经典六西格玛(6 sigma)培训内部资料A_07_Testing of Variances(3

1

Testing of Variances

A Test Single Std Dev Against Target

单一标准差与标准比较

Situation 1

情况1

Testing of Variances 方差检验

2

Testing of Variances

Inferences About Variances 方差的推论

“Most often we are interested in possible differences in the mean level of response produced by different

methods or treatments. Sometimes, however, it is the degree of variation of the data that is of interest ….

Process modifications that reduce variance, even though they leave the mean unchanged, can be of great

importance. Again it may be of interest to compare the variation of two or more analytical methods.”

“通常我们对由不同方法或处理过程产生的响应的均值差异感兴趣。有时对数据的变异程度感兴趣。通过制程改善以减少过程的方差非常重要，即使没有改变均值。同样，两种或多种分析方法的变异也是感兴趣的话题。”

-Box, Hunter & Hunter, Statistics for Experimenters, 1978 Section 5.4

3

Testing of Variances

Example: Test Single Std Dev Against Target 例：单一标准差与标准比较

The Coca-Cola Bottling Company puts 16 ounces of Coke in every can. Now they wants to reduce the

variation. After changes implemented, staff claim that the standard deviation of the process is 0.02. Data: Coca-Cola. MTW

可口可乐装瓶公司希望减少装瓶重量的变异。经过一些变更后，他们声称过程的标准差为0.02

Note: Minitab does not provide an individual χ2test for standard deviations. Instead, it is necessary to look at the confidence interval on the standard deviation and determine if the CI contains the claimed value.

注：Minitab 没提供单独标准差的χ2检验。作为替代，必须检查标准差的置信区间并确定置信区间是否包含声明值。

4

Testing of Variances

Example: Single Standard Deviation 例：单一标准差

统计> 基本统计量> 图形汇总

5

Testing of Variances

16.03

16.0216.0116.0015.9915.9815.97中位数

平均值16.015

16.010

16.005

16.000

15.995

15.990

第一四分位数15.983中位数

16.000第三四分位数16.018最大值16.03015.99016.01215.988

16.0120.015

0.032

A 平方0.35P 值0.427平均值16.001标准差0.020方差0.000偏度0.06379峰度-1.12966

N

16最小值

15.970Anderson-Darling 正态性检验95% 平均值置信区间95% 中位数置信区间95% 标准差置信区间

95% 置信区间

ounces 摘要

Minitab output Minitab 输出

Sigma CI

6

Testing of Variances

Our Conclusion Statement 结论陈述

Because the confidence interval on the standard deviation contains the value consistent with the claim, we can make the following statement:

由于标准差的置信区间包含了声明值，我们可以作出下述陈述：

“We have insufficient evidence to reject the null hypothesis that σ= 0.02.”

我们没有足够的证据拒绝零假设σ= 0.02

7

Testing of Variances

Situation 2情况2

Testing the Equality of Two Population Variances

检验两个抽样对象的方差是否相等

8

Testing of Variances

¾Are Two Variances the “Same ”or Different?两方差相同还是不同？¾Use the F Test 使用F 检验

¾If F > F crit , then reject equality 如果F > F crit , 那么否定两者相等

Testing Two (Normal) Variances 检验两方差（正态）

Compare 比较

to F crit , with s 1> s 2

2

2

21s s F

=

F crit

α

F crit is based on α, with n 1-1 df in numerator and n 2-1 df in denominator

F crit 基于α，分子自由度为n 1-1 ，分母自由度为n 2-1

9

Testing of Variances

¾It is normally assumed that the samples have the same variance when performing Hypothesis Testing on population means.¾Test for variance should be conducted to verify this assumption.

Test For Equal Variances 等方差检验

In Minitab, we use what we select.

等方差检验

统计方差分析10

Testing of Variances

Test For Equal Variances 等方差检验

Levene ’s Test

No 否

Bartlett ’s Test Yes 是More than 2大于2

Levene ’s Test No 否

2

F-Test Yes

是Appropriate Hypothesis Test

假设检验Normally Distributed?是否正态

Number of Populations 抽样对象数量2

221122210:H :H σ≠σσ=σdifferent is variance one least at :1H 232221:0H σ=σ=σ…If P<α, reject H 0

If P>α, Do not reject H 0

11

Testing of Variances

¾Bartlett ’s and Levene ’s test.Bartlett 和Levene

Bartlett’s test can only used when the data is normally

distributed when testing two samples, it is equivalent to F test.Bartlett 检验仅用于正态数据，且对两样本与F 检验是等同的。 Levene’s test does not require the data to be normally distributed.

Levene 在检验方差的差异时并不要求数据是正态的

Test For Equal Variances 等方差检验

12

Testing of Variances

Test For Equal Variance: 2 Samples 等方差检验：2样本

Assume we’re working on a new composite material and we desire to test the effect of two different processing

temperature setting on the variance of the strength of the material. The temperature settings to be tested are 200 F and 300 F. Samples of 9 materials are taken at each setting.Data is in: Material strength.MTW.

假定我们正在改善一种复合材料。我们希望检验两种加工温度对材料强度的方差产生的影响。温度设定分别为200F 和300F 。每种温度下测量了9个材料。. . .