Statistics 简明教程

Statistics - Skewness

如果离散性度量了差异量，则离散的方向由偏斜度度量。偏斜度最常用的度量是 Karl Pearson 的度量，用符号 Skp 表示。它是一个相对的偏斜度度量。

If dispersion measures amount of variation, then the direction of variation is measured by skewness. The most commonly used measure of skewness is Karl Pearson’s measure given by the symbol Skp. It is a relative measure of skewness.

Formula

当分布对称时，偏斜系数的值为零，因为平均值、中位数和众数相同。如果偏斜系数为正值，则分布为正偏态；如果它为负值，则分布为负偏态。根据矩，偏斜度表示如下：

When the distribution is symmetrical then the value of coefficient of skewness is zero because the mean, median and mode coincide. If the co-efficient of skewness is a positive value then the distribution is positively skewed and when it is a negative value, then the distribution is negatively skewed. In terms of moments skewness is represented as follows:

如果 ${\mu_3}$ 的值为零，则表示分布是对称的。${\mu_3}$ 的值越高，对称性就越大。然而，${\mu_3}$ 并未告诉我们偏斜性的方向。

If the value of ${\mu_3}$ is zero it implies symmetrical distribution. The higher the value of ${\mu_3}$, the greater is the symmetry. However ${\mu_3}$ do not tell us about the direction of skewness.

Example

Problem Statement:

收集到两所大学的 IT 课程的学生平均成绩的信息如下：

Information collected on the average strength of students of an IT course in two colleges is as follows:

Measure	College A	College B
Mean	150	145
Median	141	152
S.D	30	30

我们可以得出结论，这两个分布在差异方面是相似的吗？

Can we conclude that the two distributions are similar in their variation?

Solution:

对可用信息的一瞥即可发现，两所大学的学生人数差异均为 30 人。但是，要确定这两个分布是否相似，则需要进行更全面的分析，即我们需要计算出一个偏斜度度量。

A look at the information available reveals that both the colleges have equal dispersion of 30 students. However to establish if the two distributions are similar or not a more comprehensive analysis is required i.e. we need to work out a measure of skewness.

未给出众数的值，但可以使用以下公式计算：

Value of mode is not given but it can be calculated by using the following formula: