Year 11 WJEC Statistics: Quick Reference Formula and Theorem Handbook | WJEC 统计 Year 11：公式定理速查手册

📚 Year 11 WJEC Statistics: Quick Reference Formula and Theorem Handbook | WJEC 统计 Year 11：公式定理速查手册

This handbook brings together all the essential formulas, rules and theorems you will need for the WJEC GCSE Statistics examination. It is designed for rapid revision, with every concept presented in a clear, bilingual format. Keep this page bookmarked and test yourself regularly on each section.

本手册汇集了 WJEC GCSE 统计考试所需的所有核心公式、法则和定理。手册采用双语对照的形式，方便快速查阅和复习。建议将本页面加入书签，并定期对每个小节进行自测。

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

The mean of a set of values is the sum of the values divided by the number of values. For grouped data, use the midpoints of the class intervals.

一组数据的平均值等于所有数值之和除以数值的个数。对于分组数据，使用组中值进行计算。

• Mean (ungrouped): x̄ = Σx / n
• 平均数（未分组）：x̄ = Σx / n
• Mean (grouped): x̄ = Σ fx / Σ f
• 平均数（分组）：x̄ = Σ fx / Σ f

The median is the middle value when the data are arranged in order. If n is even, it is the average of the two middle values. For grouped data, use linear interpolation within the median class interval.

中位数是将数据按大小顺序排列后中间位置的值。如果数据个数 n 为偶数，则取中间两个数的平均值。对于分组数据，在中位数组内进行线性插值。

The mode is the value that occurs most frequently. For grouped data, the modal class is the class interval with the highest frequency.

众数是出现次数最多的数值。分组数据的众数组是频数最高的组距。

2. Measures of Dispersion | 离散程度的度量

The range is the difference between the largest and smallest values. It is a simple measure of spread but is affected by outliers.

极差是最大值与最小值之差。它是一个简单的离散程度度量，但容易受异常值影响。

Variance and standard deviation measure how far the data values are spread out from the mean.

方差和标准差用于衡量数据值相对于平均值的分散程度。

• Variance (population): σ² = Σ (x – μ)² / n
• 总体方差：σ² = Σ (x – μ)² / n
• Standard deviation (sample): s = √ [ Σ (x – x̄)² / (n – 1) ]
• 样本标准差：s = √ [ Σ (x – x̄)² / (n – 1) ]

For the WJEC course, you may use the divisor n when working with a complete data set, and n-1 when the data is a sample. Always check the question context.

在 WJEC 课程中，对于完整数据集可使用除数 n，当数据为样本时使用 n-1。作答时务必根据题目背景选择合适的公式。

3. Quartiles and Interquartile Range | 四分位数与四分位距

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.

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。

The interquartile range (IQR) = Q₃ – Q₁. It measures the spread of the middle 50% of the data and is resistant to outliers.

四分位距 (IQR) = Q₃ – Q₁。它衡量中间 50% 数据的分散程度，且不受异常值影响。

To construct a box-and-whisker plot, you need the minimum, Q₁, median, Q₃ and the maximum.

绘制箱线图需要最小值、Q₁、中位数、Q₃ 和最大值。

• IQR = Q₃ – Q₁
• 四分位距：IQR = Q₃ – Q₁
• Lower fence = Q₁ – 1.5 × IQR, Upper fence = Q₃ + 1.5 × IQR (for identifying outliers)
• 下边缘 = Q₁ – 1.5 × IQR，上边缘 = Q₃ + 1.5 × IQR（用于识别异常值）

4. Probability Rules | 概率基本法则

Probability is a measure of the likelihood that an event will occur, given as a number between 0 and 1, or as a percentage.

概率是事件发生可能性的度量，用 0 到 1 之间的数字或百分数表示。

For any event A, the probability that A does not occur is P(A’) = 1 – P(A).

对于任何事件 A，A 不发生的概率为 P(A’) = 1 – P(A)。

If two events A and B are mutually exclusive (they cannot happen at the same time), then P(A or B) = P(A ∪ B) = P(A) + P(B).

如果事件 A 和 B 互斥（不能同时发生），则 P(A 或 B) = P(A ∪ B) = P(A) + P(B)。

For any two events A and B, the general addition rule is P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

对于任意两个事件 A 和 B，一般加法法则为 P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。

Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, P(A ∩ B) = P(A) × P(B).

