2.69.2.2 Algorithm for Capability Analysis

1 Normal Capability Analysis
2 Between/Within Capability Analysis
3 Non-normal Capability Analysis
4 Binomial Capability Analysis
5 Poisson Capability Analysis

Normal Capability Analysis

Standard Deviation Estimation

Within-subgroup and overall standard deviations are estimated for normal capability analysis.

Within-subgroup Standard Deviation ()
According to the subgroup size (bigger than 1, or equal to 1), estimating method is different.
- Subgroup Size > 1
  - Average of Subgroup Ranges (Rbar)
    $\sigma_{within}=S_r=\frac{\sum_{i=1}^N\frac{f_ir_i}{d_2(n_i)}}{\sum_{i=1}^Nf_i}$ , where $f_i = \frac{(d_2(n_i))^2}{(d_3(n_i))^2}$
    
    $N$ : Number of subgroups
    
    $r_i$ : The range of the $ith$ subgroup, $r_i=\max(ith\_subgroup\_observations)-\min(ith\_subgroup\_observations)$
    
    $n_i$ : Number of observations in the $ith$ subgroup
    
    $d_2(n_i), d_3(n_i)$ : Unbiasing constant, $d2(), d3()$
  - Average of Subgroup Standard Deviations (Sbar)
    Unbiased $\sigma_{within}=\bar{S}=\frac{\sum_{i=1}^N\frac{h_iS_i}{c_4(n_i)}}{\sum_{i=1}^Nh_i}$ , where $h_i=\frac{(c_4(n_i))^2}{1-(c_4(n_i))^2}$
    
    Not use unbiasing constant, $\sigma_{within}=\frac{\sum_{i=1}^NS_i}{N}$
    
    $N$ : Number of subgroups
    
    $S_i$ : The standard deviation of the $ith$ subgroup
    
    $n_i$ : Number of observations in the $ith$ subgroup
    
    $c_4(n_i)$ : Unbiasing constant, $c4()$
  - Pooled Standard Deviation
    Unbiased $\sigma_{within}=\frac{S_p}{c_4(d+1)}$ , where $S_p=\sqrt{\frac{\sum_{i=1}^N\sum_{j=1}^{n_i}(X_{ij}-\bar{X}_i)^2}{\sum_{i=1}^N(n_i-1)}}$ , $c_4(d+1)=\frac{\Gamma{(\frac{d+1}{2})}}{\Gamma{(\frac{d}{2})}}\sqrt{\frac{2}{d}}$
    
    Not use unbiasing constant, $\sigma_{within}=S_p$
    
    $N$ : Number of subgroups
    
    $n_i$ : Number of observations in the $ith$ subgroup
    
    $X_{ij}$ : The $jth$ observation in the $ith$ subgroup
    
    $\bar{X_i}$ : The Mean of the $ith$ subgroup
    
    $d$ : Degrees of freedom for $S_p, d=\sum_{i=1}^N(n_i-1)$
    
    $c_4(d+1)$ : Unbiasing constant, $c4()$
    
    $\Gamma()$ : Gamma function
- Subgroup Size = 1
  - Average of Moving Range
    $\sigma_{within}=\frac{R_w+\cdots+R_N}{(N-w+1)d_2(w)}$
    
    $N$ : The number of all observations
    
    $w$ : The number of observations used in the moving range
    
    $R_i$ : The $ith$ moving range. $R_i=\max{(X_i,\cdots,X_{i-w+1})}-\min{(X_i,\cdots,X_{i-w+1})}, i=w,\cdots,N$ , and $X_i$ is the $ith$ observation
    
    $d_2(w)$ : Unbiasing constant, $d2()$
  - Median of Moving Range
    $\sigma_{within}=\frac{\overline{MR}}{d_4(w)}$
    
    $N$ : The number of all observations
    
    $w$ : The number of observations used in the moving range
    
    $MR_i$ : The $ith$ moving range. $MR_i=\max{(X_i,\cdots,X_{i-w+1})}-\min{(X_i,\cdots,X_{i-w+1})}, i=w,\cdots,N$ , and $X_i$ is the $ith$ observation
    
    $\overline{MR}$ : The median of the $MR_i, i=w,\cdots,N$
    
    $d_4(w)$ : Unbiasing constant, $d4()$
  - Square Root of Mean Squared Successive Differences (MSSD)
    Unbiased $\sigma_{within}=MSSD=\frac{\sqrt{\frac{\sum_{i=1}^{N-1}d_i^2}{2(N-1)}}}{c_4'(N)}$
    
