毫升 |原始和中心时刻

矩是一组统计参数，用于描述频率分布的不同特征和特征，即频率曲线的集中趋势、离散度、对称性和峰度（驼峰）。

对于未分组数据（即离散数据），变量 X 的观测值可通过以下方式获得$x_1, x_2, x_3, ...., x_n$ , 对于分组数据（即连续数据），获得变量 X 的观测值，并在频率表中按 K 类间隔制表。 Inervals 的中点表示为$x_1, x_2, x_3, ...., x_n$频率发生$f_1, f_2, f_3, ...., f_n$分别和$n=f_1, f_2, f_3, ...., f_n$ .

Class Intervals	Mid Points ($x_i$)	Absolute Frequency ($f_i$)
$c_1 - c_2$	$x_1 = (c_1 + c_2)/2$	$f_1$
$c_2 - c_3$	$x_2 = (c_2 + c_3)/2$	$f_2$
$c_3 - c_4$	$x_3 = (c_3 + c_4)/2$	$f_3$
…	…	…
…	…	…
$c_k_-_1 - c_k$	$x_k = (c_k_-_1 + c_k)/2$	$f_k$

关于任意点 A 的矩
这$r^t^h$变量 X 关于观测值上任意点 A 的矩$x_1, x_2, x_3, ...., x_n$定义为：

For ungrouped data $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}(x_i - A)^r$

For grouped data $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}f_i(x_i - A)^r$

where
$n = \sum_{i=1}^{k}f_i$

关于Python中任意点的时刻——

考虑给定的数据点。以下是 20 位不同的人每周在 GeeksforGeeks 门户网站上花费的时间（以小时为单位）。

15, 25, 18, 36, 40, 28, 30, 32, 23, 22, 21, 27, 31, 20, 14, 10, 33, 11, 7, 13

# data points
time = [15, 25, 18, 36, 40, 28, 30, 32, 23, 22, 
        21, 27, 31, 20, 14, 10, 33, 11, 7, 13]
  
# Arbitrary point 
A = 22
  
# Moment for r = 1
moment = (sum([(item-A) for item in time]))/len(time)

原始时刻 –

这$r^t^h$原点 A = 0 周围的矩称为原始矩，定义为：

For ungrouped data, $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}x_i^r$
For grouped data, $\mu^'_r = \frac{1}{n}\sum_{i=1}^{n}f_i x_i^r$

where, $n = \sum_{i=1}^{k}f_i$

笔记：

-> We can find first raw moment ($\mu^'_1$) just by replacing r with 1 and second raw moment ($\mu^'_2$) just by replacing r with 2 and so on.
-> When r = 0 the moment $\mu^'_0 = 1$ for both grouped and ungrouped data.

Python中的原始时刻——

# data points
time = [15, 25, 18, 36, 40, 28, 30, 32, 23,
       22, 21, 27, 31, 20, 14, 10, 33, 11, 7, 13]
  
  
# Moment for r = 1
moment = sum(time)/len(time)

中心时刻——

变量 X 关于算术平均值的矩 ( $\overline{x}$ ) 称为中心矩，定义为：

For ungrouped data,$\mu_r = \frac{1}{n}\sum_{i=1}^{n}(x_i-\overline{x})^r$

For grouped data,$\mu_r = \frac{1}{n}\sum_{i=1}^{n}f_i (x_i-\overline{x})^r$

where $n = \sum_{i=1}^{k}f_i$ and $\overline{x} = \sum_{i=1}^{k}f_ix_i$

笔记：

-> We can find first raw moment ($\mu_1$) just by replacing r with 1 and second raw moment ($\mu_2$) just by replacing r with 2 and so on.
-> When r = 0 the moment $\mu_0 = 1$, and when r = 1 the moment $\mu_1 = 0$ for both grouped and ungrouped data.

# data points
time = [15, 25, 18, 36, 40, 28, 30, 32, 23, 22,
       21, 27, 31, 20, 14, 10, 33, 11, 7, 13]
  
# Mean 
A = sum(time)/len(time)
  
# Moment for r = 1
moment = (sum([(item-A) for item in time]))/len(time)

原始时刻和中心时刻之间的关系 –