📚 Measures of Dispersion: Range, Interquartile Range, Variance and Standard Deviation | 离散程度度量：极差、四分位距、方差与标准差
In statistics, a measure of location such as the mean or median tells you where the centre of a data set lies, but it says nothing about how spread out the values are. Measures of dispersion, also called measures of variability, describe this spread. This article explains the most important ones — range, interquartile range, variance and standard deviation — with worked examples and exam advice.
在统计学中，位置度量（如平均数或中位数）告诉我们一组数据的中心在哪里，但它无法说明数据之间的分散程度。离散程度度量（也称变异程度度量）正是用来描述这种分散性的。本文将讲解最常用的一组度量——极差、四分位距、方差和标准差，并结合例题与考试指导帮助你掌握。
1. What Is Dispersion? | 什么是离散程度？
Two data sets can have the same mean and median but look completely different. Consider Set A {4, 5, 6} and Set B {1, 5, 9}. Both have a mean of 5, yet Set B has much greater variation. Dispersion measures capture this difference, so they are essential when comparing data sets.
两组数据集可能有完全相同的平均数和相同的中位数，但形态却截然不同。例如：数据集 A {4, 5, 6} 与数据集 B {1, 5, 9}，两者的平均数都为 5，但 B 的波动明显更大。离散程度度量正是用来捕捉这种差异的，因此它是比较数据集时不可或缺的工具。
2. Range | 极差
The range is the simplest measure of dispersion. It is the difference between the largest and smallest values in a data set.
极差是最简单的离散程度度量，它是一组数据中最大值与最小值之差。
Range = Maximum − Minimum
For example, if a data set has a minimum of 3 and a maximum of 18, the range is 18 − 3 = 15. The range is quick to calculate and easy to understand.
例如，若一组数据的最小值为 3，最大值为 18，则极差为 18 − 3 = 15。极差计算快捷，也容易理解。
3. The Limitation of the Range | 极差的局限性
The range only uses two values from the whole data set. This means it can be heavily distorted by a single extreme value, called an outlier. For example, in the data set {1, 2, 3, 4, 100}, the range is 99, but most of the values are actually very close to each other.
极差只使用了整个数据集中的两个数值。这意味着它很容易被一个极端值（即异常值）严重扭曲。例如，在数据集 {1, 2, 3, 4, 100} 中，极差为 99，但大部分数值实际上彼此非常接近。
Because the range ignores all intermediate values, it gives no information about how the data are distributed between the minimum and maximum. For this reason, statisticians often prefer more robust measures.
由于极差忽略了所有中间数值，它无法提供数据在最小值与最大值之间如何分布的信息。因此，统计学家通常更青睐更稳健的度量。
4. Quartiles | 四分位数
Quartiles divide an ordered data set into four roughly equal parts. The lower quartile, Q₁, marks the point below which about 25% of the data lie. The middle quartile, Q₂, is the median, with about 50% of the data below it. The upper quartile, Q₃, has about 75% of the data below it.
四分位数将一组有序数据大致分成四个相等的部分。下四分位数 Q₁ 是约有 25% 的数据位于其下的点；中位数 Q₂ 是约有 50% 的数据位于其下的点；上四分位数 Q₃ 是约有 75% 的数据位于其下的点。
There are different quartile conventions. A common classroom method is: if the median is one of the data values, exclude it from both halves; then take the median of the lower half as Q₁ and the median of the upper half as Q₃.
四分位数的计算方法有多种约定。课堂中常用的一种方法是：若中位数正好是原数据中的某个值，则将其从上下两半分出；然后取下半部分的中位数为 Q₁，取上半部分的中位数为 Q₃。
5. Interquartile Range | 四分位距
The interquartile range, often written as IQR, is the difference between the upper and lower quartiles.
四分位距通常写作 IQR，它是上四分位数与下四分位数之差。
IQR = Q₃ − Q₁
For the ordered data set {3, 5, 7, 8, 12, 14, 18}, the median Q₂ is 8. The lower half is {3, 5, 7}, so Q₁ = 5. The upper half is {12, 14, 18}, so Q₃ = 14. Therefore, IQR = 14 − 5 = 9.
对于有序数据集 {3, 5, 7, 8, 12, 14, 18}，中位数 Q₂ 为 8。下半部分为 {3, 5, 7}，因此 Q₁ = 5；上半部分为 {12, 14, 18}，因此 Q₃ = 14。所以 IQR = 14 − 5 = 9。
The IQR describes the spread of the middle 50% of the data. Because it ignores extreme values, it is much more robust than the range when outliers are present.
