). Continuous data can take any value within a range, measured on a continuous scale (e.g. height, mass, time).
离散数据只能取特定的、分隔的数值，通常是整数（如鞋码、宠物数量）。连续数据可以在一个区间内取任意值，在连续尺度上测量（如身高、质量、时间）。
| Type | Examples |
|---|---|
| Qualitative | Gender, favourite sport, blood type |
| Quantitative Discrete | Number of goals, number of children in a family |
| Quantitative Continuous | Temperature, length of time, weight |
2. Sampling Techniques | 抽样方法
A sample is a subset of a population. Sampling methods determine how the sample is selected. The three core methods at Year 10 level are random, stratified, and systematic sampling.
样本是总体的一个子集。抽样方法决定了样本如何选取。Year 10 阶段核心的三种抽样方法是随机抽样、分层抽样和系统抽样。
Random sampling: every member of the population has an equal chance of being chosen. This reduces bias but requires a complete sampling frame.
随机抽样：总体中的每个成员有均等的机会被选入样本。这能减少偏差，但需要一个完整的抽样框。
Stratified sampling: the population is divided into groups (strata) and a random sample is taken from each group in proportion to its size. Formula for the number to select from a stratum:
分层抽样：将总体划分为若干层（组），然后按各层大小比例从每层中随机抽取样本。从某层抽取数量的公式：
Number from stratum = (stratum size ÷ total population) × sample size
某层抽取数 = (该层大小 ÷ 总体大小) × 样本容量
Systematic sampling erval k = population size ÷ sample size. A random starting point is selected, then every k‑th member is taken.
系统抽样：从名单中按固定间隔选取样本。间隔 k = 总体大小 ÷ 样本容量。随机选择一个起点，然后每隔 k 个成员抽取一个。
3. Frequency Distributions | 频率分布
A frequency table lists data values (or groups) alongside their frequencies. The sum of all frequencies is the total number of observations n.
频率表列出数据值（或组）及其对应的频数。所有频数之和为总观测数 n。
t overlap. Midpoints are used in calculations: midpoint = (lower bound + upper bound) ÷ 2.
对分组数据而言，组距不应重叠。计算时使用组中点：组中点 = (下界 + 上界) ÷ 2。
The cumulative frequency of a class is the sum of its frequency and all earlier frequencies. This is the foundation of cumulative frequency graphs.
某一组的累积频数是该组及其之前所有组频数之和。这是累积频率图的基础。
4. Averages: Mean, Median, Mode | 平均数：均值、中位数、众数
Three measures of central tendency summarise a data set with a typical value.
三种中心趋势度量用一个典型值概括数据集。
Mean (arithmetic average):
均值（算术平均数）：
Mean = ∑x ÷ n
data, use midpoints: Mean = ∑(f × midpoint) ÷ ∑f, where f is the frequency of each class.
对于分组数据，使用组中点：均值 = ∑(f × 组中点) ÷ ∑f，其中 f 是各组频数。
Median of : the middle value when data are ordered. Position of median = (n + 1) ÷ 2. If n is even, the median is the average of the two middle values.
中位数：数据排序后位于中间的值。中位数位置 = (n + 1) ÷ 2。若 n 为偶数，中位数是中间两个值的平均数。
Mode: the value that appears most often. For grouped data, the modal class is the class interval with the highest frequency.
众数：出现次数最多的值。对于分组数据，众数组是频数最高的组距。
5. Quartiles and Interquartile Range | 四分位数与四分位距
Quartiles split an ordered data set into four equal parts.
四分位数将有序数据分为四个等份。
1) ÷ 4. If the positions are not integers, take the average of the two surrounding values.
下四分位数 (Q₁) 的位置 = (n + 1) ÷ 4。上四分位数 (Q₃) 的位置 = 3(n + 1) ÷ 4。如果得到的位置不是整数，取相邻两个值的平均数。
Interquartile range (IQR) = Q₃ − Q₁. IQR measures the spread of the middle 50% of data and is resistant to outliers.
四分位距 (IQR) = Q₃ − Q₁。IQR 衡量中间 50% 数据的离散程度，不受极端值影响。
Range = maximum value − minimum value. It is sensitive to extreme values.
极差 = 最大值 − 最小值。极差易受极端值影响。
6. Box Plots | 箱线图
A box plot displays the five‑number summary: minimum, Q₁, median, Q₃, and maximum.
箱线图显示五数概括：最小值、Q₁、中位数、Q₃ 和最大值。
The box spans from Q₁ to Q₃, with a vertical line inside at the median. Whiskers extend to the minimum and maximum, unless there are outliers.