如果两个事件相互独立，即一个事件的发生不影响另一个事件发生的概率。独立事件满足 P(A ∩ B) = P(A) × P(B)。

5. Conditional Probability | 条件概率

Conditional probability deals with the probability of an event given that another event has already occurred.

条件概率研究在已知另一事件已经发生的条件下，某事件发生的概率。

The conditional probability of event A given event B is denoted P(A | B) and calculated as:

在事件 B 发生的条件下，事件 A 发生的条件概率记作 P(A | B)，计算公式为：

P(A | B) = P(A ∩ B) / P(B)

This formula only applies when P(B) ≠ 0. Conditional probabilities are often represented using tree diagrams or two-way tables.

该公式仅在 P(B) ≠ 0 时适用。条件概率常用树形图或双向表来表示。

The multiplication rule in terms of conditional probability is: P(A ∩ B) = P(A) × P(B | A). This is useful for solving problems involving successive events.

条件概率的乘法法则为：P(A ∩ B) = P(A) × P(B | A)。该法则常用于解决涉及连续事件的题目。

6. Discrete Random Variables | 离散随机变量

A discrete random variable X takes a countable number of values. Its probability distribution lists each value x together with its probability P(X = x).

离散随机变量 X 取可数个值。其概率分布列出了每个取值 x 及其概率 P(X = x)。

The sum of all probabilities in the distribution must equal 1: Σ P(X = x) = 1.

分布中所有概率之和必须等于 1：Σ P(X = x) = 1。

The expected value (mean) of X is E(X) = Σ x · P(X = x). This represents the long-run average outcome.

X 的期望值（均值）为 E(X) = Σ x · P(X = x)。它代表长期的试验平均结果。

The variance of X is Var(X) = Σ (x – μ)² P(X = x), which can also be calculated as Var(X) = E(X²) – [E(X)]².

X 的方差为 Var(X) = Σ (x – μ)² P(X = x)，也可以使用 Var(X) = E(X²) – [E(X)]² 进行计算。

7. Binomial Distribution | 二项分布

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p.

二项分布适用于在固定次数的独立试验中，每次试验具有相同成功概率 p 的条件下，成功次数的概率模型。

If X ~ B(n, p), then the probability of exactly r successes is given by:

若 X ~ B(n, p)，则恰好 r 次成功的概率为：

P(X = r) = ⁿCᵣ × p^r × (1 – p)^(n – r)

where ⁿCᵣ = n! / [r! (n – r)!] is the binomial coefficient, found on your calculator.

其中 ⁿCᵣ = n! / [r! (n – r)!] 为二项式系数，可在计算器上直接计算。

The mean of a binomial distribution is E(X) = n p, and the variance is Var(X) = n p (1 – p).

二项分布的均值 E(X) = n p，方差 Var(X) = n p (1 – p)。

8. Normal Distribution | 正态分布

The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is fully described by its mean μ and standard deviation σ.

正态分布是一种连续型概率分布，图像对称且呈钟形。它完全由均值 μ 和标准差 σ 决定。

About 68% of the data lies within one standard deviation of the mean, 95% within two, and 99.7% within three. These are known as the empirical rules.

大约 68% 的数据落在均值的一个标准差范围内，95% 落在两个标准差内，99.7% 落在三个标准差内。这被称为经验规则。

To find probabilities using standard normal tables, we first standardize the variable X by computing the z-score:

为了使用标准正态分布表求概率，需要先通过计算 z 分数将变量 X 标准化：

z = (X – μ) / σ

The z-score tells us how many standard deviations a value is above or below the mean. You then use the standard normal table to find the corresponding probability.

z 分数表示一个值高于或低于均值多少个标准差。然后查找标准正态分布表，得到相应的概率。

When using the inverse normal function, you are given a probability (area) and are required to find the corresponding z-value, and then X = μ + z σ.

在使用逆正态分布函数时，已知概率（面积），需要求出对应的 z 值，再由 X = μ + z σ 得到 X 值。

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

A scatter diagram (or scatter graph) is used to display the relationship between two quantitative variables. Each point represents a pair of values (x, y).

散点图（或散点图）用于展示两个定量变量之间的关系。每个点代表一对数值 (x, y)。

Correlation describes the strength and direction of a linear relationship between the variables. Positive correlation means as x increases, y tends to increase. Negative correlation means as x increases, y tends to decrease.