    Not use unbiasing constant, $\sigma_{within}=\sqrt{\frac{\sum_{i=1}^{N-1}d_i^2}{2(N-1)}}$
    
    $N$ : Number of observations
    
    $d_i$ : Succesive differences of observations
    
    $c_4'(N)$ : Unbiasing constant, $c4'()$

Ovarall Standard Deviation ( $\sigma_{overall}$ )
Unbiased $\sigma_{overall}=\frac{S}{c_4(N)}$ , where $S=\sqrt{\frac{\sum_{i=1}^N(X_{i}-\bar{X})^2}{N-1}}$

Not use unbiasing constant, $\sigma_{within}=S$

$N$ : Number of all observations

$X_{i}$ : The $ith$ observation

$\bar{X}$ : The Mean of the all observations

$c_4(N)$ : Unbiasing constant, $c4()$

Potential Capability

Cp
$Cp=\frac{USL-LSL}{Toler*\sigma_{within}}$

$USL, LSL$ : Upper and lower specification limits respectively

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{within}$ : Within-subgroup standard deviation

Confidence Interval Bounds for Cp
$LowerBound = Cp\sqrt{\frac{\chi_{\alpha/2,\nu}^2}{\nu}}$

$LowerBound = Cp\sqrt{\frac{\chi_{1-\alpha/2,\nu}^2}{\nu}}$

$\alpha$ : Alpha for the confidence level

$\nu$ : Degrees of freedom

$\chi_{\alpha,\nu}$ : $\alpha$ percentile of chi-square distribution with $\nu$ degrees of freedom
is calculated differently based on the method used for standard deviation.
- Average of Subgroup Ranges (Rbar): $\nu=0.9k(n-1)$
- Average of Subgroup Standard Deviations (Sbar): $\nu=f_nk(n-1)$
- Pooled standard deviation: $\nu=\sum(n_i-1)$
- Average of Moving Range or Median of Moving Range: $\nu\approx k-R_{span}+1$
- Square Root of MSSD: $\nu=k-1$
  where $n_i$ is the $ith$ subgroup size, $k$ is number of subgroups, $R_{span}$ is the length of the moving range, $n$ is the mean of subgroup size, $n=\frac{\sum n_i}{k}$ , and $f_n$ is calculated according to $n$ as follows:

n	2	3	4	5	6,7	8,9	10-17	18-64	> 64
$f_n$	0.88	0.92	0.94	0.95	0.96	0.97	0.98	0.99	1

CPL
$CPL=\frac{\bar{X}-LSL}{\frac{Toler}{2}*\sigma_{within}}$

$\bar{X}$ : Process mean estimated from observations or historical value

$LSL$ : Lower specification limit

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{within}$ : Within-subgroup standard deviation

CPU
$CPU=\frac{USL-\bar{X}}{\frac{Toler}{2}*\sigma_{within}}$

$\bar{X}$ : Process mean estimated from observations or historical value

$USL$ : Upper specification limit

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{within}$ : Within-subgroup standard deviation

Cpk
$Cpk=\min(CPU, CPL)$

$(1-\alpha)100\%$ Confidence Interval Bounds for Cpk
$LowerBound = Cpk - Z_{1-\alpha/2}\sqrt{\frac{1}{N(\frac{Toler}{2})^2}+\frac{Cpk^2}{2\nu}}$

$UpperBound = Cpk + Z_{1-\alpha/2}\sqrt{\frac{1}{N(\frac{Toler}{2})^2}+\frac{Cpk^2}{2\nu}}$

$N$ : Total number of observations

$\alpha$ : Alpha for the confidence level

$\nu$ : Degrees of freedom, for details, please refer to $(1-\alpha)100\%$ Confidence Interval Bounds for Cp above

$Toler$ : Multiplier of the sigma tolerance

$Z_{1-\alpha/2}$ : $1-\alpha/2$ percentile from the standard normal distribution

CCpk
$CCpk=\left\{\begin{array}{ll}\frac{USL-\hat{\mu}}{\frac{Toler}{2}*\sigma_{within}} &Only\;USL\;Valid\cr\frac{\hat{\mu}-LSL}{\frac{Toler}{2}*\sigma_{within}} &Only\;LSL\;Valid\cr\frac{\min{(USL-\hat{\mu}, \hat{\mu}-LSL)}}{\frac{Toler}{2}*\sigma_{within}} &Both\;USL\;and\;LSL\;Valid\end{array}\right.$

