https://study.163.com/course/introduction.htm?courseId=1005269003&utm_campaign=commission&utm_source=cp-400000000398149&utm_medium=share

# -*- coding: utf-8 -*-
'''
Author：Toby
QQ：231469242，all right reversed,no commercial use'''import scipy
from scipy.stats import f
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
# additional packages
from statsmodels.stats.diagnostic import lilliforsgroup1=[2,3,7,2,6]
group2=[10,8,7,5,10]
group3=[10,13,14,13,15]
list_groups=[group1,group2,group3]
list_total=group1+group2+group3#正态分布测试
def check_normality(testData):#20<样本数<50用normal test算法检验正态分布性if 20<len(testData) <50:p_value= stats.normaltest(testData)[1]if p_value<0.05:print"use normaltest"print "data are not normal distributed"return  Falseelse:print"use normaltest"print "data are normal distributed"return True#样本数小于50用Shapiro-Wilk算法检验正态分布性if len(testData) <50:p_value= stats.shapiro(testData)[1]if p_value<0.05:print "use shapiro:"print "data are not normal distributed"return  Falseelse:print "use shapiro:"print "data are normal distributed"return Trueif 300>=len(testData) >=50:p_value= lillifors(testData)[1]if p_value<0.05:print "use lillifors:"print "data are not normal distributed"return  Falseelse:print "use lillifors:"print "data are normal distributed"return Trueif len(testData) >300:  p_value= stats.kstest(testData,'norm')[1]if p_value<0.05:print "use kstest:"print "data are not normal distributed"return  Falseelse:print "use kstest:"print "data are normal distributed"return True#对所有样本组进行正态性检验
def NormalTest(list_groups):for group in list_groups:#正态性检验status=check_normality(group1)if status==False :return False#对所有样本组进行正态性检验
NormalTest(list_groups)

http://baike.baidu.com/link?url=o9Z7vr6VdvGAtTRO3RYxQbVu56U_XDaSdibPeVcidMJQ7B6LcAUBHcIro4tLf5BSI5Pu-59W4SPNZ-zRFJ8_FgL3dxJLaUdY0JiB2xUmqie

QQPlot图是用于直观验证一组数据是否来自某个分布，或者验证某两组数据是否来自同一（族）分布。在教学和软件中常用的是检验数据是否来自于正态分布。

# -*- coding: utf-8 -*-
import numpy as np
import statsmodels.api as sm
import pylabtest = np.random.normal(0,1, 1000)sm.qqplot(test, line='45')
pylab.show()

QQ图显示1000个点很好落在y=x直线附近，所以这些数据有很好正态性。

# -*- coding: utf-8 -*-
from scipy import stats
import matplotlib.pyplot as plt
import numpy as np
nsample = 100
np.random.seed(7654321)ax1 = plt.subplot(221)#A t distribution with small degrees of freedom:
#生成自由度3，样本量100的t分布数据，自由度太小，正态分布性较差
x = stats.t.rvs(3, size=nsample)
res = stats.probplot(x, plot=plt)#A t distribution with larger degrees of freedom:
#自由度大，数据接近正态分布
ax2 = plt.subplot(222)
x2 = stats.t.rvs(25, size=nsample)
res1 = stats.probplot(x2, plot=plt)#A mixture of two normal distributions with broadcasting:
ax3 = plt.subplot(223)
x3 = stats.norm.rvs(loc=[0,5], scale=[1,1.5],size=(nsample/2.,2)).ravel()
res = stats.probplot(x3, plot=plt)#A standard normal distribution:标准正太分布，pp-plot正态性较好
ax4 = plt.subplot(224)
x4 = stats.norm.rvs(loc=0, scale=1, size=nsample)
res = stats.probplot(x4, plot=plt)#Produce a new figure with a loggamma distribution, using the dist and sparams keywords:fig = plt.figure()
ax = fig.add_subplot(111)
x = stats.loggamma.rvs(c=2.5, size=500)
stats.probplot(x, dist=stats.loggamma, sparams=(2.5,), plot=ax)
ax.set_title("Probplot for loggamma dist with shape parameter 2.5")
plt.show()

