Year 10 Edexcel Statistics: Formula & Theorem Quick Reference Guide | Year 10 Edexcel 统计：公式定理速查手册

📚 Year 10 Edexcel Statistics: Formula & Theorem Quick Reference Guide | Year 10 Edexcel 统计：公式定理速查手册

This quick reference guide brings together the essential formulas, theorems and key concepts you must have at your fingertips for Year 10 Edexcel Statistics. Each section presents the core rule, its notation and a concise example, helping you revise effectively for assessments and build a solid foundation for further study.

本速查手册汇集了 Year 10 Edexcel 统计学科必须熟记的核心公式、定理与关键概念。每个小节均呈现核心规则、符号说明与简明示例，帮助你高效备考、为后续学习打下扎实基础。

1. Mean, Median and Mode (Averages) | 平均数、中位数与众数

The three measures of central tendency summarise a data set with a single representative value. The mean is the sum of all values divided by the number of values. The median is the middle value when the data are ordered. The mode is the value that occurs most often.

三种集中趋势度量用单一代表性数值概括数据集。平均数为所有数据之和除以数据个数。中位数是将数据排序后位于中间的值。众数是出现次数最多的值。

Sample mean: x̄ = (∑x) / n

Median position for raw data: (n + 1) / 2

For a list of n numbers, first arrange them in ascending order. If n is odd, the median is the value at position (n+1)/2. If n is even, the median is the mean of the two central values.

对于有 n 个数据的列表，先按升序排列。若 n 为奇数，中位数为位于第 (n+1)/2 个位置的值；若 n 为偶数，中位数为中间两个数的平均数。

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

Spread tells us how spread out the data are. The range is the simplest measure of spread. The interquartile range (IQR) focuses on the middle 50% of the data and is less affected by outliers.

离散程度描述数据的分散情况。极差是最简单的离散度量。四分位距 (IQR) 则关注中间 50% 的数据，受异常值影响较小。

Range = Maximum − Minimum

IQR = Upper quartile (Q₃) − Lower quartile (Q₁)

To find Q₁ and Q₃ for discrete data, use positions (n+1)/4 and 3(n+1)/4. For grouped data, quartiles are read from a cumulative frequency graph or calculated using linear interpolation.

对于离散数据，Q₁ 和 Q₃ 的位置分别为 (n+1)/4 和 3(n+1)/4。对于分组数据，四分位数可从累积频数图中读取，或通过线性插值计算。

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

Standard deviation measures the average distance of data values from the mean. A low standard deviation indicates data points are clustered closely around the mean; a high one suggests wide spread. Variance is the square of the standard deviation.

标准差衡量数据值与平均数的平均距离。标准差较小说明数据点紧密聚集在均值附近，较大则表示分散程度高。方差为标准差的平方。

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

Alternative form: s = √[ (∑x² − (∑x)²/n) / (n − 1) ]

Variance = s²

The denominator (n − 1) is used for a sample. When the whole population is considered, the denominator becomes n and the standard deviation is denoted by σ.

样本标准差的分母为 (n − 1)。若为总体数据，分母则为 n，总体标准差用 σ 表示。

4. Estimated Mean from Grouped Data | 分组数据的平均值估算

When data are summarised in a frequency table with class intervals, we cannot calculate the exact mean. Instead we approximate the mean by using the midpoints of the intervals.

当数据以分组频数表呈现时，无法计算精确平均值。此时我们使用各区间的组中点来估算平均数。

Estimated mean: x̄ ≈ (∑ f × m) / ∑ f

Here f is the frequency of each class and m is the midpoint (upper boundary + lower boundary) ÷ 2. Multiplying each midpoint by its frequency, summing these products and dividing by the total frequency gives the estimate.

其中 f 为各组频数，m 为组中点（即上限与下限的平均数）。将各中点与相应频数相乘、求和，再除以总频数，即得估算平均值。

5. Frequency Density for Histograms | 直方图频率密度

In a histogram, the area of each bar is proportional to the frequency. Because class widths may be unequal, we use frequency density to determine the height of each bar.

直方图中，每个条形的面积与频数成正比。由于组距可能不等，我们采用频率密度来确定条形的高度。

Frequency density = Frequency / Class width

Once frequency density is calculated, the height of each bar on the histogram equals that density. The area (frequency density × class width) then returns the frequency for the interval. Always check that bars touch and the horizontal axis is continuous.

计算出频率密度后，直方图中各条形的垂直高度即为相应的频率密度。面积（频率密度 × 组距）即为该区间的频数。务必确保条形相连、横轴连续。

6. Probability Rules: Addition and Multiplication | 概率运算：加法与乘法法则

Probability values range from 0 (impossible) to 1 (certain). Two fundamental rules allow us to combine probabilities: the addition rule for ‘or’ and the multiplication rule for ‘and’.

概率取值在 0（不可能）到 1（必然）之间。两条基本法则用于组合概率：“或”的加法法则与“且”的乘法法则。

General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive (cannot happen together), then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

若 A 与 B 互斥（不会同时发生），则 P(A ∩ B) = 0，故 P(A ∪ B) = P(A) + P(B)。

Multiplication rule for independent events: P(A ∩ B) = P(A) × P(B)

Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, multiply their probabilities to find P(A and B).

