Year 9 Edexcel Statistics: Formula & Theorem Quick Reference Handbook | 九年级 Edexcel 统计：公式定理速查手册

📚 Year 9 Edexcel Statistics: Formula & Theorem Quick Reference Handbook | 九年级 Edexcel 统计：公式定理速查手册

This quick reference handbook brings together all the essential formulas and theorems you need for Year 9 Edexcel Statistics. From averages and measures of spread to probability rules and data representation, each topic is presented with clear definitions, worked examples, and bilingual explanations. Use it as your go-to revision guide to master the fundamentals and build confidence for assessments.

本速查手册汇集了九年级 Edexcel 统计课程中所有的核心公式和定理。从平均数、离散量数到概率规则和数据表示，每个主题都配有清晰的定义、示例和中英文双语讲解。把它当作你的首选复习指南，掌握基础，自信应对考试。

1. Measures of Central Tendency | 集中趋势的度量

The mean is the arithmetic average, calculated by summing all data values and dividing by the number of values. For a data set x₁, x₂, …, xₙ, the formula for the mean is:

算数平均数是通过将所有数据值相加后除以数据个数计算得出的。对于数据集 x₁, x₂, …, xₙ，平均数的公式为：

Mean (x̄) = Σx / n

The symbol Σx represents the sum of all data points, and n is the total frequency (sample size).

符号 Σx 表示所有数据点的总和，n 是总频数（样本容量）。

The median is the middle value when data are arranged in order. For an odd number of values, median = value at position (n+1)/2. For an even number, median = average of the two middle values at positions n/2 and n/2 + 1.

中位数是将数据按顺序排列后处于中间的值。当数据个数为奇数时，中位数 = 第 (n+1)/2 个位置的值；当个数为偶数时，中位数 = 第 n/2 和 n/2 + 1 两个位置的值的平均数。

The mode is the value with the highest frequency. In a grouped frequency table, the modal class is the class interval with the greatest frequency.

众数是出现频数最高的值。在分组频数表中，众数所在的组是频数最大的组区间。

2. Range and Interquartile Range | 极差与四分位距

The range measures how spread out the data are. It is the difference between the maximum and minimum values.

极差衡量数据的离散程度，它是最大值与最小值之差。

Range = Largest value − Smallest value

The interquartile range (IQR) focuses on the middle 50% of the data, so it is not affected by extreme values. First find the lower quartile Q₁ (25th percentile) and upper quartile Q₃ (75th percentile).

四分位距（IQR）着眼于中间 50% 的数据，因此不受极端值影响。首先找出下四分位数 Q₁（第 25 百分位数）和上四分位数 Q₃（第 75 百分位数）。

IQR = Q₃ − Q₁

To find quartiles from a list, locate the median, then the median of the lower half (for Q₁) and the median of the upper half (for Q₃). For grouped data, linear interpolation may be used.

从列表中找四分位数：先求中位数，再分别求出下半部分的中位数（Q₁）和上半部分的中位数（Q₃）。对于分组数据，可能需要使用线性插值。

3. Variance and Standard Deviation | 方差与标准差

Variance quantifies the average squared deviation from the mean. For a population, the formula is:

方差量化了各个数据与平均数之差的平方的平均数。对于总体，公式为：

Population variance σ² = Σ(x − μ)² / N

where μ is the population mean and N is the population size. For a sample, we usually divide by (n−1) to get an unbiased estimate:

其中 μ 是总体平均数，N 是总体大小。对于样本，我们通常除以 (n−1) 来得到无偏估计：

Sample variance s² = Σ(x − x̄)² / (n−1)

Standard deviation is the square root of the variance. It has the same units as the original data and is often used to compare spreads.

标准差是方差的平方根。它与原始数据具有相同的单位，常用于比较离散程度。

Sample standard deviation s = √[ Σ(x − x̄)² / (n−1) ]

When data are given in a frequency table, use Σf(x − x̄)² instead of Σ(x − x̄)², where f is the frequency.

当数据以频数表给出时，用 Σf(x − x̄)² 代替 Σ(x − x̄)²，其中 f 是频数。

4. Basic Probability | 基础概率

The probability of an event A is the proportion of favourable outcomes to all possible equally likely outcomes.

事件 A 的概率是事件 A 包含的结果数与所有等可能结果总数之比。

P(A) = Number of outcomes in A / Total number of outcomes

Probability values always lie between 0 and 1 inclusive. P(A) = 0 means the event is impossible; P(A) = 1 means the event is certain.

概率值总是在 0 到 1 之间（含 0 和 1）。P(A) = 0 表示事件不可能发生；P(A) = 1 表示事件必然发生。

The complement rule states that the probability of not A is 1 minus the probability of A:

互补规则指出，非 A 的概率等于 1 减去 A 的概率：

P(not A) = 1 − P(A)

In Year 9, you will also meet expected frequency: Expected = probability × number of trials.

在九年级，你还会接触到期望频数：期望值 = 概率 × 试验次数。

5. Mutually Exclusive and Independent Events | 互斥事件与独立事件

Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B, the probability that either A or B occurs is the sum of their individual probabilities.

如果两个事件不能同时发生，则它们互斥。对于互斥事件 A 和 B，A 或 B 发生的概率等于各自概率之和。

P(A or B) = P(A) + P(B) (if A and B are mutually exclusive)

Two events are independent if the occurrence of one does not affect the probability of the other. Multiplication rule for independent events:

如果一件事的发生不影响另一件事的概率，则这两个事件独立。独立事件的乘法法则：

P(A and B) = P(A) × P(B) (if A and B are independent)

Be careful: for events that are not mutually exclusive, use the general addition rule P(A or B) = P(A) + P(B) − P(A and B). The concepts of independence and mutual exclusivity should not be confused.

