小馬哥課堂-統計學-方差

Variance(方差)

衡量隨機變量或一組數據的 離中 的離散程度

引入例子

假設有兩組整體:A={2,2,3,3}, B={0,0,5,5}

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

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

顯然,$\mu$並不能反映這兩組數據集的差別。

數據集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

數據集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

從上面兩組數據集能夠看出,儘管兩組數據集 指望 相同,可是,方差並不相同。數據集A 表現得更集中，而數據集B 表現得相對分散。

定義

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

在機率論和統計學中，方差的定義: 隨機變量與指望的差 的平方值 的指望，衡量隨機變量與其指望的偏離程度。

表示符號

$\displaystyle \sigma ^{2}$：通常表示整體的方差

$\displaystyle s^{2}$: 通常表示抽樣分佈的方差

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

公式

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

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

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

對於離散型隨機變量,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

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

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

對於連續型隨機變量,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))#均值線

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()