[分享]about Definitions,Formulas,Occasion of Cpk/Ppk/StDev

**Definitions
定义**
T=target
T=目标值
USL=upper specification limit
USL=规格上限
LSL=lower specification limit
LSL=规格下限
CL=center line
CL=中心值
UCL=upper control line
UCL=管制上限
LCL=low control line
LCL=管制下限
Xbar=平均值
Xbar=estimate of process mean
Rbar=average of subgroup range
Rbar=组内全距的平均值
Sbar=average of subgroup standard deviation
Sbar=每组标准差的平均值
ni = number of (nonmissing) observations in subgroup i
ni=组内观察次数(此处不讨论组内观察次数不相等的情况)
n=∑ni=total number of (nonmissing) observations
n=∑ni=总观测次数
StDev(Within)=estimate of within subgroup process standard deviation
StDev(Within)=组内标准差
StDev(Betwn)= of between subgroup process standard deviation
StDev(Betwn)=组间标准差
StDev(Total)=Total estimate of standard deviation
StDev(Total)=合并标准差
StDev(Overall)=estimate of overall process standard deviation
StDev(Overall)=总标准差
Xi=the ith observation
Xi=观测值

**Foreword
前言**
1 The reasons there are two main expressions for "population" and "sample"” is that one is "biased estimator" and the other is not.
“母体”与“样本”之所以有两种表达公式，是因为一个是有偏估计，另一个则不是。
How much real practical difference is there between using a standard deviation with n in the denominator versus one with (n-1).
计算标准差使用分母为n和n-1的两个公式，现实究竟有多大的差异？
With thirty points you are under 2% in disagreement. So in industrial situations it is always to use the (n-1) formula.
30个数据就会存在2%的差异，所以在工业生产中，通常使用n-1的公式。
2 Most of the data we've collected are individual data or multiple readings for subgroup data.
我们收集的数据通常是单独的数据或是多次测量的多组数据。
3 PROCESS CAPABILITY is defined as the 6 sigma range of a process's inherent variation, where sigma is usually estimated by R-bar/d2 and where inherent variation is defined as that portion of process variation due to common causes only.
过程能力定义为6sigma（3sigma控制线）过程的固有偏差，这里的sigma通常使用R-bar/d2计算，而固有偏差则仅是一般原因造成的，为过程偏差的一部分。
3 PROCESS PERFORMANCE is defined as the 6 sigma range of a process's total variation where sigma is usually estimated by s, the sample standard deviation
过程表现则定义为6sigma（3sigma控制线）过程的全部差异，这里的sigma是用样本的标准差来计算的。
4 Cpk---Process capability index attempts to answer the question "does my process in the long run meet specification?"
CpK---过程能力指数用来回答“过程长期运行时是否能满足要求？”
5 Ppk---Process performance index attempts to answer the question "does my current production sample meet specification ?"
PpK---过程表现能力用来回答“目前过程生产的样品能否满足要求？”
6 "Long Term" versus "Within" and "Short Term" versus "Overall" it seems antinomic.
“长期”“组内”和“短期”“全部”看起来是自相矛盾的。，
To my opinion, process should be stable in the long run if not we calls the alternate potential process.
在我看来，长期运行下的过程应该是稳定的，如果不是那就称为潜在过程。
Meanwhile, process capability evaluation can only be done after the process is brought into statistical control, that’s to say, the process you studied is stable without BETWEEN VARIATION.
同时，过程能力的研究只有在过程已处于统计管制之下才能进行，也就是你所研究的过程已消除了组间差异。
The reason is simple: Cpk is a prediction, and one can only predict something that is stable. And process performance indices should only be used when statistical control cannot be evaluated.
原因很简单：Cpk只是一个预测，只能对稳定的东西作出预测，而过程表现PpK则应该用在还没有进行管制的过程。
Then comes to the problem or misunderstand: process in long run (potential process) only has within variation, i.e. short term variability. Also, overall variation is equal to long term variability we often say.
于是问题或误解就产生了：长期运行下的过程（潜在过程）只有组内变异，也就是短时间内的变化，而总变异就是我们通常讲的长时间的变化。
So we often assosiate "Cpk"with"Long Term","Within"and"Potential" and "PpK"with"Short Term"and"Overall".
所以，我们就经常把Cpk与“长期”“组内”“潜在”联系在一起，而Ppk则与“短期”“全部”联系在一起。
Avoid confusion of "Long Term" and "Short Term", Even MINITAB no longer uses long-term & short-term. As of Release 13 they changed their terminology so it now associates Cpk with the descriptor "Potential (Within)" and Ppk with the descriptor "Overall".
为了避免混淆“长期”与“短期”，就连MINITAB也不再使用“长期”与“短期”，13版本就更改了相关术语，Cpk使用潜在（组内），Ppk使用全部。