箱子从 Q₁ 延伸到 Q₃，箱内有一条垂直于中位数的直线。触须延伸至最小值和最大值，除非存在异常值。
ove Q₃ + 1.5 × IQR. For GCSE Statistics, you may be told to draw a box plot using the five‑number summary directly.
异常值通常定义为小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值。在 GCSE 统计中，题目可能要求你直接使用五数概括绘制箱线图。
7. Cumulative Frequency Graphs | 累积频率图
Cumulative frequency is plotted against the upper boundary of each class. Points are joined with a smooth curve, not straight lines.
累积频数相对于每组的上界作图。各点用平滑曲线连接，而非直线。
To estimate the median, read the value on the horizontal axis corresponding to n/2 on the cumulative frequency axis. For Q₁, use n/4; for Q₃, use 3n/4.
估计中位数时，在累积频率轴上找到 n/2，对应到水平轴上的值。Q₁ 使用 n/4，Q₃ 使用 3n/4。
The cumulative frequency curve also allows you to find percentiles and to read the number of observations below a given value.
累积频率曲线还可以用来查找百分位数，并读取低于某一给定值的观测值数量。
8. Histograms and Frequency Density | 直方图与频率密度
Histograms display grouped continuous data. The area of each bar is proportional to frequency. When class widths are unequal, frequency density must be calculated.
直方图用于展示分组连续数据。每个条形的面积与频数成正比。当组距不相等时，必须计算频率密度。
Frequency density = frequency ÷ class width
频率密度 = 频数 ÷ 组距
To draw a histogram, plot frequency density on the vertical axis and the variable on the horizontal axis. The bar for each class extends from the lower bound to the upper bound.
绘制直方图时，纵轴为频率密度，横轴为变量。每个组距的条形从下界延伸到上界。
| Class width | Frequency | Frequency density |
|---|---|---|
| 0 ≤ t < 10 | 20 | 20 ÷ 10 = 2 |
| 10 ≤ t < 15 | 15 | 15 ÷ 5 = 3 |
ity × class width = frequency. Use this to calculate missing frequencies from a histogram.
记住：条形面积 = 频率密度 × 组距 = 频数。利用这一关系可以从直方图中计算缺失的频数。
9. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph displays bivariate data, showing the relationship between two variables.
散点图展示双变量数据，反映两个变量之间的关系。
Correlation describes the strength and direction of a linear relationship: positive, negative, or zero. Correlation is not causation.
相关性描述线性关系的强度和方向：正相关、负相关或零相关。相关不代表因果。
A line of best fit can be drawn by eye, passing as close as possible to all points. The line of best fit must go through the mean point (x̄, ȳ).
通过目测可以画出一条最佳拟合线，使其尽可能靠近所有点。最佳拟合线必须通过均值点 (x̄, ȳ)。
The mean point formula: x̄ = ∑x ÷ n, ȳ late (within the data range) but avoid extrapolation (outside the range) unless confident.
均值点公式：x̄ = ∑x ÷ n, ȳ = ∑y ÷ n。可以用该线进行内插（在数据范围内），但除非有把握，应避免外推（范围之外）。
10. Probability Formulas and Theorems | 概率公式与定理
Probability measures the chance of an event occurring. It always lies between 0 and 1 inclusive.
概率衡量某一事件发生的机会，取值范围为 0 到 1 （含）。
umber of favourable outcomes ÷ total number of outcomes, assuming equally likely outcomes.
基本概率：P(A) = 有利结果数 ÷ 总结果数，假定结果等可能。
on: P(A) = frequency of event A ÷ total trials, used when outcomes are not equally likely.
相对频率近似：P(A) = 事件 A 发生的频数 ÷ 试验总次数，用于结果不等可能的情况。
Key formulas:
P(not A) = 1 − P(A)
P(A or B) = P(A) + P(B) − P(A and B)
For mutually exclusive events: P(A or B) = P(A) + P(B)
For independent events: P(A and B) = P(A) × P(B)
中文关键公式：
P(非 A) = 1 − P(A)
P(A 或 B) = P(A) + P(B) − P(A 且 B)
互斥事件：P(A 或 B) = P(A) + P(B)
独立事件：P(A 且 B) = P(A) × P(B)
A probability tree diagram helps visualise multi‑stage experiments. Multiply along branches for combined events; add probabilities of different outcomes if they are mutually exclusive.
概率树图有助于可视化多步试验。沿分支相乘得到组合事件的概率；若不同结果互斥，则将它们的概率相加。
cy of an event = probability × number of trials; this comes from the law of large numbers.
事件的期望频率 = 概率 × 试验次数；这源自大数定律。
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