# -*- coding: utf-8 -*-
#bug report 231469242@qq.com
'''
Graphical and quantitative check, if a given distribution is normal.
- For small sample-numbers (<50), you should use the Shapiro-Wilk test or the "normaltest"
- for intermediate sample numbers, the Lilliefors-test is good since the original Kolmogorov-Smirnov-test is unreliable when mean and std of the distribution
are not known.
- the Kolmogorov-Smirnov(Kolmogorov-Smirnov) test should only be used for large sample numbers (>300)
'''# Copyright(c) 2015, Thomas Haslwanter. All rights reserved, under the CC BY-SA 4.0 International License# Import standard packages
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
import pandas as pd# additional packages
from statsmodels.stats.diagnostic import lilliforsdef check_normality():'''Check if the distribution is normal.'''# Set the parametersnumData = 1000myMean = 0mySD = 3# To get reproducable values, I provide a seed valuenp.random.seed(1234)   # Generate and show random datadata = stats.norm.rvs(myMean, mySD, size=numData)fewData = data[:100]plt.hist(data)plt.show()# --- >>> START stats <<< ---# Graphical test: if the data lie on a line, they are pretty much# normally distributed_ = stats.probplot(data, plot=plt)plt.show()pVals = pd.Series()pFewVals = pd.Series()# The scipy normaltest is based on D-Agostino and Pearsons test that# combines skew and kurtosis to produce an omnibus test of normality._, pVals['Omnibus']    = stats.normaltest(data)_, pFewVals['Omnibus'] = stats.normaltest(fewData)# Shapiro-Wilk test_, pVals['Shapiro-Wilk']    = stats.shapiro(data)_, pFewVals['Shapiro-Wilk'] = stats.shapiro(fewData)# Or you can check for normality with Lilliefors-test_, pVals['Lilliefors']    = lillifors(data)_, pFewVals['Lilliefors'] = lillifors(fewData)# Alternatively with original Kolmogorov-Smirnov test_, pVals['Kolmogorov-Smirnov']    = stats.kstest((data-np.mean(data))/np.std(data,ddof=1), 'norm')_, pFewVals['Kolmogorov-Smirnov'] = stats.kstest((fewData-np.mean(fewData))/np.std(fewData,ddof=1), 'norm')print('p-values for all {0} data points: ----------------'.format(len(data)))print(pVals)print('p-values for the first 100 data points: ----------------')print(pFewVals)if pVals['Omnibus'] > 0.05:print('Data are normally distributed')# --- >>> STOP stats <<< ---return pVals['Kolmogorov-Smirnov']if __name__ == '__main__':p = check_normality()    print(p)

- For small sample-numbers (<50), you should use the Shapiro-Wilk test or the "normaltest"

# -*- coding: utf-8 -*-
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
x = stats.norm.rvs(loc=5, scale=3, size=49)
stats.shapiro(x)
'''
p值大于0.05，H0成立，数据呈现正态分布
Out[9]: (0.9735164046287537, 0.3322194814682007)
'''
plt.hist(x)

Lilliefors-test

适用于中等样本数据- for intermediate sample numbers, the Lilliefors-test is good since the original Kolmogorov-Smirnov-test is unreliable when mean and std of the distribution are not known.

