小马哥课堂-统计学-方差

Variance(方差)

衡量随机变量或一组数据的 离中 的离散程度

引入例子

假设有两组整体:A={2,2,3,3}, B={0,0,5,5}

$$\mu_A=\frac {2+2+3+3}{4}=2.5$$

$$\mu_B=\frac{0+0+5+5}{4}=2.5$$

显然,$\mu$并不能反映这两组数据集的差别。

数据集A：$ Var[A]=\frac { \displaystyle \sum_{i=1}^N (x_i - \mu)^2}{N}=\frac{0.25 \cdot 4}{4}=0.25$

i	$x_i$	$\mu$	$x_i-\mu$	$(x_i-\mu)^2$
1	$x_1=2$	2.5	-0.5	0.25
2	$x_1=2$	2.5	-0.5	0.25
3	$x_1=3$	2.5	0.5	0.25
4	$x_1=3$	2.5	0.5	0.25

数据集B：$ Var[B]=\frac { \displaystyle \sum_{i=1}^N (x_i - \mu)^2}{N}=\frac{6.25 \cdot 4}{4}=6.25$

i	$x_i$	$\mu$	$x_i-\mu$	$(x_i-\mu)^2$
1	$x_1=0$	2.5	-2.5	6.25
2	$x_1=0$	2.5	-2.5	6.25
3	$x_1=5$	2.5	2.5	6.25
4	$x_1=5$	2.5	2.5	6.25

从上面两组数据集能够看出,尽管两组数据集 指望 相同,可是,方差并不相同。数据集A 表现得更集中，而数据集B 表现得相对分散。

定义

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by $\displaystyle \sigma ^{2}$,$\displaystyle s^{2}$ , or $\displaystyle \operatorname {Var} (X)$.

在几率论和统计学中，方差的定义: 随机变量与指望的差 的平方值 的指望，衡量随机变量与其指望的偏离程度。

表示符号

$\displaystyle \sigma ^{2}$：通常表示整体的方差

$\displaystyle s^{2}$: 通常表示抽样分布的方差

$\displaystyle \operatorname {Var} (X)$

公式

The variance of a random variable $\displaystyle X$ is the expected value of the squared deviation from the mean of $\displaystyle X$ , $\displaystyle \mu =\operatorname {E} [X]$:

$$\displaystyle \operatorname {Var} (X)=\operatorname {E} [(X-\mu)^2]$$

$$\displaystyle \begin{array}{rcl} \operatorname {Var} (X)&=&\operatorname {E} [(X-\operatorname{E}[X])^2] \ &=&\operatorname {E} [X^2-2X\operatorname{E}[X] +\operatorname{E}[X]^2] \ &=& \operatorname{E}[X^2]-2\operatorname{E}[X]\operatorname{E}[X]+\operatorname{E}[X]^2 \ &=& \operatorname{E}[X^2]-\operatorname{E}[X]^2 \end{array} $$

对于离散型随机变量,方差的公式:

If the generator of random variable $\displaystyle X$ is discrete with probability mass function $\displaystyle x_{1}\mapsto p_{1},x_{2}\mapsto p_{2},\ldots ,x_{n}\mapsto p_{n}$ then

$$\displaystyle Var(X)=\sum_{i=1}^{n}p_i\cdot(x_i-\mu)^2$$

$$\displaystyle \mu = \sum_{i=1}^np_ix_i$$

对于连续型随机变量,方差的公式:

$$\displaystyle Var(X)=\sigma^2=\int (x-\mu)^2f(x)dx = \int x^2 f(x)dx - \mu^2$$

$$\displaystyle \mu=\int xf(x)dx$$

#!/usr/bin/env python3                       
#-*- coding:utf-8 -*-
#############################################
#File Name: variance.py
#Brief:
#Author: frank
#Email: frank0903@aliyun.com
#Created Time:2018-08-06 22:40:11
#Blog: http://www.cnblogs.com/black-mamba
#Github: https://github.com/xiaomagejunfu0903/statistic_notes
#############################################
import numpy as np
import matplotlib.pyplot as plt

A = [2, 2, 3, 3]
B = [0, 0, 5, 5]

mean_A = np.mean(A)
print("mean_A:{}".format(mean_A))
mean_B = np.mean(B)
print("mean_B:{}".format(mean_B))

var_A = np.var(A)
print("var_A:{}".format(var_A))
var_B = np.var(B)
print("var_B:{}".format(var_B))

y_A = [0,0,0,0]
plt.scatter(A,y_A,c='r',s=25,marker='o')
plt.scatter(B,y_A,c='b',s=25,marker='*')
plt.plot(var_A, 2, 'k+')
#A_handle, = plt.plot((var_A,mean_A), (2,2))
plt.plot((var_B), (1), 'gD')
#A_handle, = plt.plot((var_B,mean_B), (1,1))

plt.plot((mean_A,mean_A),(0.0,2.5))#均值线

plt.plot((2,mean_A),(0.25,0.25),'peru')
plt.plot((3,mean_A),(0.50,0.50),'seagreen')

plt.plot((0,mean_B),(1.5,1.5),'magenta')
plt.plot((5,mean_B),(2.0,2.0),'hotpink')

plt.grid(True)
plt.show()