📚 Measures of Spread | 离散程度的度量
In any data set, knowing the average is only half the story. Measures of spread tell us how concentrated or dispersed the values are around that central value. In Edexcel A-Level Mathematics, you need to calculate and interpret range, interquartile range, variance, standard deviation and interpercentile ranges, and to choose the most appropriate measure for a given context.
在任何数据集中，了解平均值只是故事的一半。离散程度的度量告诉我们数据值在中心值周围是集中还是分散。在Edexcel A-Level数学中，你需要计算并解读极差、四分位距、方差、标准差和百分位数间距，并且能够根据具体情境选择最合适的度量方式。
1. Introduction to Measures of Spread | 离散程度度量简介
Measures of spread (also called dispersion) complement measures of central tendency. While the mean or median locates the centre, the spread describes the variability. A small spread means data are tightly clustered; a large spread indicates high variability. Common measures at this level are the range, interquartile range (IQR), variance and standard deviation.
离散程度度量（也称变异度量）补足了集中趋势的度量。均值或中位数定位了中心，而离散程度描述了变异性。小的离散程度意味着数据紧密聚集；大的离散程度则表明变异性高。在这个层次上常见的度量包括极差、四分位距（IQR）、方差和标准差。
Understanding spread is crucial for comparing data sets and making reliable inferences. For instance, two classes might have the same mean test score, yet one class might have widely varying marks while the other has very consistent performance.
理解离散程度对于比较数据集和做出可靠推断至关重要。例如，两个班级可能有相同的平均考试分数，但一个班级的成绩波动很大，而另一个班级的表现则非常一致。
2. Range and Interquartile Range | 极差与四分位距
The range is the simplest measure of spread: it is the difference between the largest value (max) and the smallest value (min). Range = max − min. It is easy to compute but highly sensitive to extreme values or outliers.
极差是最简单的离散程度度量：它是最大值与最小值之差。 极差 = 最大值 − 最小值。它计算简单，但对极端值或离群值高度敏感。
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁). IQR = Q₃ − Q₁. It covers the middle 50% of the data and is more resistant to outliers than the range.
四分位距（IQR）是上四分位数（Q₃）与下四分位数（Q₁）之差。 IQR = Q₃ − Q₁。它覆盖了中间50%的数据，比极差更能抵抗离群值的影响。
3. Calculating Quartiles for Discrete Data | 离散数据的四分位数计算
For a small data set listed in order, the median (Q₂) is the middle value. Q₁ is the median of the lower half of the data, and Q₃ is the median of the upper half. When the number of data points n is odd, the median is not included in either half.
对于一个有序排列的小数据集，中位数（Q₂）是中间的值。Q₁ 是数据下半部分的中位数，Q₃ 是数据上半部分的中位数。当数据点个数 n 为奇数时，中位数不包含在任一半中。
Using positional formulae: the position of Q₁ is (n+1)/4, the median is at (n+1)/2 and Q₃ is at 3(n+1)/4. If the position is not an integer, linear interpolation is used between the surrounding data values.
使用位置公式：Q₁ 的位置是 (n+1)/4，中位数在 (n+1)/2，Q₃ 在 3(n+1)/4。如果位置不是整数，则在相邻数据值之间进行线性插值。
4. Box Plots and Identifying Outliers | 箱线图与识别离群值
A box plot (or box-and-whisker diagram) displays the minimum, Q₁, median, Q₃ and maximum. The box spans Q₁ to Q₃ with the median marked inside. Whiskers extend to the smallest and largest values that are not outliers.
箱线图（或称盒须图）展示最小值、Q₁、中位数、Q₃ 和最大值。箱体从 Q₁ 延伸至 Q₃，中位数在内部标出。须线延伸至不是离群值的最小值和最大值。
Outliers are typically defined as values lying more than 1.5 × IQR below Q₁ or above Q₃. Lower fence = Q₁ − 1.5 × IQR, upper fence = Q₃ + 1.5 × IQR. Any data point outside these fences is plotted as an individual point and considered an outlier.
离群值通常定义为位于 Q₁ 以下 1.5 × IQR 或 Q₃ 以上 1.5 × IQR 的值。下边界 = Q₁ − 1.5 × IQR，上边界 = Q₃ + 1.5 × IQR。任何位于这些边界之外的数据点被单独画出，并被视为离群值。
5. Variance and Standard Deviation: Concepts | 方差与标准差的概念
Variance measures the average squared deviation of each data point from the mean. Standard deviation is the square root of the variance, bringing the measure back to the original units. For a population, variance is denoted by σ² (sigma squared), and for a sample, we use s².
方差度量每个数据点与均值之偏差平方的平均值。标准差是方差的平方根，使度量回到原始单位。对于总体，方差用 σ² 表示；对于样本，我们使用 s²。
A key distinction: when calculating the variance of a sample, we divide by n − 1 (Bessel’s correction) to give an unbiased estimate of the population variance. When the entire population is available, we divide by n. In Edexcel questions, you will often work with sample data and use s².
一个关键区别：计算样本方差时，我们除以 n − 1（贝塞尔校正），以给出总体方差的无偏估计。当拥有整个总体时，我们除以 n。在Edexcel考题中，你通常会处理样本数据并使用 s²。
6. Variance Formula for Ungrouped Data | 未分组数据的方差公式
For a sample of n values x, with mean x̄, the sample variance s² is given by:
s² = ∑(x − x̄)² / (n − 1)
对于 n 个值的样本 x，均值为 x̄，样本方差 s² 由下式给出：
s² = ∑(x − x̄)² / (n − 1)
An equivalent computational formula is:
s² = [∑x² − (∑x)² / n] / (n − 1)
等价的简便计算公式为：
s² = [∑x² − (∑x)² / n] / (n − 1)
If the data represent the entire population, replace n − 1 with n and s² with σ². Standard deviation is simply the positive square root: s = √s².
如果数据代表整个总体，则将 n − 1 替换为 n，并用 σ² 替换 s²。标准差就是正的平方根：s = √s²。
7. Standard Deviation for Frequency Distributions | 频数分布的标准差
For grouped data or frequency tables, each class is represented by its midpoint x, with frequency f. The sample variance becomes:
s² = ∑f(x − x̄)² / (∑f − 1)
对于分组数据或频数表，每个组用其中点 x 表示，频率为 f。样本方差变为：
s² = ∑f(x − x̄)² / (∑f − 1)
The more efficient computational formula is:
s² = [∑fx² − (∑fx)² / ∑f] / (∑f − 1)
更高效的计算公式为：
s² = [∑fx² − (∑fx)² / ∑f] / (∑f − 1)
Here x̄ is the estimated mean = ∑fx / ∑f. If you are told the data represents the whole population, divide by ∑f instead of ∑f − 1. Remember to take the square root to obtain the standard deviation.
这里 x̄ 是估计均值 = ∑fx / ∑f。如果题目说明数据代表整个总体，则除以 ∑f 而非 ∑f − 1。记得取平方根以获得标准差。
8. Interpercentile Range | 百分位数间距
The kth percentile Pₔ is the value below which k% of the data fall. The interpercentile range is the difference between two percentiles, most commonly P₉₀ − P₁₀ (the 10th to 90th percentile range). It gives a wider central range than the IQR and is useful for skewed data.
第 k 百分位数 Pₔ 是有 k% 的数据低于它的值。百分位数间距是两个百分位数之差，最常见的是 P₉₀ − P₁₀（第10至第90百分位数间距）。它提供了比 IQR 更宽的中央范围，对于偏斜数据很有用。
To estimate a percentile for grouped continuous data, use linear interpolation based on cumulative frequencies. The position of Pₔ is (k/100) × n, where n is the total frequency. Then apply the standard interpolation formula within the appropriate class interval.
要估计分组连续数据的百分位数，可使用基于累积频率的线性插值。Pₔ 的位置是 (k/100) × n，其中 n 是总频率。然后在适当的组区间内应用标准插值公式。
9. Comparing Measures of Spread | 离散度量的比较
Each measure has strengths and limitations. The table below summarises key characteristics.
每种度量都有优点和局限性。下表总结了关键特征。
| Measure | Advantages | Disadvantages |
|---|---|---|
| Range | Quick to calculate; easy to understand | Affected by outliers; ignores most data |
| IQR | Robust to outliers; shows middle spread | Ignores extreme 50% of data |
| Standard Deviation | Uses all data values; forms basis for further statistics | Sensitive to outliers; harder to compute manually |
| Interpercentile Range | Flexible central coverage; robust to extreme tails | Choice of percentiles can be arbitrary |
For symmetric distributions without outliers, the standard deviation is usually the best summary because it incorporates every observation. For skewed distributions or those containing outliers, the IQR or interpercentile range is more representative.
对于没有离群值的对称分布，标准差通常是最好的概括，因为它纳入了每一个观测值。对于偏斜分布或包含离群值的分布，IQR 或百分位数间距更具代表性。
10. Choosing the Appropriate Measure | 选择合适的度量
When answering exam questions, justify your choice. If a data set contains an extreme value, the range would be misleading but the IQR remains stable. If the data are symmetric and you need to carry out further analysis like hypothesis testing, the standard deviation is essential.
在回答考试题目时，要说明选择的理由。如果数据包含极端值，极差会误导，但 IQR 保持稳定。如果数据对称，且你需要进行进一步的分析（如假设检验），标准差是必不可少的。
Always read the context: financial returns might focus on downside risk using lower percentiles; quality control often uses standard deviation. The Edexcel specification expects you to discuss the relative advantages of different measures.
始终要结合上下文：金融回报可能侧重于使用较低百分位数的下行风险；质量控制通常使用标准差。Edexcel 教学大纲期望你讨论不同度量的相对优点。
11. Worked Examples | 例题解析
Example 1 – Ungrouped data: The heights (cm) of five students are: 162, 167, 170, 173, 178. Calculate the sample standard deviation.
例题1 – 未分组数据：五名学生的身高（cm）为：162, 167, 170, 173, 178。计算样本标准差。
First find the mean: x̄ = (162+167+170+173+178)/5 = 850/5 = 170. Then compute ∑(x − x̄)²: (162−170)²=64, (167−170)²=9, (170−170)²=0, (173−170)²=9, (178−170)²=64. Sum = 146. Sample variance s² = 146/(5−1) = 146/4 = 36.
Published by TutorHao | A-Level Mathematics 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Cancel reply