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# Variance(方差)

## 引入例子

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

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

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

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

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

## 表示符号

$\displaystyle \sigma ^{2}$：一般表示总体的方差

$\displaystyle s^{2}$: 一般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 $\displaystyle X$ , $\displaystyle \mu =\operatorname {E} [X]$:

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

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

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

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

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

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

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

#!/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()


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