$\hat{\mu}$ : Estimated mean, $\hat{\mu} = \left\{\begin{array}{ll}Target & Target\;is\;specified\cr\frac{USL+LSL}{2}&USL\;and\;LSL\;Valid,Target\;is\;not\;specified\cr \bar{X}&Otherwise\end{array}\right.$

$USL, LSL$ : Upper and lower specification limits respectively

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{within}$ : Within-subgroup standard deviation

$\bar{X}$ : Mean of observations

Overall Capability

Pp
$Pp = \frac{USL-LSL}{Toler*\sigma_{overall}}$

$USL, LSL$ : Upper and Lower specification limits respectively

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{overall}$ : Overall standard deviation

$(1-\alpha)100\%$ Confidence Interval Bounds for Pp
$LowerBound = Pp\sqrt{\frac{\chi_{\alpha/2,\nu}^2}{\nu}}$

$UpperBound = Pp\sqrt{\frac{\chi_{1-\alpha/2,\nu}^2}{\nu}}$

$\alpha$ : Alpha for the confidence level

$\nu$ : Degrees of freedom, $\nu=N-1$

$N$ : Number of observations

$\chi_{\alpha, \nu}^2$ : $\alpha$ percentile of the chi-square distribution with $\nu$ degrees of freedom

PPL
$PPL = \frac{\bar{X}-LSL}{(Toler/2)*\sigma_{overall}}$

$\bar{X}$ : Process mean, can be historical value, or calculated from observations

$LSL$ : Lower specification limit

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{overall}$ : Overall standard deviation

PPU
$PPU = \frac{USL-\bar{X}}{(Toler/2)*\sigma_{overall}}$

$\bar{X}$ : Process mean, can be historical value, or calculated from observations

$USL$ : Upper specification limit

$Toler$ : Multiplier of the sigma tolerance

$\sigma_{overall}$ : Overall standard deviation

Ppk
$Ppk = \min(PPU, PPL)$

$(1-\alpha)100\%$ Confidence Interval Bounds for Ppk
$LowerBound = Ppk - Z_{1-\alpha/2}\sqrt{\frac{1}{N(\frac{Toler}{2})^2} + \frac{Ppk^2}{2\nu}}$

$UpperBound = Ppk + Z_{1-\alpha/2}\sqrt{\frac{1}{N(\frac{Toler}{2})^2} + \frac{Ppk^2}{2\nu}}$

$N$ : Number of observations

$\alpha$ : Alpha for the confidence level

$\nu$ : Degrees of freedom, $\nu=N-1$

$Toler$ : Multiplier of the sigma tolerance

$Z_{1-\alpha/2}$ : The $1-\alpha/2$ percentile from the standard normal distribution

Cpm
When $Target$ is specified, it is able to calculate Cpm using $USL, LSL$ and $Target$ .

$Cpm = \left\{\begin{array}{ll}\frac{USL-LSL}{Toler*\sqrt{\frac{\sum_{i=1}^K\sum_{j=1}^{n_i}(X_{ij}-Target)^2}{\sum_{i=1}^Kn_i}}}&USL,LSL\;Valid\;and\;Target=m\cr\frac{\min(Target-LSL, USL-Target)}{\frac{Toler}{2}*\sqrt{\frac{\sum_{i=1}^K\sum_{j=1}^{n_i}(X_{ij}-Target)^2}{\sum_{i=1}^Kn_i}}}&USL,LSL\;Valid\;and\;Target\neq m\cr\frac{USL-Target}{\frac{Toler}{2}*\sqrt{\frac{\sum_{i=1}^K\sum_{j=1}^{n_i}(X_{ij}-Target)^2}{\sum_{i=1}^Kn_i}}}&Only\;USL\;and\;Target\;Valid\cr\frac{Target-LSL}{\frac{Toler}{2}*\sqrt{\frac{\sum_{i=1}^K\sum_{j=1}^{n_i}(X_{ij}-Target)^2}{\sum_{i=1}^Kn_i}}}&Only\;LSL\;and\;Target\;Valid\cr NANUM &Otherwise\end{array}\right.$

$USL, LSL$ : Upper and Lower specification limits respectively

$Target$ : Target value

$Toler$ : Multiplier of the sigma tolerance

$m$ : Midpoint between $USL$ and $LSL$

$n_i$ : Number of observations in $ith$ subgroup

$X_{ij}$ : The $jth$ observation in the $ith$ subgroup

$K$ : Number of subgroups

$NANUM$ : Missing value

Confidence Interval Bounds for Cpm
- Two-Sided
  $LowerBound = Cpm\sqrt{\frac{\chi_{\alpha/2, \nu}^2}{\nu}}$
  
