📚 IGCSE WJEC Statistics: Quick Reference Handbook of Formulas and Theorems | IGCSE WJEC 统计：公式定理速查手册
This quick-reference handbook summarises the essential formulas and theorems required for the IGCSE WJEC Statistics examination. Use it as a concise revision aid to reinforce key concepts, from descriptive statistics and probability to correlation and sampling.
这本速查手册总结了IGCSE WJEC统计考试所需的基本公式与定理。可将其作为简明复习工具，巩固从描述性统计与概率到相关性与抽样等关键概念。
1. Measures of Central Tendency | 集中趋势度量
The mean, median and mode are the three most common measures of central tendency, each describing the ‘centre’ of a data set in a different way.
均值、中位数和众数是三种最常见的集中趋势度量，各自以不同方式描述数据集的“中心”。
Arithmetic Mean (ungrouped data): x̄ = Σx / n
For ungrouped data, add all values (Σx) and divide by the total number of items n. The mean is sensitive to extreme values (outliers).
对于未分组数据，将所有数值求和 (Σx)，再除以总个数 n。均值易受极端值（异常值）影响。
Mean for a Frequency Table (grouped data): x̄ = Σfx / Σf
When data are grouped in a frequency table, use the midpoint x of each class interval. Multiply each midpoint by its frequency f, sum these products (Σfx), and divide by the total frequency Σf.
当数据以频数表分组时，用每个组中值 x。将每个中值乘以对应频数 f，求和 (Σfx) 后再除以总频数 Σf。
Weighted Mean: x̄w = Σwx / Σw
If some values are more important than others, assign weights w. The weighted mean is the sum of each value multiplied by its weight, divided by the sum of the weights.
若某些数值更重要，可赋予权重 w。加权平均数为各值乘对应权重之和除以权重总和。
Median: The median is the middle value when the data are arranged in order. For n observations, the position of the median is (n+1)/2. If n is even, the median is the average of the two middle numbers.
中位数： 中位数是将数据按顺序排列后的中间值。对于 n 个观测值，中位数的位置为 (n+1)/2。如果 n 为偶数，中位数是中间两个数的平均数。
Mode: The mode is the most frequently occurring value(s) in a data set. A data set may have one mode (unimodal), more than one (bimodal or multimodal), or no mode if all values occur equally often.
众数： 众数是数据集中出现频率最高的值。数据集可以有单众数、多众数，或在所有值出现频率相同时无众数。
2. Measures of Dispersion | 离散度量
Measures of dispersion describe how spread out the data are. Key measures include range, interquartile range (IQR), variance and standard deviation.
离散度量描述数据分散程度。关键指标包括范围、四分位距 (IQR)、方差和标准差。
Range = Maximum value – Minimum value
The range is the simplest measure of spread. It is easily affected by outliers because it uses only the two extreme values.
范围是最简单的散布度量，只用两个极值，因此极易受异常值影响。
Interquartile Range (IQR) = Q3 – Q1
The IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1). It covers the middle 50% of the data and is resistant to outliers.
四分位距是上四分位数 (Q3) 与下四分位数 (Q1) 之差，涵括中间50%数据，不易受异常值影响。
Population Variance: σ² = Σ(x – μ)² / N
Population Standard Deviation: σ = √[ Σ(x – μ)² / N ]
If the data represent the entire population (size N, mean μ), each deviation from the mean is squared, summed, and divided by N. The square root gives the standard deviation.
若数据代表整个总体（容量 N，均值为 μ），每个离差平方后求和并除以 N，开方即得标准差。
Sample Standard Deviation: s = √[ Σ(x – x̄)² / (n – 1) ]
When working with a sample drawn from a larger population, dividing by (n-1) provides an unbiased estimate of the population standard deviation.
当处理来自更大总体的样本时，除以 (n-1) 可提供总体标准差的无偏估计。
Alternative Formula for Grouped Data: σ = √[ Σfx² / Σf – (Σfx / Σf)² ]
For grouped frequency tables, this computational formula avoids calculating each deviation from the mean. Here x is the class midpoint, and f the frequency.
对于分组频数表，此计算式避免了逐一计算离差。其中 x 为组中值，f 为频数。
3. Quartiles, Percentiles and Box Plots | 四分位数、百分位数与箱线图
Quartiles divide an ordered data set into four equal parts. The lower quartile Q1 is the 25th percentile, the median Q2 is the 50th percentile, and the upper quartile Q3 is the 75th percentile.
四分位数将有序数据分为四等份。下四分位数 Q1 为第25百分位数，中位数 Q2 为第50百分位数，上四分位数 Q3 为第75百分位数。
To find the position of the kth percentile in a small raw data set, use L = (k/100) × (n+1). If L is not an integer, linearly interpolate between the two surrounding values.
要在小型原始数据集中找到第 k 百分位数的位置，使用 L = (k/100) × (n+1)。若 L 非整数，则在相邻两个数值间进行线性插值。
Five-number summary: Minimum, Q1, Median, Q3, Maximum
A box-and-whisker plot (or boxplot) displays the five-number summary. The box spans from Q1 to Q3, with a line at the median. Whiskers extend to the minimum and maximum values (or to 1.5×IQR boundaries, beyond which points are considered outliers).
箱线图（或盒须图）展示五数汇总。箱体从 Q1 延伸至 Q3，中位线在内。胡须延伸至最小值和最大值（或至 1.5×IQR 边界，超出者为异常值）。
The IQR is the width of the box. Outliers are often defined as values below Q1 – 1.5×IQR or above Q3 + 1.5×IQR.
IQR 是箱体的宽度。异常值通常定义为小于 Q1 – 1.5×IQR 或大于 Q3 + 1.5×IQR 的值。
4. Cumulative Frequency and Interpolation | 累积频率与插值
A cumulative frequency graph (ogive) plots the running total of frequencies against the upper class boundaries. It is useful for estimating medians, quartiles and percentiles from grouped data without raw values.
累积频率图（肩形图）根据上组界绘制频数累计值。它适用于在无原始数据时从分组数据估计中位数、四分位数和百分位数。
To estimate the median, draw a horizontal line from 50% of the total cumulative frequency on the vertical axis to the curve, then vertically down to read the value on the horizontal axis.
要估计中位数，从纵轴上总累积频率的50%处作水平线与曲线相交，再垂直向下在横轴上读取数值。
Linear Interpolation for Median (grouped table):
Median = L + [ (n/2 – F) / f ] × c
Where L = lower boundary of the median class, n = total frequency, F = cumulative frequency up to the class before the median class, f = frequency of the median class, and c = class width.
其中 L = 中位数组的下组界，n = 总频数，F = 中位数组之前的累积频数，f = 中位数组的频数，c = 组距。
The same approach can be used for any percentile by replacing n/2 with the appropriate value, e.g., n/4 for Q1 and 3n/4 for Q3.
对于任何百分位数，将 n/2 替换为适当的值即可应用相同方法，例如 Q1 用 n/4，Q3 用 3n/4。
5. Probability Rules | 概率法则
Probability (P) measures the likelihood of an event, ranging from 0 (impossible) to 1 (certain). The sum of probabilities of all mutually exclusive and exhaustive outcomes is 1.
概率 (P) 衡量事件发生的可能性，范围从0（不可能）到1（必然）。所有互斥且完备结果的概率之和为1。
Complement Rule: P(A’) = 1 – P(A)
The probability of an event not occurring (A’) is 1 minus the probability of it occurring.
事件不发生 (A’) 的概率等于 1 减去事件发生的概率。
Addition Rule (general): P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
The
Published by TutorHao | IGCSE 统计 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导