Measures of Dispersion | 数据离散程度的度量方法

📚 Measures of Dispersion | 数据离散程度的度量方法

In statistics, measures of dispersion describe how spread out a set of data is. While measures of central tendency such as the mean, median and mode summarise the typical value, dispersion measures tell us about the variability, consistency and reliability of the data.

在统计学中，离散程度的度量用于描述一组数据的分散情况。平均数、中位数和众数等集中趋势度量概括了数据的典型值，而离散程度的度量则告诉我们数据的变异性、一致性和可靠性。

1. Why Dispersion Matters | 为什么离散程度很重要

Two datasets can have the same mean but very different distributions. For example, the sets {5, 5, 5, 5} and {1, 3, 5, 7, 9} both have a mean of 5, yet the first has no variability while the second is highly spread. Relying only on averages can be misleading in decision-making.

两组数据集的平均数可能相同，但分布却截然不同。例如，集合{5,5,5,5}和{1,3,5,7,9}的平均数都是5，但前者没有任何变异性，而后者则高度分散。仅依赖平均数做决策可能会产生误导。

Measures of dispersion help us assess risk, quality control, and consistency. In finance, higher dispersion implies higher risk; in manufacturing, lower dispersion means more consistent products. Thus, understanding dispersion is essential for interpreting data meaningfully.

离散程度的度量帮助我们评估风险、质量控制和一致性。在金融领域，离散程度越高意味着风险越大；在制造业中，离散程度越低意味着产品越稳定。因此，理解离散程度对于有意义地解读数据至关重要。

2. Range | 极差

The range is the simplest measure of dispersion. It is the difference between the largest and smallest values in a dataset: Range = Maximum − Minimum.

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：极差 = 最大值 − 最小值。

Range = x_max − x_min

For example, for the data 4, 7, 9, 12, 20, the range is 20 − 4 = 16. Although easy to calculate, the range is heavily affected by outliers. A single extreme value can make the range unrealistically large.

例如，对于数据4, 7, 9, 12, 20，极差为20 − 4 = 16。虽然极差计算简单，但极易受异常值影响。一个极端值就能使极差变得不切实际地大。

3. Interquartile Range (IQR) | 四分位距

The interquartile range measures the spread of the middle 50% of the data. It is calculated as the difference between the third quartile Q₃ and the first quartile Q₁: IQR = Q₃ − Q₁.

四分位距度量的是数据中间50%的分散程度。它等于第三四分位数Q₃与第一四分位数Q₁之差：IQR = Q₃ − Q₁。

IQR = Q₃ − Q₁

To find Q₁ and Q₃, arrange the data in ascending order. Q₁ is the median of the lower half, and Q₃ is the median of the upper half. The IQR is robust to outliers because it ignores extreme values.

求Q₁和Q₃时，先将数据按升序排列。Q₁是下半部分的中位数，Q₃是上半部分的中位数。IQR不受异常值影响，因为它忽略了极端值。

In a box-and-whisker plot, the box spans from Q₁ to Q₃, and the IQR is the box length. Outliers are often defined as values below Q₁ − 1.5×IQR or above Q₃ + 1.5×IQR.

在箱线图中，箱体从Q₁延伸到Q₃，箱体的长度就是IQR。异常值通常定义为小于Q₁ − 1.5×IQR或大于Q₃ + 1.5×IQR的值。

4. Variance | 方差

Variance measures the average squared deviation of each data point from the mean. It gives a comprehensive picture of dispersion because it uses all data values. The population variance σ² is given by:

方差度量每个数据点与平均数之差的平方的平均值。由于使用了所有数据值，方差能全面反映离散程度。总体方差σ²的计算公式为：

σ² = Σ(xᵢ − μ)² / N

where μ is the population mean, xᵢ are the data values, and N is the population size. Squaring the deviations ensures that positive and negative deviations do not cancel each other out.

其中μ是总体平均数，xᵢ是数据值，N是总体容量。对偏差进行平方可以确保正负偏差不会相互抵消。

For a sample, we use a slightly different formula with n − 1 in the denominator, denoted as s²:

对于样本，我们使用分母为n − 1的略有不同的公式，记为s²：

s² = Σ(xᵢ − x̄)² / (n − 1)

The sample variance uses n − 1 to provide an unbiased estimate of the population variance. This correction is known as Bessel’s correction.

样本方差使用n − 1是为了对总体方差进行无偏估计。这一修正称为贝塞尔修正。

5. Standard Deviation | 标准差

The standard deviation is the square root of the variance. It is expressed in the same units as the original data, making it easier to interpret than variance. The population standard deviation is σ, and the sample standard deviation is s.