Formulas(1)---Following is sample estimates of MINITAB
公式（1）---以下是MINITAB的计算公式，且都为估计值(sample)

>>Subgroup Size>1(Betwn/Within)
(1)StDev(with)
Pooled standard deviation---Ddefault
StDev(Within)=SQRT/C4(d)
d=∑(ni-1)+1
Rbar---Average of subgroup ranges
StDev(Within)=Rbar/d2(ni)
Sbar---Average of subgroup standard deviation
StDev(Within)=Sbar/C4(n)
(2) StDev(Betwn)
StDev(Betwn)=SQRT(max(0, StDev(Xibar)^2- StDev(within)^2/Subgroup Size)
for StDev(Xibar) use the formula of Average moving range(Subgroup Size=1)
(3) StDEV(Total)
StDEV(Total)=SQRT(StDev(with)^2+ StDev(Betwn)^2)
(4) StDEV(Overall)
StDEV(Overall)=SQRT(∑∑(Xi-Xbar)^2/(n-1))/c4(n)
>>>Subgroup Size=1(Normal)
(1)StDev(within)
Average moving range---Default
StDev(Within)=MRbar/d2(w)
MRi=max(Xi,…, Xi-w+1)-min(Xi,…, Xi-w+1), for i=w,…, n
MRbar=(MRw+…+MRn)/(n-w+1)
w=2 by default
Median moving range
StDev(Within)=MR-Median/d4(w)
MRi=max(Xi,…, Xi-w+1)-min(Xi,…, Xi-w+1), for i=w,…, n
MR-Median= the median of all MRi
w=2 by default
Square root of MSSD
StDev(Within)=SQRT/C4(ni)
di=successive differences
(2)StDEV(Overall)
StDEV(Overall)=SQRT(∑∑(Xi-Xbar)^2/(n-1))/c4(n)

Formulas(2)---simply & usually
StDev(Population)=SQRT(∑(Xi-u)^2/n)
StDev(Sample)=SQRT(∑(Xi-Xbar)^2/(n-1))
StDev(Within)=Rbar/d2
StDEV(Overall)=SQRT(∑(Xi-Xbar)^2/(n-1))

Formulas(3)---Capability Statistics
(1)Population
Cp=(USL-LSL)/(6*StDev(Within))
CPU=(USL-u)/(3*StDev(Within))
CPL=(u -LCL)/(3*StDev(Within))
CpK=min(CPU,CPL)
Pp=(USL-LSL)/(6*StDev(Overall))
PPU=(USL-u)/(3*StDev(Overall))
PPL=(u -LCL)/(3*StDev(Overall))
PpK=min(PPU,PPL)
(2)Sample
Cp=(USL-LSL)/(6*StDev(Within))
CPU=(USL-Xbar)/(3*StDev(Within))
CPL=(Xbar-LCL)/(3*StDev(Within))
CpK=min(CPU,CPL)
Pp=(USL-LSL)/(6*StDev(Overall))
PPU=(USL-Xbar)/(3*StDev(Overall))
PPL=(Xbar-LCL)/(3*StDev(Overall))
PpK=min(PPU,PPL)
(3)Cpm(estimates)
Cpm=(USL-LSL)/(6*SQRT(∑(Xi-T)^2/(n-1)))
Cp>=CpK>=PpK
Note: If the indices are for sample or estimates we should add the hats(^)m on top of the letter.

when use Cp/CpK & Pp/PpK
for Cp/Cpk
1 Data from control charts
2 no less than 100 data points
for Pp/PpK
1 process is chronically unstable but meeting the specifications and in a predictable pattern.
2 Data from short term（短时间） studies: a snap shot of the process
3 less than 100 data points
4 when we do PPAP