四分位距描述的是中间 50% 数据的分布范围。由于它忽略了极端值，因此在存在异常值时，它比极差稳健得多。
6. Box Plots and Outliers | 箱线图与异常值
A box plot is a useful visual representation of dispersion. It is based on the five-number summary: minimum, Q₁, median, Q₃ and maximum. The box is drawn from Q₁ to Q₃, with a line at the median, and whiskers extend to the minimum and maximum.
箱线图是一种非常直观的离散程度展示方式。它以五数概括为基础：最小值、Q₁、中位数、Q₃ 和最大值。箱体从 Q₁ 画到 Q₃，中位数处画一条线，触须延伸至最小值和最大值。
A common outlier rule is: any value below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR is treated as an outlier. When outliers exist, the whiskers usually extend only to the most extreme values that are not classified as outliers, and the outliers are plotted as separate points.
一个常用的异常值判定规则是：任何小于 Q₁ − 1.5 × IQR 或大于 Q₃ + 1.5 × IQR 的数值都被视为异常值。当存在异常值时，触须通常只延伸到未被视为异常值的最远数值，异常值则用单独的点标出。
7. Variance | 方差
Variance measures the average squared distance of each value from the mean. It uses all the data, so it gives a fuller picture of dispersion than the range or IQR.
方差衡量的是每个数值与平均数的平均平方距离。它使用了全部数据，因此比极差或四分位距更能完整地反映离散程度。
For a population with size N and mean μ, the variance σ² is defined as follows.
对于容量为 N、平均数为 μ 的总体，方差 σ² 的定义如下。
σ² = Σ(x − μ)² / N
When the data are a sample with size n and sample mean x̄, the sample variance is often calculated using n − 1 in the denominator.
当数据是容量为 n、样本平均数为 x̄ 的样本时，样本方差通常使用分母 n − 1 计算。
s² = Σ(x − x̄)² / (n − 1)
The symbol Σ means “sum of all values”. Squaring the deviations makes every contribution positive, but it also means the variance is measured in square units of the original data.
符号 Σ 表示“对所有值求和”。将离差平方可以让每一项都为正数，但这也意味着方差的单位是原始数据单位的平方。
8. Standard Deviation | 标准差
The standard deviation is the square root of the variance. It is the most widely used measure of dispersion because it is expressed in the same units as the original data.
标准差是方差的平方根。它是最常用的离散程度度量，因为它与原始数据具有相同的单位。
σ = √(Σ(x − μ)² / N) for a population
s = √(Σ(x − x̄)² / (n − 1)) for a sample
Worked example: for the data {5, 7, 8, 10, 15}, the mean is 9. The squared deviations are 16, 4, 1, 1 and 36, giving a total of 58. The population variance is 58 ÷ 5 = 11.6, so the population standard deviation is √11.6 ≈ 3.41.
例题：对于数据 {5, 7, 8, 10, 15}，平均数为 9。各离差平方分别为 16、4、1、1 和 36，总和为 58。总体方差为 58 ÷ 5 = 11.6，因此总体标准差为 √11.6 ≈ 3.41。
A smaller standard deviation means the values are clustered closely around the mean; a larger standard deviation means they are spread out. For many roughly symmetric distributions, about 68% of the data lie within one standard deviation of the mean.
标准差越小，表示数值越紧密地聚集在平均数附近；标准差越大，表示数值越分散。对于许多大致对称的分布，约有 68% 的数据位于平均数的一个标准差范围之内。
9. Choosing the Right Measure | 如何选择恰当的离散程度度量
Different situations call for different measures. The table below summarises their strengths and weaknesses.
不同情境需要不同的度量方法。下表总结了它们各自的优点和缺点。
| Measure | 度量 | Strength | 优点 | Weakness | 缺点 |
| Range | 极差 | Very simple to calculate and understand. 计算和理解都非常简单。 | Uses only two values; easily distorted by outliers. 只使用两个数值，易受异常值影响。 |
| IQR | 四分位距 | Robust to outliers; focuses on the middle 50%. 不受异常值影响，聚焦中间 50%。 | Ignores the smallest and largest quarters of the data. 忽略数据中最小的四分之一和最大的四分之一。 |
| Variance | 方差 | Uses all values; important in further statistical theory. 使用全部数据，在后续统计理论中很重要。 | Units are squared; sensitive to outliers. 单位是平方，对异常值敏感。 |
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