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兩個配對樣本，均勻分布，非正太分布

Wilcoxon signed-rank test

曼-惠特尼U檢驗Mann–Whitney Test

兩個獨立樣本，均勻分布，非正太分布

https://en.wikipedia.org/wiki/Frank_Wilcoxon

ranking method

這是a=0.05對應得樣本數

critical value 對應樣本數，20樣本對應的關鍵值為52

positive sum:

差異為正值的排名和

negative sum：

差異為負值的排名和

如果negative sum小于52，表明兩組數有顯著差異，推翻原假設

http://blog.sina.com.cn/s/blog_4bcf5ebd0101422n.html

SPSS學習筆記之——兩配對樣本的非參數檢驗（Wilcoxon符號秩檢驗）

一、概述

非參數檢驗對于總體分布沒有要求，因而使用范圍更廣泛。對于兩配對樣本的非參數檢驗，首選Wilcoxon符號秩檢驗。它與配對樣本t檢驗相對應。

二、問題

為了研究某放松方法（如聽音樂）對于入睡時間的影響，選擇了10名志愿者，分別記錄未進行放松時的入睡時間及放松后的入睡時間（單位為分鐘），數據如下筆。請問該放松方法對入睡時間有無影響。

本例可以采用配對樣本t檢驗，但由于樣本量少，數據可能不符合正太分布，所以考慮用非參數檢驗。

三、統計操作

數據視圖

菜單選擇

打開如下的對話框

該對話框有三個選項卡，第一個選項卡會根據第三個選項卡的設置自動設置，故一般不用手動設定。點擊進入“字段”選項卡。將“放松前”、“放松后”均選入右邊“檢驗字段”框中。

點擊進入“設置”對話框，選擇檢驗方法，切換為“自定義檢驗”，選擇“Wilcoxon匹配樣本對符號秩（二樣本）”復選框。“檢驗選項”可以設定顯著性水平。

點擊“運行”按鈕，輸出結果

四、結果解讀

這就是輸出結果。原假設示放松前好放松后差值的中位數等于0，P=0.015<0.05，拒絕原假設，認為放松前后有統計學差異。

雙擊該表格，會彈出如下的“模型瀏覽器”窗口，可以看到更詳細的信息。如下圖。

# -*- coding: utf-8 -

import scipy.stats as stats

data1=[21,12,12,23,19,13,20,17,14,19]

data2=[12,11,8,9,10,15,16,17,10,16]

stats.wilcoxon(data1,data2)

'''

Out[2]: WilcoxonResult(statistic=2.0, pvalue=0.01471359242280415)

p值小于0.05，兩組數據有顯著差異

'''

https://study.com/academy/lesson/non-parametric-inferential-statistics-definition-examples.html

In this lesson, you're going to learn about the major differences between parametric and non-parametric methods for dealing with inferential statistics, as well as see an example of the non-parametric method.

Normal

What is normal? At least in the world of statistics, this has nothing to do with how someone dresses, acts, or what their beliefs are. Normal data comes from a population with a normal distribution. A normal distribution is a distribution that has a symmetrical bell-shaped curve to it, which you're probably well aware of.

Keep this concept in mind as we go over the major differences between parametric and non-parametric statistics.

Parametric Methods

First, let's define our terms really simply. When we talk about parameters in statistics, what are we actually hinting at? Parameters are descriptive measures of population data, such as the population mean or population standard deviation.

When the variable we are considering is approximately (or completely) normally distributed, or the sample size is large, we can use two inferential methods that are concerned with parameters - appropriately called parametric methods - when performing a hypothesis test for a population mean. For instance, if we find that the distribution of the average salary of a sample looks like a bell curve, then parametric methods may be used.

These two methods are probably ones you've heard of before. They are the z-test, which we'd use when the population standard deviation is known to us; or the t-test, which we'd use when the population standard deviation is not known to us.

Non-Parametric Methods

Inferential methods that are not concerned with parameters are known, easily enough, as non-parametric methods. However, this term is also more broadly used to refer to many methods that are applied without assuming normality. So, for instance, if we find that the distribution of the average salary of a sample looks like the histogram you see on the screen now [see video], which is nothing close to that of a bell curve, then we will have to turn to non-parametric methods.

Such non-parametric methods have their pros and cons. On the pro side, these methods are usually simpler to compute and are more resistant to extreme values when compared to parametric methods. On the con side, if the requirements for the use of a parametric method are actually met, non-parametric methods do not have as much power as the z-test or t-test.

By power, I simply mean the probability of avoiding a type II error, which is an error where we fail to reject the false null hypothesis.

Example of a Non-Parametric Method

One example of a non-parametric method is the Wilcoxon signed-rank test. This is a test that assumes the variable under consideration does not need a specific shape and doesn't have to be normally distributed, but is symmetric in its distribution nonetheless. This means that it can be sliced in half to produce two mirror images.

So, for example, a right-skewed or left-skewed distribution would not be appropriate for this test since it's not symmetric. But a normal, symmetric bimodal, triangular, or uniform distribution would be a fit for this test since any one of those can be sliced in half to produce two mirror images of one another.

Other non-parametric tests include the likes of:

The Kruskal-Wallis test
The Mann-Whitney U test
The sign test

Lesson Summary

Normal data comes from a population with a normal distribution. A normal distribution is a distribution that has a symmetrical bell-shaped curve to it, which you're probably well aware of.

Inferential methods that are concerned with parameters are appropriately called parametric methods, and include the z-test and t-test. Parameters are descriptive measures of population data.

Inferential methods that are not concerned with parameters are known as non-parametric methods. This term is also more broadly used to refer to many methods that are applied without assuming normality.

While non-parametric methods are easier to compute than parametric ones, they do not have as much power as parametric methods if the requirements for the use of a parametric method are met. By power, I simply mean the probability of avoiding a type II error, which is an error where we fail to reject the false null hypothesis.

An example of a non-parametric method is the Wilcoxon signed-rank test. This is a test that assumes the variable under consideration does not need a specific shape and doesn't have to be normally distributed, but is symmetric in its distribution nonetheless.

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