  $UpperBound = Cpm\sqrt{\frac{\chi_{1-\alpha/2, \nu}^2}{\nu}}$
- One-Sided
  $LowerBound = Cpm\sqrt{\frac{\chi_{\alpha, \nu}^2}{\nu}}$
  
  $\nu$ : Degrees of freedom, $\nu = \frac{N((1+a^2)^2}{1+2a^2}$ , where $a = (Mean-Target)/\sigma_{overall}$ , and $N$ is the number of observations
  
  $\alpha$ : Alpha for the confidence level
  
  $\chi_{\alpha,\nu}^2$ : $\alpha$ quantile of the chi-square distribution with $\nu$ degrees of freedom

Benchmark Zs for Potential Capability

Z.LSL, Z.USL, and Z.Bench
$Z.LSL = \frac{\bar{X}-LSL}{\sigma_{within}}$

$Z.USL = \frac{USL-\bar{X}}{\sigma_{within}}$

$Z.Bench = \Phi^{-1}(1-P_1-P_2)$

$\bar{X}$ : Process mean, estiimated from data, or historical mean

$LSL, USL$ : Lower and upper specification limits

$P_1=Prob(X<LSL)=1-\Phi(Z.LSL)$

$P_2=Prob(X>USL)=1-\Phi(Z.USL)$

$\Phi(X)$ : Cumulative distribution function of standard normal distribution

$\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution

$\sigma_{within}$ : Within subgroups standard deviation

Confidence Intervals for Z.Bench With Two Specification Limits
- Two-Sided
  $LowerBound = -\Phi^{-1}(U)$
  
  $UpperBound = -\Phi^{-1}(L)$
  
  where
  
  $L=\frac{\exp(L(p)-Z_{1-\alpha/2}\sqrt{V})}{1+\exp(L(p)-Z_{1-\alpha/2}\sqrt{V})}$
  
  $U=\frac{\exp(L(p)+Z_{1-\alpha/2}\sqrt{V})}{1+\exp(L(p)+Z_{1-\alpha/2}\sqrt{V})}$
  
  $L(p)=\ln(p/(1-p))$
  
  $p=\left(1-\Phi(\frac{\bar{X}-LSL}{s})\right)+\left(1-\Phi(\frac{USL-\bar{X}}{s})\right)$
  
  $V=V(p)=\left(\frac{\partial{L(p)}}{\partial{\mu}}\right)^2\frac{\sigma^2}{N}+\left(\frac{\partial{L(p)}}{\partial{\sigma^2}}\right)^2\frac{2\sigma^4}{\nu}$
  
  $\frac{\partial{L(p)}}{\partial{\mu}}=\frac{1}{p(1-p)}\frac{\partial{p}}{\partial{\mu}}=\frac{1}{p(1-p)}\frac{1}{\sigma}\left(\varphi(\frac{USL-\mu}{\sigma})-\varphi(\frac{LSL-\mu}{\sigma})\right)$
  
  $\frac{\partial{L(p)}}{\partial{\sigma^2}}=\frac{1}{p(1-p)}\frac{\partial{p}}{\partial{\sigma^2}}=\frac{1}{2p(1-p)}\left(\frac{USL-\mu}{\sigma^3}\varphi(\frac{USL-\mu}{\sigma})-\frac{LSL-\mu}{\sigma^3}\varphi(\frac{LSL-\mu}{\sigma})\right)$
  
  $\sigma=s, \mu=\bar{X}$
  
  $N$ : Total number of obsevations
  
  $p$ : Tail probabilities outside of the specification limits
  
  $Z_{1-\alpha/2}$ : $(1-\alpha/2)^{th}$ percential of standard normal distribution
  
  $\alpha$ : Alpha for the confidence level
  
  $\bar{X}$ : Process mean, estimated from data, or historical mean
  
  $LSL, USL$ : Lower and upper specification liimits
  
  $s$ : Within subgroups standard deviation
  
  $\nu$ : Degrees of freedom for $s$
  
  $\Phi(X)$ : Cumulative distribution function of standard normal distribution
  
  $\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution
  
  $\varphi(X)$ : Probability density function of standard normal distribution
- One-Sided
  Refer to the Two-Sided above, and change $1-\alpha/2$ to $1-\alpha$ in the definition of $L$ for $UpperBound$ .