# -*- coding: utf-8 -*-
'''
样本量必须大于等于20
UserWarning: kurtosistest only valid for n>=20
'''import numpy
from numpy import random
from statsmodels.stats.diagnostic import lilliforsd = numpy.random.normal(size=200)
n = lillifors(d)
print (n)
'''
(0.047470987201221337, 0.3052490552871156)
'''

Kolmogorov-Smirnov(Kolmogorov-Smirnov) test

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kstest.html#scipy.stats.kstest

- the Kolmogorov-Smirnov(Kolmogorov-Smirnov) test should only be used for large sample numbers (>300)

# -*- coding: utf-8 -*-
'''
样本量必须大于等于20
UserWarning: kurtosistest only valid for n>=20
'''import numpy
from numpy import random
from scipy import statsd = numpy.random.normal(size=1000)
n = stats.kstest(d,'norm')
print (n)'''
KstestResult(statistic=0.028620435047503723, pvalue=0.38131540630243177)
'''

http://jingyan.baidu.com/article/86112f135cf84c27379787cb.html

在统计学中，柯尔莫可洛夫-斯米洛夫检验基于累计分布函数，用以检验两个经验分布是否不同或一个经验分布与另一个理想分布是否不同。

在进行累计概率（cumulative probability）统计的时候，你怎么知道组之间是否有显著性差异？有人首先想到单因素方差分析或双尾检验（2 tailedTEST）。其实这些是不准确的，最好采用Kolmogorov-Smirnov test（柯尔莫诺夫-斯米尔诺夫检验）来分析变量是否符合某种分布或比较两组之间有无显著性差异。

分类：

1、Single sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.

采用柯尔莫诺夫-斯米尔诺夫检验来分析变量是否符合某种分布，可以检验的分布有正态分布、均匀分布、Poission分布和指数分布。指令如下：

>> H = KSTEST(X,CDF,ALPHA,TAIL) % X为待检测样本，CDF可选：如果空缺，则默认为检测标准正态分布；

如果填写两列的矩阵，第一列是x的可能的值，第二列是相应的假设累计概率分布函数的值G(x)。ALPHA是显著性水平（默认0.05）。TAIL是表示检验的类型（默认unequal,不平衡）。还有larger，smaller可以选择。

如果，H=1 则否定无效假设； H=0，不否定无效假设（在alpha水平上）

例如，

x = -2:1:4
x =
  -2  -1   0   1   2   3   4

[h,p,k,c] = kstest(x,[],0.05,0)
h =
   0
p =
   0.13632
k =
   0.41277
c =
   0.48342

The test fails to reject the null hypothesis that the values come from a standard normal distribution.

2、Two-sample Kolmogorov-Smirnov test

检验两个数据向量之间的分布的。

>>[h,p,ks2stat] = kstest2(x1,x2,alpha,tail)

% x1,x2都为向量，ALPHA是显著性水平（默认0.05）。TAIL是表示检验的类型（默认unequal,不平衡）。

例如，x = -1:1:5
y = randn(20,1);
[h,p,k] = kstest2(x,y)
h =
     0
p =
    0.0774
k =
    0.5214

Kolmogorov–Smirnov test (K–S test)

wiki翻译起来太麻烦，还有可能曲解本意，最好看原版解释。

In statistics, the Kolmogorov–Smirnov test (K–S test) is a form of minimum distance estimation used as a nonparametric test of equality of one-dimensional probability distributions used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). The Kolmogorov–Smirnov statistic quantifies a distance between theempirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples. The null distribution of this statistic is calculated under the null hypothesis that the samples are drawn from the same distribution (in the two-sample case) or that the sample is drawn from the reference distribution (in the one-sample case). In each case, the distributions considered under the null hypothesis are continuous distributions but are otherwise unrestricted.

The two-sample KS test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples.

The Kolmogorov–Smirnov test can be modified to serve as a goodness of fit test. In the special case of testing for normality of the distribution, samples are standardized and compared with a standard normal distribution. This is equivalent to setting the mean and variance of the reference distribution equal to the sample estimates, and it is known that using the sample to modify the null hypothesis reduces the power of a test. Correcting for this bias leads to theLilliefors test. However, even Lilliefors' modification is less powerful than the Shapiro–Wilk test or Anderson–Darling test for testing normality.^[1]