相关性描述变量之间线性关系的强度和方向。正相关意味着 x 增加时 y 趋于增加。负相关意味着 x 增加时 y 趋于减少。

Correlation does not imply causation. A strong correlation between two variables does not prove that a change in one causes a change in the other.

相关性并不意味着因果关系。两个变量之间存在强相关性并不能证明一个变量的变化导致了另一个变量的变化。

The strength of linear correlation can be measured using the product-moment correlation coefficient (Pearson’s r) or, for ranked data, Spearman’s rank correlation coefficient.

线性相关性的强度可以用积矩相关系数（皮尔逊 r）衡量，对于等级数据可以使用斯皮尔曼等级相关系数。

10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

Spearman’s rank correlation coefficient, rs, measures the strength and direction of association between two ranked variables. It is particularly useful when the data does not meet the assumptions for Pearson’s r or when the data is ordinal.

斯皮尔曼等级相关系数 rs 用于衡量两个定序变量之间关联的强度和方向。当数据不满足皮尔逊 r 的假设条件或数据为有序数据时，该方法特别有用。

First, rank the data for each variable separately. For tied values, assign the average rank. Then find the difference d between the ranks for each pair, and square it.

首先，对每个变量的数据分别进行排名。对于并列值，则赋予平均排名。然后计算每对排名之差 d，并求其平方 d²。

The formula for Spearman’s rank coefficient is:

斯皮尔曼等级相关系数公式为：

rs = 1 – [ 6 Σ d² ] / [ n (n² – 1) ]

where n is the number of data pairs. The value of rs ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation).

其中 n 为数据对的个数。rs 的取值范围从 -1（完全负相关）到 +1（完全正相关）。

11. Regression Line (Least Squares Method) | 回归线（最小二乘法）

A regression line can be used to model the relationship between two variables and to make predictions. The equation of the regression line of y on x is y = a + b x, where b is the gradient and a is the y-intercept.

回归线可用于对两个变量之间的关系建模并进行预测。y 对 x 的回归线方程为 y = a + b x，其中 b 是斜率，a 是 y 截距。

The least squares regression line minimises the sum of the squares of the vertical distances from the data points to the line. The formulas for a and b are:

最小二乘回归线使所有数据点到直线的竖直距离的平方和最小。a 和 b 的计算公式如下：

b = Σ (x – x̄)(y – ȳ) / Σ (x – x̄)²

a = ȳ – b x̄

The formula for b can also be expressed as b = (n Σ xy – Σ x Σ y) / (n Σ x² – (Σ x)²), which is often easier for calculator work.

b 的公式也可写成 b = (n Σ xy – Σ x Σ y) / (n Σ x² – (Σ x)²)，在使用计算器时更为方便。

Once the line is found, you can substitute a given x-value to predict the corresponding y-value, but be careful with extrapolation beyond the range of the data.

得到回归线后，可以通过代入 x 值预测对应的 y 值，但需要注意在数据范围之外进行外推时要格外谨慎。

12. Index Numbers and Time Series | 指数与时间序列

An index number is a statistical measure designed to show changes in a variable or group of variables over time. It is often expressed relative to a base year, which is given the value 100.

指数是一种统计量度，用于显示一个或一组变量随时间的变化。通常以基准年为 100，相对于基准年表示。

The simple index for a given year is calculated as: Index = (Value in current year / Value in base year) × 100.

某年的简单指数计算方法为：指数 =（当年数值 / 基准年数值）× 100。

A time series is a sequence of data points recorded at regular time intervals. Its components are trend, seasonal variation, cyclical variation, and random variation.

时间序列是按固定时间间隔记录的一系列数据点。其组成部分包括趋势、季节变动、循环变动和随机变动。

A moving average is used to smooth out short-term fluctuations and highlight the long-term trend. For example, a 4-point moving average can be calculated for quarterly data.

移动平均数用于平滑短期波动并突出长期趋势。例如，对于季度数据可以计算 4 点移动平均。

To calculate a 4-point moving average: (x₁ + x₂ + x₃ + x₄)/4, then shift one time period forward and repeat.

计算 4 点移动平均的方法为：(x₁ + x₂ + x₃ + x₄) / 4，然后向前移动一个时期重复计算。

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