Confidence Intervals for Z.Bench With One Specification Limit
- Lower Specification Limit and Two-Sided
  $LowerBound=Z.LSL-Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.LSL^2}{2\nu}}$
  
  $UpperBound=Z.LSL+Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.LSL^2}{2\nu}}$
  
  $N$ : Total number of obsevations
  
  $Z_{1-\alpha/2}$ : $(1-\alpha/2)^{th}$ percential of standard normal distribution
  
  $\alpha$ : Alpha for the confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
- Lower Specification Limit and One-Sided
  $LowerBound = -\Phi^{-1}(p_1)$
  
  $p_1$ : Root of the equation: $Pr\left(T_{\nu}(-\sqrt{N}\Phi^{-1}(p_1))\le\sqrt{N}Z.LSL\right)=1-\alpha$
  
  $N$ : Total number of obsevations
  
  $\alpha$ : Alpha for the confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
  
  $\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution
  
  $T_{\nu}(\delta)$ : Random variable that is distributed as non-central t distribution with $\nu$ degrees of freedom and non-centrality parameter $\delta$
  
  $Pr(\cdot)$ : Cumulative distribution function of non-central t distribution
- Upper Specification Limit and Two-Sided
  $LowerBound=Z.USL-Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.USL^2}{2\nu}}$
  
  $UpperBound=Z.USL+Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.USL^2}{2\nu}}$
  
  $N$ : Total number of obsevations
  
  $Z_{1-\alpha/2}$ : $(1-\alpha/2)^{th}$ percential of standard normal distribution
  
  $\alpha$ : Alpha for the confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
- Upper Specification Limit and One-Sided
  $LowerBound = -\Phi^{-1}(p_2)$
  
  $p_2$ : Root of the equation: $Pr\left(T_{\nu}(-\sqrt{N}\Phi^{-1}(p_2))\le\sqrt{N}Z.USL\right)=1-\alpha$
  
  $N$ : Total number of obsevations
  
  $\alpha$ : Alpha for the confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
  
  $\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution
  
  $T_{\nu}(\delta)$ : Random variable that is distributed as non-central t distribution with $\nu$ degrees of freedom and non-centrality parameter $\delta$
  
  $Pr(\cdot)$ : Cumulative distribution function of non-central t distribution

Benchmark Zs for Overall Capability

The calculation of benchmark Zs for overall capability is similar to potential capability, by replacing the $\sigma_{within}$ by $\sigma_{overall}$ . Please refer to Benchmark Zs for Potential Capability for more details.

Expected Within Performance

PPM < LSL and % < LSL
The parts per million (PPM) less than the lower specification limit (PPM < LSL) and percentage less than the lower specification limit (% < LSL) are computed from the probability which is as follows:

$P(X < LSL)=1-\Phi(\frac{\bar{X}-LSL}{s})$

$LSL$ : Lower specification limit

$\bar{X}$ : Process mean, estimated from data, or historical mean

$s$ : Within subgroups standard deviation

$\Phi(X)$ : Cumulative distribution function of standard normal distribution

Then $PPM\;<\;LSL$ and $\%\;<\;LSL$ are multiples of the above probability:

$[PPM\;<\;LSL] = 1000000\cdot P(X<LSL)$

$[\%\;<\;LSL] = 100\cdot P(X<LSL)$
Confidence Intervals for PPM < LSL and % < LSL
- Two-Sided
  Confidence intervals for $P(X < LSL)$ are given by the following formulas
  
  $LowerBound=1-\Phi(U)$
  
  $UpperBound = 1-\Phi(L)$
  
  $U=Z.LSL+Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.LSL^2}{2\nu}}$
  
  $L=Z.LSL-Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.LSL^2}{2\nu}}$
  
  $\Phi(X)$ : Cumulative distribution function of standard normal distribution
  
  $N$ : Number of observations
  
  $\alpha$ : Alaph for confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
  
  $Z_{1-\alpha/2}$ : $(1 - \alpha/2)_{th}$ percentile of standard normal distribution
  
  Then get
  
  $LowerBound(PPM\;<\;LSL)=1000000\cdot LowerBound$
  
  $UpperBound(PPM\;<\;LSL)=1000000\cdot UpperBound$
  
  $LowerBound(\%\;<\;LSL)=100\cdot LowerBound$
  
  $UpperBound(\%\;<\;LSL)=100\cdot UpperBound$
- One-Sided
  $UpperBound(PPM\;<\;LSL)=1000000\cdot p_1$
  