标准差是方差的平方根。它与原始数据具有相同的单位，因此比方差更容易解释。总体标准差为σ，样本标准差为s。

σ = √(Σ(xᵢ − μ)² / N)

The standard deviation tells us, on average, how far each data value lies from the mean. A small standard deviation indicates that the data are clustered closely around the mean, while a large standard deviation indicates a wider spread.

标准差告诉我们，平均而言每个数据值距离平均数有多远。标准差小表示数据紧密聚集在平均数周围，标准差大则表示数据分布较分散。

In IB Mathematics, you are expected to calculate standard deviation using your GDC (graphical display calculator) as well as from raw data. Always check whether you need the sample or population version.

在IB数学中，你既需要使用图形计算器（GDC）计算标准差，也需要根据原始数据手算。务必确认你需要的是样本标准差还是总体标准差。

6. Variance and Standard Deviation for Grouped Data | 分组数据的方差与标准差

When data are grouped into intervals, we use the midpoint of each interval as a representative value. Let fᵢ be the frequency of the i-th class and xᵢ be its midpoint. The mean is x̄ = Σfᵢxᵢ / Σfᵢ.

当数据分组到区间中时，我们使用每个区间的组中值作为代表值。设fᵢ为第i组的频数，xᵢ为该组的组中值。平均数为 x̄ = Σfᵢxᵢ / Σfᵢ。

σ² = Σfᵢ(xᵢ − x̄)² / Σfᵢ

Alternatively, the computational formula σ² = Σfᵢxᵢ² / Σfᵢ − x̄² is often more convenient. Grouped data calculations are approximations because the exact raw values within each interval are lost.

此外，计算式σ² = Σfᵢxᵢ² / Σfᵢ − x̄²通常更方便。分组数据计算是近似值，因为每个区间内的原始精确值已经丢失。

7. Effect of Linear Transformations | 线性变换的影响

If every data value is transformed by yᵢ = a + bxᵢ, then the new mean, variance and standard deviation change in a predictable way. The mean becomes a + bμ, and the standard deviation becomes |b| σ.

如果每个数据值经过变换 yᵢ = a + bxᵢ，那么新的平均数、方差和标准差会以可预测的方式变化。平均数变为a + bμ，标准差变为|b| σ。

The variance becomes b² σ². Adding a constant a shifts the centre but does not affect dispersion; multiplying by b scales the dispersion by |b| for standard deviation and by b² for variance.

方差变为b² σ²。加上常数a会平移数据的中心但不影响离散程度；乘以b会使标准差缩放|b|倍，使方差缩放b²倍。

This principle is used when converting units, such as changing degrees Celsius to Fahrenheit. Understanding these transformations helps in comparing datasets measured in different scales.

这一原理用于单位转换，例如将摄氏度转换为华氏度。理解这些变换有助于比较不同尺度下测量的数据集。

8. Coefficient of Variation | 变异系数

The coefficient of variation (CV) is a relative measure of dispersion. It is defined as the ratio of the standard deviation to the mean, often expressed as a percentage: CV = (σ / μ) × 100%.

变异系数（CV）是一种相对离散程度度量。它定义为标准差与平均数之比，通常以百分比表示：CV = (σ / μ) × 100%。

The CV allows comparison of variability between datasets with different units or very different means. For example, a standard deviation of 5 with a mean of 100 (CV = 5%) indicates less relative variability than a standard deviation of 5 with a mean of 20 (CV = 25%).

变异系数允许比较不同单位或平均数差异很大的数据集之间的变异性。例如，标准差为5、平均数为100（CV=5%）所表示的相对变异性小于标准差为5、平均数为20（CV=25%）的情况。

However, the CV is only meaningful when the mean is positive and the data are measured on a ratio scale. If the mean is near zero, the CV becomes unstable and misleading.

然而，只有当平均数为正且数据为比率尺度时，变异系数才有意义。如果平均数接近零，变异系数会变得不稳定且具有误导性。

9. Empirical Rule and Chebyshev’s Inequality | 经验法则与切比雪夫不等式

For a symmetric, bell-shaped distribution (approximately normal), the empirical rule states that about 68% of the data lie within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

对于对称的钟形分布（近似正态分布），经验法则表明约68%的数据落在平均数的一个标准差范围内，约95%落在两个标准差范围内，约99.7%落在三个标准差范围内。

Chebyshev’s inequality applies to any distribution, regardless of shape. It states that for any k > 1, at least (1 − 1/k²) of the data lie within k standard deviations of the mean.

