Process Capability
For Continuous and Discrete Data
Variation
Dont worry the rope is over half an inch thick on average. average
It is not the average Im worried about!
Process Capability - Definition
Process Capability is a measurement of how the process is performing with respect to a desired outcome
Allowed Variation Actual V i i A l Variation
Lower Spec Limit LSL
determinedbythecustomer determinedbytheprocess d db h
Lower Spec Limit LSL
Voice of Customer Voice f Process V i of P
Upper Spec Limit USL
Upper Spec Limit USL
Rejects Acceptable Quality Products
Rejects
Rejects Acceptable Quality Products LCL -3s Target
Rejects
Lower Control Limit Target Upper Control Limit LCL UCL -3s +3s
VOC = USP - LSL
VOC = USP - LSL VOP = UCL - LCL = 6s
UCL +3s
VOP = UCL - LCL = 6s
GOOD CAPABILITY
POOR CAPABILITY
Short Term (Cp) vs Long Term (Pp) Capability
Over an extended period we expect to see more variation
Changes in process
Time This process is short term Capable but not long term Capable
Cp
Val of Produ Charact lue uct teristic
LSL LCL 3s -3s
Pp
UCL +3s
USL
Short term 1 cycle performance
Sustained long term performance
Time
Process Capability Ratios (Cp, Pp) & Index (Cpk, Ppk)
A centred process Cp = Pp = USL - LSL 6sshortterm USL - LSL 6s 6 longterm An off-centred process TL = TU = Cpk = Ppk =
Lower Spec Limit LSL X Upper Spec Limit USL Lower Spec Limit LSL
X - LSL 3s USL - X 3s 3 Min (TL, TU)short term 3sshortterm Min Mi (TL, TU)long term 3slongterm
Upper Spec Limit USL
Rejects Acceptable Quality Products LCL -3s Target UCL +3s
Rejects
Rejects Acceptable Quality Products LCL -3s Target UCL +3s
Rejects
VOC = USP - LSL VOP = UCL - LCL = 6s
TL
TU
Note: Standard Deviation long term is usually referred to as Sigma or
Summary of Capability Metrics
Short Term Performance Considers Centring
Long Term Performance
Cpk Cp
Ppk Pp
Does Not Consider Centring
Test Your Understanding
Sketch the histogram and specification limits for the data sets below
2.0 B 1.5 Cpk 1.0 D 0.5 0.0 0.0 B A 0.5 1.0 Cp 1.5 2.0 C E E F C
Calculating Sigma Score (Z)
Point of interest
X Z = +3
Yield Defects
+1s +2s +3s Target Mean X
For normally distributed data the standard deviation (s) and mean ( ( ) (Xbar) ) can be used to estimate the probability of defects Point of interest X can be an USL or LSL Example with X = 25, s =0.5, X=USL=26.5 then Z = (26.5-25)/0.5 = 3. Z scores can be converted to probabilities in Minitab or Excel fx NORMSDIST(Z). Note Excel returns the
left tail or yield of the distribution. If you left tail enter Z=3 you get 0.99865. The defects are 1-0.99865 = 0.00135 or 0.135% (defects = 1yield).
(X X) Z= s
(Z = the number of standard deviations from the mean. Z = 3Cp if using USL and LSL)
Calculating the Total Defect Rate (Zbench)
X=LSL = 24.0 ZL = -2 X=USL = 26.5 ZU = +3
Yield Defects
s = 0.5
Defects
The total defect rate expected from a p process can be estimated by converting y g both upper and lower Z scores to probabilities and adding them :NORMSDIST(3)=0.99865 ( ) P=1-0.99865 = 0.00135. NORMSDIST(-2)=0.02275 (left tail). Total probability PT = 0.00135+0.02275 = 0.0241 or 2.41%.
-2s Target Mean X = 25 +3s
The total sigma score called Zbench can be found from the total yield of the process YT =1 - PT. This represents the overall process capability:If total probability PT = 0.0241 then YT = 0.9759. Using Excel fx NORMSINV(0.9759) = 1.9756. 1 9756
(X X) Z= s
(Z = th number of standard deviations the b f t d d d i ti from the mean. Z = 3Cp if using USL and LSL)
We say the SIGMA of this process is ~2!
Cp/Pp Relationships
N.B. Yields and PPMs are calculated using defects at both tails of the Distribution assuming a Centred Process
Capability for Discrete/Attribute Data
Discrete data has clear boundaries between adjoining values includes names categories counts and rank names, categories, orders e.g. dates, colours, defects, defectives The data will either be Defectives (e g pass/fail or on (e.g. time or late) = Binomial (0 or 1) or the data will be Defects (e.g. scratches or number of omissions) = ( g ) Poisson. Capability can be calculated directly from Binomial or Poisson distributions using Minitab or other statistical software packages, however you can convert the data to continuous and then use the previous method e g method. e.g. convert defects to probabilities (%ages / 100).
Capability for Non-Normal Data
Normal capability analysis with non-normal d t would give l data ld i misleading results Non-normal capability can be p y found from software packages like Minitab or by transforming the data to a distribution which is Normal. Try taking logs (either Log10 or Ln) by i i the data to L ) or b raising th d t t a power (called a Box-Cox transformation) e.g. data2.
Process Capability Summary
All processes have variation P Process C Capability i a measurement of h bilit is t f how th process the is performing with respect to a desired outcome Capability is defined as the voice of the customer over the voice of the process Long term capability is not the same as short term capability Covert discrete to continuous and non-normal data to non normal normal before analysing capability or use specialist software The overall capability of a process can be defined by its Sigma value.
