# Normality Test Before the Xbar R Diagram is Made?

D

#### DavidDeng

hi,all!
Yesterday and today I have discussion about a puzzled problem below.
During the initial study phrase of APQP,we have to do SPC(usually, XbarR diagram).
Speaking of XbarR diagrma, I thought these formulas related with XbarR diagram are all based on the normality distribution study:
UCL(Xbar)=Xbarbar+A2Rbar
LCL(Xbar)=Xbarbar-A2Rbar
CL(Xbar)=Xbarbar
UCL(R)=D3Rbar
LCL(R)=D4Rbar
CL(R)=Rbar

Cpk=min{[( USL-Xbarbar)/3SIGMA(Rbar/D2)],[(Xbarbar-LSL)/3SIGMA(Rbar/D2)]}

as a result,before u begin to caculate the Cpk u had to judge whether all ur data are under the normality distribution.
I got the supporting theory from the page 57 of the AIAG SPC (version 1).it reads:
The individual measurements from the process conform to the normality distribution.
There are many techniques for assessing the capability of a process that is in statistical control.Some assume that the process output follows the bell-shaped normality distribution.If it is not known whether the distribution is normal, a test for normality should be made such as reviewing a histogram,plotting on normal probability paper,or using more precise methods(...).If nonnormality is suspected or confirmed, more flexible techniques should be used,such as data transformation to 'normalize" the distribution...

furthermore,I thought we couldn't caculate any one of these control lines on the specified sheet until we do the normality test to the collected data.
by now,I didn't find any supporting theory and/or evidence from any of an authoritative technical manual(Eg. AIAG SPC)

So, I look forward to the help from all of u.Pls be so kind to do that.
thanks million!

B

#### balajishr

Hi

As far as thoery concerned i have heard of conventional methods other than a histogram

1) Skewness & kurtosis test

Skewness is a measure of distortion

3/n
Skewness = summation fi (Xi - X bar)
___________
3
sigma

where fi is the frequency of the cell of data
c is the number of cells

when Skewness is near to Zero it indicates strong normality

Kurtosis - it is a measure how the curve peaks

for a normal distribution , kurtosis is 3

4/n
Kurtosis = summation fi (Xi - X bar)
___________
4
sigma

2) Probability plots on a special sigma scaled graph

3) Chi square test

you can refer to many good books in SPC

for eg : Statistics for Engineers by Freund & Miller
Stastistics for managers (?!) Levin

There is also one practical method after completing your X bar & R Chart before calculations prepare tally marks for the frequency of data for each plotted value. If the result is a histogram with a normal distribution then you have a normally distributed process.

for eg :

x bar 50.84 - xx
50.83 - xxxxxx
50.82 - xxxxxxxxx
50.81 - xxxxxxx
50.80 - xxxxx
50.79 - xx

Then you have a normal distribution.

Enjoy playing with SPC

Balaji

R

#### roland_lu

hi,all!
Yesterday and today I have discussion about a puzzled problem below.
During the initial study phrase of APQP,we have to do SPC(usually, XbarR diagram).
Speaking of XbarR diagrma, I thought these formulas related with XbarR diagram are all based on the normality distribution study:
UCL(Xbar)=Xbarbar+A2Rbar
LCL(Xbar)=Xbarbar-A2Rbar
CL(Xbar)=Xbarbar
UCL(R)=D3Rbar
LCL(R)=D4Rbar
CL(R)=Rbar

Cpk=min{[( USL-Xbarbar)/3SIGMA(Rbar/D2)],[(Xbarbar-LSL)/3SIGMA(Rbar/D2)]}

as a result,before u begin to caculate the Cpk u had to judge whether all ur data are under the normality distribution.
I got the supporting theory from the page 57 of the AIAG SPC (version 1).it reads:
The individual measurements from the process conform to the normality distribution.
There are many techniques for assessing the capability of a process that is in statistical control.Some assume that the process output follows the bell-shaped normality distribution.If it is not known whether the distribution is normal, a test for normality should be made such as reviewing a histogram,plotting on normal probability paper,or using more precise methods(...).If nonnormality is suspected or confirmed, more flexible techniques should be used,such as data transformation to 'normalize" the distribution...

furthermore,I thought we couldn't caculate any one of these control lines on the specified sheet until we do the normality test to the collected data.
by now,I didn't find any supporting theory and/or evidence from any of an authoritative technical manual(Eg. AIAG SPC)

So, I look forward to the help from all of u.Pls be so kind to do that.
thanks million!

welcome!

you are right, all the calculations are based on the assumption of normality. If you read the munual carefully, you will find the the preconditions for calculation are always that process is stable, or in control, or special causes have been identified and eliminated, etc., all these imply that the process shall be normal; otherwise, you just get a bunch of numbers that are difficult to interpret.

you can find tons of information in this site.

D

#### Darius

I put a file L008.pdf of an article about skewed data on a chart.

IMHO Donald Wheeler is one of the best on SPC and he show on this article that the SPC work altho skewed data are used.

D

#### DavidDeng

thx to u all.
seems I have solved my question.but I still have sth to say here and hope the further verification.
1.According to the file L008.pdf,I needn't do any normality test to those collected data. I can directly caculate the 5 control lines and then draw them on the specified sheet. AM I RIGHT?
2.Now that I have 5 control lines on the sheet,one by one I plot each single dot onto,I begin to observe what happens,if:
a.none of dot is out of the control lines
b.all dots are OK by testing with the 8 rules
therefore,I make a conclusion the process is stable and under control. AM I RIGHT?
3.If there is any dot out of the control lines,I should analyze the related cause.
if it is special cause,I may delete this dot (relative with a group data);after doing that,there is still enough dot,I can still judge the process situation.but there isn't enough,I have to do the whole data collection again.
if it is common cause,I can't delete this dot. on the contrary,I have to do the whole data collection again.
AM I RIGHT?
4.once the step 3 and 4 above are completed,I can say my process is stable and under control,now I need to caculate Cpk. because I don't still know whether my data is normality distribution( Sorry,I didn't agree with Roland Lu's opinion),I can use the histogram and so on(eg. Minitab software) to do a neceesary normality test. if ok,I can use the formula--Cpk=min{[( USL-Xbarbar)/3SIGMA(Rbar/D2)],[(Xbarbar-LSL)/3SIGMA(Rbar/D2)]} to get the Cpk,since this formula is based on the normality distribution.
if I have other method and/or formula to accomplish it,and the new method and formula aren't based on the normality distribution,I need only follow the specified rule(s) of them,maybe I needn't do the normality distribution.
AM I RIGHT?
5. the last one, if I want to caculate Ppk and now I have collected enough data, I immediately begin to use the formula Ppk=min{[( USL-Xbarbar)/3SIGMA(s)],[(Xbarbar-LSL)/3SIGMA(s)]} to get Ppk value without the need to observe the Xbar diagram and R diagram, BUT WITH the need to normality test.
AM I RIGHT?

thx again to u all.

P

#### potdar

David,

Playing with data using SPC techniques is all right. Remember, if you find that special causes exist and delete data relating to those observations, special causes dont go away. You have to dirty your hands on the machine to make them go away and bring the process under control.

Dont spend your time juggling with numbers and 'managing' to estimate control limits, ppk,... for a process thats not in control. Whats the use if after all this effort the process is going to physically generate unpredictable / unacceptable output?

Very nice statistical solutions exist to analyse all sorts of situations and data. I recognise them as solutions in search of problems.