  $UpperBound(\%\;<\;LSL)=100\cdot p_1$
  
  where $p_1$ is the root of $Pr\left(T_{\nu}(-\sqrt{N}\Phi^{-1}(p_1))\le\sqrt{N}Z.LSL\right)=1-\alpha$
  
  $N$ : Total number of obsevations
  
  $\alpha$ : Alpha for the confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
  
  $\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution
  
  $T_{\nu}(\delta)$ : Random variable that is distributed as non-central t distribution with $\nu$ degrees of freedom and non-centrality parameter $\delta$
  
  $Pr(\cdot)$ : Cumulative distribution function of non-central t distribution
PPM > USL and % > USL
The parts per million (PPM) greater than the upper specification limit (PPM > USL) and percentage greater than the upper specification limit (% > USL) are computed from the probability which is as follows:

$P(X > USL)=1-\Phi(\frac{USL-\bar{X}}{s})$

$USL$ : Upper specification limit

$\bar{X}$ : Process mean, estimated from data, or historical mean

$s$ : Within subgroups standard deviation

$\Phi(X)$ : Cumulative distribution function of standard normal distribution

Then $PPM\;>\;USL$ and $\%\;>\;USL$ are multiples of the above probability:

$[PPM\;>\;USL] = 1000000\cdot P(X>USL)$

$[\%\;>\;USL] = 100\cdot P(X>USL)$
Confidence Intervals for PPM > USL and % > USL
- Two-Sided
  Confidence intervals for $P(X > USL)$ are given by the following formulas
  
  $LowerBound=1-\Phi(U)$
  
  $UpperBound = 1-\Phi(L)$
  
  $U=Z.USL+Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.USL^2}{2\nu}}$
  
  $L=Z.USL-Z_{1-\alpha/2}\sqrt{\frac{1}{N}+\frac{Z.USL^2}{2\nu}}$
  
  $\Phi(X)$ : Cumulative distribution function of standard normal distribution
  
  $N$ : Number of observations
  
  $\alpha$ : Alaph for confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
  
  $Z_{1-\alpha/2}$ : $(1 - \alpha/2)^{th}$ percentile of standard normal distribution
  
  Then get
  
  $LowerBound(PPM\;>\;USL)=1000000\cdot LowerBound$
  
  $UpperBound(PPM\;>\;USL)=1000000\cdot UpperBound$
  
  $LowerBound(\%\;>\;USL)=100\cdot LowerBound$
  
  $UpperBound(\%\;>\;USL)=100\cdot UpperBound$
- One-Sided
  $UpperBound(PPM\;>\;USL)=1000000\cdot p_2$
  
  $UpperBound(\%\;>\;USL)=100\cdot p_2$
  
  where $p_2$ is the root of $Pr\left(T_{\nu}(-\sqrt{N}\Phi^{-1}(p_2))\le\sqrt{N}Z.USL\right)=1-\alpha$
  
  $N$ : Total number of obsevations
  
  $\alpha$ : Alpha for the confidence level
  
  $\nu$ : Degrees of freedom for standard deviation
  
  $\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution
  
  $T_{\nu}(\delta)$ : Random variable that is distributed as non-central t distribution with $\nu$ degrees of freedom and non-centrality parameter $\delta$
  
  $Pr(\cdot)$ : Cumulative distribution function of non-central t distribution
PPM Total and % Total
The parts per million that are outside the specification limits is calculated by:

$[PPM\;<\;LSL]+[PPM\;>\;USL]$ or $[\%\;<\;LSL]+[\%\;>\;USL]$
Confidence Intervals for PPM Total and % Total with Both Lower and Upper Specification Limits
- Two-Sided
  $UpperBound(PPM\;Total)=1000000\cdot U$ or $UpperBound(\%\;Total)=100\cdot U$
  
  $LowerBound(PPM\;Total)=1000000\cdot L$ or $LowerBound(\%\;Total)=100\cdot L$
  
  The calculation of $U$ and $L$ can be referred to Benchmark Zs for Potential Capability for more details.
- One-Sided
  $UpperBound(PPM\;Total)=1000000\cdot U$ or $UpperBound(\%\;Total)=100\cdot U$
  
  Here $U$ is calculated using the same method as two-sided, but replacing $\alpha/2$ by $\alpha$
Confidence Intervals for PPM Total and % Total with Only One Specification Limit (Lower Only or Upper Only)
- Lower Specification Limit Only
  Use the same calculation as the confidence interval for the PPM < LSL or % < LSL
- Upper Specification Limit Only
  Use the same calculation as the confidence interval for the PPM > USL or % > USL

Expected Overall Performance

The calculation for expected overall performance is similar as the procedure for expected within performance, but by using overall standard deviation instead. For more details, please refer to Expected Within Performance.