若两事件独立（一件发生不影响另一件的概率），则称为独立事件。对于独立事件，将它们的概率相乘即可求得 P(A 且 B)。

7. Conditional Probability | 条件概率

Conditional probability is the probability of an event occurring given that another event has already occurred. It is written as P(A | B), meaning ‘probability of A given B’.

条件概率指在另一事件已经发生的条件下，某事件发生的概率。记作 P(A | B)，表示“在 B 发生的条件下 A 发生的概率”。

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0

Tree diagrams are extremely useful for conditional probability problems: multiply along branches and add between branches. Remember that the probabilities on the second set of branches change depending on the first outcome.

树形图在处理条件概率问题时极为有用：沿分支相乘，分支间相加。注意第二层分支的概率会因第一层结果而改变。

8. Permutations and Combinations | 排列与组合

Permutations count the number of ways to arrange items where order matters. Combinations count selections where order does not matter. The factorial notation n! means n × (n−1) × … × 1.

排列计算考虑顺序时物品的排列方式数目。组合则计算不考虑顺序的选择方式数目。阶乘记号 n! 表示 n × (n−1) × … × 1。

Permutations: nPr = n! / (n − r)!

Combinations: nCr = n! / [r! (n − r)!]

nPr gives the number of ways to arrange r items chosen from n distinct items. nCr gives the number of ways to select r items from n distinct items without regard to order. These formulas underpin the binomial distribution.

nPr 表示从 n 个不同物品中选取 r 个进行排列的方法数。nCr 表示从 n 个不同物品中选取 r 个而不考虑顺序的选择方法数。这些公式是二项分布的基础。

9. Binomial Distribution | 二项分布

A binomial distribution arises when we have a fixed number of independent trials, each with only two outcomes (success or failure) and a constant probability of success, p. It is denoted X ~ B(n, p).

当进行固定次数的独立试验，每次试验只有两种结果（成功或失败）且成功概率 p 恒定时，该分布为二项分布，记作 X ~ B(n, p)。

P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ

Mean: μ = np

Variance: σ² = np(1 − p)

Here n is the number of trials, r is the number of successes, and p is the probability of success on a single trial. Use the nCr formula to find the binomial coefficient.

其中 n 为试验次数，r 为成功次数，p 为单次试验成功概率。二项式系数用 nCr 公式求得。均值为 np，方差为 np(1−p)。

10. Cumulative Frequency and Box Plots | 累积频数与箱线图

Cumulative frequency is the running total of frequencies. A cumulative frequency graph (or ogive) plots cumulative frequency against the upper class boundary, allowing you to estimate medians and quartiles for grouped data.

累积频数是频数的累加总和。累积频数图（折线图）将累积频数对组距上限描点，可以估算分组数据的中位数和四分位数。

To find median from graph: locate total frequency / 2 on the vertical axis, read across to the curve and down to the horizontal axis.

Lower quartile: total frequency / 4; Upper quartile: 3 × total frequency / 4.

A box plot (box‑and‑whisker diagram) uses the five‑number summary: minimum, Q₁, median, Q₃ and maximum. The box spans from Q₁ to Q₃, with the median marked inside. Whiskers extend to the minimum and maximum values that are not outliers.

箱线图（盒须图）利用五数概括：最小值、下四分位数 Q₁、中位数、上四分位数 Q₃ 和最大值。盒体从 Q₁ 跨到 Q₃，内部标出中位数。须线延伸至非异常值的最小和最大值。

Outliers can be identified using the 1.5 × IQR rule: an outlier is any value below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR.

异常值可用 1.5 × IQR 法则识别：低于 Q₁ − 1.5×IQR 或高于 Q₃ + 1.5×IQR 的值被视为异常。

11. Transforming Data (Coding) | 数据编码变换

When data values are large or difficult to work with, we can use coding to simplify calculations. If we transform x to y = (x − a)/b, the mean and standard deviation change in a predictable way.

当数据值较大或不便计算时，可用编码简化。若将 x 变换为 y = (x − a)/b，则平均数与标准差可按可预测的方式变化。

ȳ = (x̄ − a) / b

sᵧ = sₓ / b

Adding or subtracting a constant affects the mean but not the standard deviation. Multiplying or dividing by a constant affects both the mean and the standard deviation (or variance). Always reverse the coding after calculations.

加减常数仅影响平均数，不影响标准差。乘除常数则同时影响平均数和标准差（或方差）。计算完毕后务必反向解码。

12. Index Numbers (Foundation) | 指数基础

Index numbers compare the price or quantity of an item over time, with a base period value set to 100. They are widely used in economic statistics.

指数用于比较某物品价格或数量随时间的变化，以基期数值设为 100。在经济统计中被广泛使用。

Index number = (Value in current period / Value in base period) × 100

An index above 100 shows an increase relative to the base period; an index below 100 shows a decrease. Chain base indices use the previous period as the base, while fixed base indices always compare with the same base period.

指数高于 100 表示相对于基期上升；低于 100 表示下降。链基指数以前一期为基期，而定基指数始终与同一基期比较。

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Year 10 Edexcel Statistics: Formula & Theorem Quick Reference Guide | Year 10 Edexcel 统计：公式定理速查手册