注意：对于非互斥事件，应使用一般加法公式 P(A 或 B) = P(A) + P(B) − P(A 和 B)。独立和互斥的概念不要混淆。

6. Tree Diagrams and Conditional Probability | 树形图与条件概率

Tree diagrams are used to show all possible outcomes of two or more events, especially when events are sequential. Probabilities are written on each branch. To find the probability of a combination of events, multiply the probabilities along the branches.

树形图用于表示两个或多个事件的所有可能结果，尤其当事件有先后顺序时。概率写在每个分支上。要找出组合事件的概率，将路径上的概率相乘。

Conditional probability is the probability of event B given that event A has occurred, written as P(B|A). For independent events, P(B|A) = P(B). A common formula is:

条件概率是在事件 A 已经发生的条件下事件 B 发生的概率，记作 P(B|A)。对于独立事件，P(B|A) = P(B)。常用公式为：

P(A and B) = P(A) × P(B|A)

From a tree diagram, P(B|A) is simply the probability written on the second branch after A has occurred.

在树形图中，P(B|A) 就是 A 发生之后第二条分支上写明的概率。

7. Bar Charts, Pie Charts and Histograms | 条形图、饼图与直方图

A bar chart presents categorical data with rectangular bars. The length (or height) of each bar is proportional to the frequency or value it represents. Bars are separated to show categories are distinct.

条形图用矩形条表示分类数据。每个条的长度（或高度）与其代表的频数或数值成正比。条形之间有间隔，表示类别是离散的。

A pie chart shows data as sectors of a circle. The angle of each sector = (frequency / total frequency) × 360°.

饼图以圆心角的扇形表示数据。每个扇形的角度 = (频数 / 总频数) × 360°。

A histogram is used for grouped continuous data. Unlike bar charts, there are no gaps between bars. The area of each bar is proportional to the frequency. Frequency density = frequency / class width. On a histogram, the vertical axis is frequency density.

直方图用于分组连续数据。与条形图不同，直方图的条形之间没有间隙。每个条的面积与频数成正比。频数密度 = 频数 / 组距。在直方图中，纵轴表示频数密度。

8. Frequency Tables and Cumulative Frequency | 频数表与累积频数

A frequency table lists data values or intervals and how often they occur. It is the starting point for calculating mean, median and mode for grouped data. To estimate the mean from a grouped frequency table, use the midpoints of each class interval.

频数表列出数据值或区间及其出现的次数，它是计算分组数据平均数、中位数和众数的起点。要从分组频数表估算平均数，需要使用每个组区间的组中值。

Estimated mean = Σ(f × x) / Σf

where x is the midpoint of each class and f is the frequency.

其中 x 是每个组的组中值，f 是频数。

Cumulative frequency is the running total of frequencies. It is used to draw a cumulative frequency curve and to find the median, quartiles and percentiles. On the cumulative frequency graph, the median is found by locating the value at ½ of the total frequency, Q₁ at ¼, and Q₃ at ¾.

累积频数是频数的逐项累加。它用于绘制累积频数曲线，并求中位数、四分位数和百分位数。在累积频数图上，中位数位于总频数 ½ 处对应的值，Q₁ 位于 ¼ 处，Q₃ 位于 ¾ 处。

9. Scatter Graphs and Correlation | 散点图与相关性

A scatter graph displays the relationship between two numerical variables. Each point represents a pair of values (x, y). The pattern of points can indicate positive correlation (as x increases, y increases), negative correlation (as x increases, y decreases), or no correlation.

散点图显示两个数值变量之间的关系。每个点代表一对数值 (x, y)。点的分布模式可以表明正相关（x 增大，y 增大）、负相关（x 增大，y 减小）或无相关。

Correlation does not imply causation. Outliers are points that lie far from the main pattern and can affect the interpretation.

相关性并不意味着因果关系。离群值是远离主要模式的点，会影响对相关性的解读。

You can draw a line of best fit on a scatter graph to make predictions. The line should go through the mean point (x̄, ȳ) and follow the general trend. Interpolation (within the data range) is more reliable than extrapolation (outside the data range).

你可以在散点图上画出最佳拟合线来进行预测。该直线应穿过均值点 (x̄, ȳ) 并遵循总体趋势。内插（在数据范围内）比外推（超出数据范围）更可靠。

10. Sampling Methods | 抽样方法

A population is the entire group you want to study. A sample is a subset of the population. Good sampling methods aim to produce a representative sample without bias.

总体是你想研究的整个群体。样本是总体的一个子集。良好的抽样方法旨在产生一个无偏的代表性样本。

Random sampling: every member of the population has an equal chance of being selected. This can be done using random number generators or drawing lots.

随机抽样：总体中的每个成员都有相等的机会被选中。可以通过随机数生成器或抽签来实现。
Stratified sampling: the population is divided into groups (strata), and a random sample is taken from each group in proportion to its size. This ensures all groups are fairly represented.

分层抽样：将总体分成不同的层，然后按各层在总体中的比例随机抽取样本。这样可以确保所有群体都能公平体现。
Systematic sampling: select every k‑th member after a random starting point. Useful for large populations.

系统抽样：在随机起点后，每隔 k 个成员选取一个。适用于大规模总体。
Convenience sampling: choose members who are easiest to reach. This is quick but often biased and not representative.

便利抽样：选择最容易接触到的成员。这种方法快捷但往往有偏误，缺乏代表性。

In Year 9, understanding the advantages and disadvantages of each method helps you evaluate statistical conclusions.

在九年级，理解每种方法的优缺点有助于你评判统计结论的可靠性。

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Year 9 Edexcel Statistics: Formula & Theorem Quick Reference Handbook | 九年级 Edexcel 统计：公式定理速查手册