Observed Performance

PPM < LSL for Observed Performance
$[PPM\;<\;LSL(Observed)]=\frac{1000000\cdot(NumberOfObservations\;<\;LSL)}{N}$ , where $LSL$ is lower specification limit, and $N$ is the total number of observations
PPM > USL for Observed Performance
$[PPM\;>\;USL(Observed)]=\frac{1000000\cdot(NumberOfObservations\;>\;USL)}{N}$ , where $USL$ is upper specification limit, and $N$ is the total number of observations
PPM Total for Observed Performance
$[PPM\;Total]=[PPM\;<\;LSL(Observed)]+[PPM\;>\;USL(Observed)] = \frac{1000000\cdot(NumberOfObservations\;<\;LSL)}{N}+\frac{1000000\cdot(NumberOfObservations\;>\;USL)}{N}$ , where $LSL$ is lower specification limit, $USL$ is upper specification limit, and $N$ is the total number of observations

Between/Within Capability Analysis

Standard Deviation Estimation

Within Subgroup Standard Deviation ( $\sigma_{within}$ )
Please refer to Standard Deivation Estimation for more details about pooled standard deviation, average of subgroup ranges, and average of subgroup standard deviation.
Between Subgroup Standard Deviation ( $\sigma_{between}$ )
$\sigma_{between} = \max\left(0, \sqrt{\sigma_{xbar}^2-\frac{\sigma_{within}^2}{SubgroupSize}}\right)$

$\sigma_{xbar}$ is calculated by average of moving range, median of moving range or square root of mean squared successive differences. For more details, please refer to Standard Deivation Estimation.
Between/Within Standard Deviation ( $\sigma_{b/w}$ )
$\sigma_{b/w}=\sqrt{\sigma_{within}^2+\sigma_{between}^2}$
Ovarall Standard Deviation ( $\sigma_{overall}$ )
Please refer to Overall Standard Deviation subsection in Standard Deivation Estimation.

Between/Within Capability

Please refer to Potential Capability section for the calculations of Cp, CPL, CPU, Cpk, and CCpk. The difference is that the $\sigma_{within}$ is replaced by $\sigma_{b/w}$ . And, the calculation of $\nu$ in the formula of Cp confidence interval is also different. Here, $\nu$ is computed by:

$\nu=\frac{\left(\frac{MS_B+(m-1)MS_E}{m}\right)^2}{\frac{(m-1)^2}{m^2}\frac{MS_E^2}{df_E}+\frac{1}{m^2}\frac{MS_B^2}{df_B}}$

$MS_B=\frac{\sum_{i=1}^Ln_i(\bar{X_i}-\bar{X})^2}{df_B}$

$MS_E=\frac{\sum_{i=1}^L\sum_{j=1}^{n_i}(X_{ij}-\bar{X_i})^2}{df_E}$

$m = \frac{N^2-\sum_{i=1}^Ln_i^2}{N(L-1)}$

$df_E=\sum_{i=1}^L(n_i-1)$

$df_B=L-1$

$N$ : Total number of observations

$L$ : Number of subgroups

$n_i$ : The $i^{th}$ subgroup size

$\bar{X}$ : Mean across all subgroups

$\bar{X_i}$ : Mean of the $i^{th}$ subgroup

$X_{ij}$ : The $j^{th}$ observation in the $i^{th}$ subgroup

Overall Capability

Please refer to Overall Capability section for more details.

Benchmark Zs for Between/Within Capability

The calculation of benchmark Zs for between/within capability is similar to potential capability, by replacing the $\sigma_{within}$ by $\sigma_{b/w}$ . Please refer to Benchmark Zs for Potential Capability for more details.

Benchmark Zs for Overall Capability

Expected Between/Within Performance

The calculation of expected between/within performance is similar to expected within performance, by replacing the $\sigma_{within}$ by $\sigma_{b/w}$ . Please refer to Expected Within Performance for more details. And the following confidence intervals have different calculations:

Confidence Intervals for PPM < LSL and % < LSL
- One-Sided
  $LowerBound(PPM\;<\;LSL)=1000000\cdot p_1$
  
  $LowerBound(\%\;<\;LSL)=100\cdot p_1$
Confidence Intervals for PPM > USL and % > USL
- One-Sided
  $LowerBound(PPM\;>\;USL)=1000000\cdot p_2$
  
  $LowerBound(\%\;>\;USL)=100\cdot p_2$

Expected Overall Performance

Observed Performance

For more details, please refer to Observed Performance.

Non-normal Capability Analysis

Overall Capability

Pp: Pp is computed by the parameters of the distribution used. Two methods are used for the Pp calculation, Z-Score method and ISO method.
- Z-Score Method
  $Pp = \frac{Z_{usl}-Z_{lsl}}{6}$
  
  $Z_{usl}=\Phi^{-1}(p_2), Z_{lsl}=\Phi^{-1}(p_1)$
  
  $\Phi^{-1}(p)$ : Inverse cumulative distribution function of standard normal distribution, $p*100^{th}$ percentile of standard normal distribution
  
  $p_1=Prob(X\le LSL), p_2=Prob(X\le USL)$ : Cumulative distribution function of the used distribution
  
  $USL, LSL$ : Upper and Lower specification limits respectively
- ISO Method
  $Pp = \frac{USL-LSL}{X_{0.99865}-X_{0.00135}}$
  
  $USL, LSL$ : Upper and Lower specification limits respectively
  
  $X_{0.99865}, X_{0.00135}$ : The $99.865^{th},0.135^{th}$ percentile of the used distribution
PPL
- Z-Score Method
  $PPL= \frac{-\Phi^{-1}(p_1)}{3}$
  
  $\Phi^{-1}(p)$ : Inverse cumulative distribution function of standard normal distribution, $p*100^{th}$ percentile of standard normal distribution
  
  $p_1=Prob(X\le LSL)$ : Cumulative distribution function of the used distribution
  
  $LSL$ : Lower specification limit
- ISO Method
  $PPL = \frac{X_{0.5}-LSL}{X_{0.5}-X_{0.00135}}$
  
  $LSL$ : Lower specification limit
  
  $X_{0.5}, X_{0.00135}$ : The $50^{th},0.135^{th}$ percentile of the used distribution
PPU
- Z-Score Method
  $PPU= \frac{\Phi^{-1}(p_2)}{3}$
  
  $\Phi^{-1}(p)$ : Inverse cumulative distribution function of standard normal distribution, $p*100^{th}$ percentile of standard normal distribution
  
  $p_2=Prob(X\le USL)$ : Cumulative distribution function of the used distribution
  
  $USL$ : Upper specification limit
- ISO Method
  $PPU = \frac{USL-X_{0.5}}{X_{0.99865}-X_{0.5}}$
  
  $LSL$ : Lower specification limit
  
  $X_{0.99865}, X_{0.5}$ : The $99.865^{th},50^{th}$ percentile of the used distribution
Ppk
$Ppk = \min(PPU, PPL)$

Overall Benchmark Zs for Non-normal Capability

Z.LSL, Z.USL, and Z.Bench
$Z.LSL = 3*PPL$

$Z.USL = 3*PPU$

$Z.Bench = \Phi^{-1}(1-P_1-P_2)$

$P_1=Prob(X<LSL)$ : Cumulative distribution function of the used distribution, probability (X < LSL) based on the used nonnormal distribution

$P_2=Prob(X>USL)$ : Cumulative distribution function of the used distribution, probability (X > USL) based on the used nonnormal distribution

$\Phi(X)$ : Cumulative distribution function of standard normal distribution

$\Phi^{-1}(X)$ : Inverse cumulative distribution function of standard normal distribution

Expected Overall Performance

PPM < LSL
The parts per million (PPM) less than the lower specification limit (PPM < LSL) is computed from the probability which is as follows:

$[PPM\;<\;LSL]=1000000*F(LSL)$

$PPM$ : Parts per million

$LSL$ : Lower specification limit

$F(X)$ : Cumulative distribution function of the used nonnormal distribution
PPM > USL
The parts per million (PPM) greater than the upper specification limit (PPM > USL) is computed from the probability which is as follows:

$[PPM\;>\;USL]=1000000*(1-F(USL))$

$PPM$ : Parts per million

$USL$ : Upper specification limit

$F(X)$ : Cumulative distribution function of the used nonnormal distribution
PPM Total
$[PPM\;Total] = [PPM\;<\;LSL] + [PPM\;>\;USL]$