切比雪夫不等式适用于任何分布，无论其形状如何。它表明对于任意k > 1，至少有(1 − 1/k²)的数据落在平均数的k个标准差范围内。

For k = 2, Chebyshev’s inequality guarantees that at least 75% of the data lie within two standard deviations, which is less precise than the 95% from the empirical rule but always valid. These concepts help interpret standard deviation in context.

当k = 2时，切比雪夫不等式保证至少有75%的数据落在两个标准差范围内，这不如经验法则中的95%精确，但总是成立。这些概念有助于在具体情境中解读标准差。

10. Choosing an Appropriate Measure | 选择合适的度量方法

The range is quick but sensitive to outliers. The IQR is robust and best used with skewed distributions or when outliers are present. Variance and standard deviation use all data points and are essential for further statistical inference.

极差计算快速但对异常值敏感。IQR稳健，最适合偏态分布或存在异常值时使用。方差和标准差使用了所有数据点，是进一步统计推断的基础。

If the data are symmetric without outliers, standard deviation is preferred. If the data are skewed, the median and IQR often provide a better summary. Always consider the shape of the distribution before choosing a measure of dispersion.

如果数据对称且无异常值，优先选择标准差。如果数据偏态，中位数和IQR通常能提供更好的概括。在选择离散程度度量之前，务必考虑分布的形态。

In IB Mathematics, you may be asked to justify your choice of measure based on the context. Mention robustness, use of all values, and suitability for further calculations in your reasoning.

在IB数学中，你可能会被要求根据情境证明你的度量选择。在推理中应提及稳健性、是否使用所有数值以及是否适合进一步计算。

11. Worked Example | 例题详解

Consider the dataset: 2, 5, 7, 8, 10, 12, 15. Find the range, IQR, variance, and standard deviation (sample).

考虑数据集：2, 5, 7, 8, 10, 12, 15。求极差、IQR、方差和样本标准差。

Range = 15 − 2 = 13. The median is 8 (the 4th value). The lower half is 2, 5, 7, so Q₁ = 5. The upper half is 10, 12, 15, so Q₃ = 12. Thus IQR = 12 − 5 = 7.

极差 = 15 − 2 = 13。中位数为8（第4个值）。下半部分为2, 5, 7，故Q₁ = 5。上半部分为10, 12, 15，故Q₃ = 12。因此IQR = 12 − 5 = 7。

The sample mean is x̄ = (2+5+7+8+10+12+15)/7 = 59/7 ≈ 8.429. The squared deviations are: (2−8.429)² = 41.327, (5−8.429)² = 11.755, (7−8.429)² = 2.041, (8−8.429)² = 0.184, (10−8.429)² = 2.469, (12−8.429)² = 12.755, (15−8.429)² = 43.184.

样本平均数 x̄ = (2+5+7+8+10+12+15)/7 = 59/7 ≈ 8.429。各偏差平方为：(2−8.429)² = 41.327，(5−8.429)² = 11.755，(7−8.429)² = 2.041，(8−8.429)² = 0.184，(10−8.429)² = 2.469，(12−8.429)² = 12.755，(15−8.429)² = 43.184。

The sum of squared deviations is approximately 113.714. Dividing by n − 1 = 6 gives s² ≈ 18.952. The sample standard deviation is s = √18.952 ≈ 4.353.

偏差平方和约为113.714。除以n − 1 = 6得到s² ≈ 18.952。样本标准差为 s = √18.952 ≈ 4.353。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

A frequent error is confusing sample and population variance. When a question says “the data represent a sample”, use division by n − 1; when it refers to the entire population, divide by N. Always read the question carefully.

常见错误是混淆样本方差和总体方差。当题目说明“数据代表一个样本”时，除以n − 1；当涉及整个总体时，除以N。务必仔细审题。

Another mistake is forgetting to square the deviations when calculating variance. Standard deviation is the square root of the variance, so it is not simply the average of absolute deviations.

另一个错误是在计算方差时忘记对偏差求平方。标准差是方差的平方根，所以它并不是绝对偏差的简单平均值。

For grouped data, always use midpoints. When using your GDC, check which standard deviation symbol is being used: σₙ for population and σₙ₋₁ for sample. In IB exams, give answers to three significant figures unless otherwise specified.

对于分组数据，务必使用组中值。使用图形计算器时，注意检查标准差符号：σₙ表示总体，σₙ₋₁表示样本。在IB考试中，除非另有说明，答案应保留三位有效数字。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Measures of Dispersion | 数据离散程度的度量方法