Table of Contents
 1 What is normal process capability?
 2 Data requirements & recommendations
 3 Potential capability indices formulae
 4 Performance capability indices formulae.
 5 Notation
 6 What is the difference between within and overall standard deviation?
 7 Formulas for estimation of within and overall standard deviations.
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In this article we will explore in brief about normal process capability indices used in Statistical Process Control, and the formulas used in popular statistical software.
What is normal process capability?
Normal process capability analysis is used to evaluate the potential and overall capability of manufacturing processes, assuming the data in the processes are normally distributed. The goals are primarily:
 To determine how well process outputs meet your customer or internal specifications.
 To identify improvement opportunity, i.e:
 Stability: The difference between process performance (Pp, Ppk) vs process potential (Cp, Cpk) is small for stable processes, and high for processes with too much of adjustments(scope of improvement).
 Centring: The difference between Pp vs Ppk, or between Cp vs Cpk is small for centred processes.
Data requirements & recommendations
 Required: You need at least one of the specification limits, i.e a lower specification limit (LSL) or a upper specification limit (USL), or both.
 The data collected should be continuous (numeric / decimal) in nature.
 Each subgroup should be collected as a rational subgroup if possible.
 A rational subgroup consists of samples produced over a short enough time window / sequence such that the chances of external factors (special causes) are low.
 Multiple subgroups can be spaced apart in time.
 Data should be approximately normal. If the data is nonnormal, then:
 Subgroup size should be increased, as that way distribution of means tends to become normal
 Or use a transformation like boxcox transformation.
 100 or more data points is recommended. If data is collected in subgroups, enough subgroups to sumup 100 data points; say 25 subgroups of subgroup size 4.
Potential capability indices formulae
Apart from above, CCpk is calculated based on the "known" values of the LSL, USL or the Target.
Known value  CCpk 
USL only  
LSL only  
LSL and USL 
Performance capability indices formulae.
Apart from above, Cpm is calculated when a target is specified.
Known value  Cpm 

LSL and USL only  * 
LSL, USL, and T = m  
LSL, USL, and T ≠ m  
USL and T only  
LSL and T only  
Either LSL or USL only  * 
Notation
Term  Description  
USL  Upper specification limit  
LSL  Lower specification limit  
Toler  Multiplier of the sigma tolerance (6 is the default value)  
m  Midpoint between USL and LSL  
T  Target value  

Withinsubgroup standard deviation  

Overall standard deviation  

Process means (estimated from the sample data or a historical value)  

Estimator for the population mean. T if T is specified. If T is not specified, but LSL and USL are specified, then m. Otherwise process mean.  
*  Missing value  
X_{ij}  jth observation in the ith subgroup  
n_{i}  Number of observations in the ith subgroup 
What is the difference between within and overall standard deviation?
For normal process capability estimation, two "estimated" standard deviations are used.
 Within standard deviation is estimated as a measure of variation /spread by trying to filter out external influences to the process. It can be assumed as the inherent "natural" variation of a process. The estimation methods uses the variation / spread of data within subgroups or between consecutive readings.
 Overall standard deviation is estimated as a measure of variation / spread asis, including any external influences and instabilities in the process.
Formulas for estimation of within and overall standard deviations.
Depending on the subgroup size, within standard deviation can be estimated with either of the options below:
 When subgroup size > 1, estimation can be done using either of the following methods:
 Using pooled standard deviation (Pooled sd)
 Using average subgroup range (Rbar)
 Using average of subgroup standard deviation (Sbar)
 When subgroup size = 1, estimation can be done using either of the following methods:
Using pooled sd method(n>1):
Where,
Using Rbar method(n>1):
Where,
Using Sbar method(n>1):
where:
Using average moving range method(n=1):
Where,
Using median moving range method(n=1):
where: