Statistics 简明教程
Statistics - Kurtosis
分布的尾数程度由峰度测量。它告诉我们该分布比正态分布在多大程度上更容易或不太容易出现异常值（更重或更轻的尾）。来自 Investopedia 提供的三种不同类型的曲线如下所示：
The degree of tailedness of a distribution is measured by kurtosis. It tells us the extent to which the distribution is more or less outlier-prone (heavier or light-tailed) than the normal distribution. Three different types of curves, courtesy of Investopedia, are shown as follows −
从密度图（左面板）中很难看出不同类型的峰度，因为所有分布的尾部都接近于零。但是，在正态分位数-分位数图（右面板）中很容易看出尾部差异。
It is difficult to discern different types of kurtosis from the density plots (left panel) because the tails are close to zero for all distributions. But differences in the tails are easy to see in the normal quantile-quantile plots (right panel).
正态曲线称为中峰度曲线。如果某个分布的曲线比正态分布或中峰度曲线更容易出现异常值（或更重），则它被称为厚尾分布曲线。如果某个曲线比正态分布更不容易出现异常值（或更轻），则它被称为扁尾分布曲线。峰度通过矩进行测量，由以下公式给出：
The normal curve is called Mesokurtic curve. If the curve of a distribution is more outlier prone (or heavier-tailed) than a normal or mesokurtic curve then it is referred to as a Leptokurtic curve. If a curve is less outlier prone (or lighter-tailed) than a normal curve, it is called as a platykurtic curve. Kurtosis is measured by moments and is given by the following formula −
Formula
其中——
Where −
-
${\mu_4 = \frac{\sum(x- \bar x)^4}{N}}$
\beta_2 值越大，曲线越尖锐或越厚尾。正态曲线的数值为 3，厚尾分布的 \beta_2 大于 3，扁尾分布的 \beta_2 小于 3。
The greater the value of \beta_2 the more peaked or leptokurtic the curve. A normal curve has a value of 3, a leptokurtic has \beta_2 greater than 3 and platykurtic has \beta_2 less then 3.
Example
Problem Statement:
Problem Statement:
给出某工厂 45 名工人的每日工资数据。使用围绕均值的矩计算 \beta_1 和 \beta_2。对结果发表评论。
The data on daily wages of 45 workers of a factory are given. Compute \beta_1 and \beta_2 using moment about the mean. Comment on the results.
Wages(Rs.) |
Number of Workers |
100-200 |
1 |
120-200 |
2 |
140-200 |
6 |
160-200 |
20 |
180-200 |
11 |
200-200 |
3 |
220-200 |
2 |
Solution:
Solution:
Wages (Rs.) |
Number of Workers (f) |
Mid-pt |
m-${\frac{170}{20}}$ |
${fd}$ |
${fd^2}$ |
${fd^3}$ |
${fd^4}$ |
100-200 |
1 |
110 |
-3 |
-3 |
9 |
-27 |
81 |
120-200 |
2 |
130 |
-2 |
-4 |
8 |
-16 |
32 |
140-200 |
6 |
150 |
-1 |
-6 |
6 |
-6 |
6 |
160-200 |
20 |
170 |
0 |
0 |
0 |
0 |
0 |
180-200 |
11 |
190 |
1 |
11 |
11 |
11 |
11 |
200-200 |
3 |
210 |
2 |
6 |
12 |
24 |
48 |
220-200 |
2 |
230 |
3 |
6 |
18 |
54 |
162 |
${N=45}$ |
|
${\sum fd = 10}$ |
||
${\sum fd^2 = 64}$ |
${\sum fd^3 = 40}$ |
${\sum fd^4 = 330}$ |
由于这些偏差取自一个假设均值，因此我们首先计算围绕任意原点的矩，然后再计算围绕均值的矩。围绕任意原点“170”取矩
Since the deviations have been taken from an assumed mean, hence we first calculate moments about arbitrary origin and then moments about mean. Moments about arbitrary origin '170'
Moments about mean
现在，我们可以利用围绕均值的矩值计算 ${\beta_1}$ 和 ${\beta_2}$：
From the value of movement about mean, we can now calculate ${\beta_1}$ and ${\beta_2}$:
从以上计算中可以得出结论，测量偏度的 ${\beta_1}$ 近乎为零，因此表明该分布几乎是对称的。测量峰度的 ${\beta_2}$ 大于 3，因此表明该分布呈厚尾分布。
From the above calculations, it can be concluded that ${\beta_1}$, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic.