Cpk Formula | Cp Formula | Process Capability Analysis Formulas

Cp, Cpk and Other Process Capability Metrics
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The QI Macros Histogram calculates several process capability analysis metrics. Descriptions and formulas for these metrics are described below.

Cp and Cpk Process Capability Metricsx
Pp and Ppk Process Performance Metrics
- Pp and Ppk formulas
Cp, Cpk vs Pp, Ppk
Other Process Metrics
- Skewness (Excel's skew function)
- Standard Deviation (Excel's stdev function)
- Min and Max
- Z score
- Z bench
- Z Target
- % Defects
- Parts Per Million and Expected Parts Per Million
- Sigma

Cp and Cpk Process Capability Metrics

C_p measures how well the data fits within the spec limits (USL, LSL)
C_pk measures how centered the data is between the spec limits.
A C_pk of 1.33 is considered to be at 4-Sigma.

Use C_p when you have a sample, not the population, and are testing the potential capability of a process to meet customer needs. C_p and C_pk use Sigma estimator.

Sigma estimator Formula

sigma estimator formulas
d₂ is a constant based on subgroup size
c₄ is a constant based on subgroup size

Xbar = Σ(X_i)/n
Rbar = Average(R_i) (Average of the Ranges in samples)

Cp, Cpk Formula

Cp, Cpk formulas

C_p = (USL-LSL)/(6*sigest) (Cp > 1.33 is desired)
Note: Cp cannot be calculated if there is no USL or no LSL.
C_pU (upper) = (USL-Xbar)/(3*sigest)
C_pL (lower) = (Xbar-LSL)/(3*sigest)

C_pk = Min(C_pU,C_pL) (Cpk > 1.33 is desired)

Cr Formula

The Capability Ratio (Cr) indicates what proportion of tolerance is occupied by the data.

Cr = 1/Cp Cp = 1.33 ~ Cr = 0.75 (data fits 75% of tolerance)

Cpm Formula

C_pm = (USL-LSL)/(6√(sigma²+(Xbar-Target)²))
(C_pm can be used when you have a target value.)

One-Sided Specifications or Unilateral Spec Limits

These equations assume that the process has both upper and lower specification limits.
Until 10/2008, the macros used either C_pU (USL) or C_pL (LSL) for both C_p and C_pk.
After 10/2008 Cp, Cpk, Cpm and other measures relying on USL/LSL will show "*".

Pp, Ppk Process Performance Metrics

Use P_p when you have the total population and are testing the performance of a system to meet customer needs. Pp, Ppk use standard deviation.

Pp, Ppk formulas

stdev = stdev(X_i)
P_p(sigma) = (USL-LSL)/(6*stdev) (Pp > 1.33 is desired)
Note: Pp cannot be calculated if there is no USL or no LSL.
P_pu (upper) = (USL-Xbar)/(3*stdev)
P_pl (lower) = (Xbar-LSL)/(3*stdev)

P_pk(sigma) = Min(P_pu,P_pl) (Ppk > 1.33 is desired)

Cp, Cpk vs Pp, Ppk

Use Cp, Cpk when you have a sample and are testing the potential capability of a process to meet customer needs. Cp Cpk use sigma estimator.

Use Pp, Ppk when you have the total population and are testing the performance of a process to meet customer needs. Pp, Ppk use standard deviation.

For example, if you test a sample of 10 widgets from a batch of 100 widgets:

Use Cp, Cpk to measure and make a statement about the whole batch of 100 widgets.
Use Pp, Ppk to measure and make a statement about just the 10 widgets you measured.

Other Process Metrics

Min-Max

Min = Min(X_i)
Max = Max(X_i)

Z Score

Z scores help estimate the non-conforming PPM.
Z scores standardize +/-3*stdev values into +/-3.

Zlower = (Xbar-LSL)/stdev
Zupper = (USL-Xbar)/stdev

Zbench

Zbench = normsinv(1-(Expected PPM/1,000,000)) Zbench is the Z score for the Expected PPM

ZTarget/Delta Z

ZT (target) = ΔZ= |Xbar-Target|/(stdev)

Ztarget measures how far the average varies from the Target.

ZT < 0.5 is desired (less than 1/2 standard deviation from target)

The target is assumed to be the midpoint between the USL and LSL.

Sample Size Calculator

Variable
Sample Size (SS) = (ZStdev)/Confidence²
Where:
Z = 1.96 for 95% confidence level
Stdev = standard deviation (or 0.5 Default)
Confidence interval expressed as decimal (e.g., .05 = ±5%)

Attribute
Sample Size (SS) = (Z²p(1-p))/Confidence²
Where:
Z = 1.96 for 95% confidence level
p = percent defects (0-0.5 = 0-50%)
Confidence interval expressed as decimal (e.g., .05 = ±5%)

Correction for Finite Population
new sample size = SS/(1+ (SS-1)/population)

Defects

defects = number of points outside USL-LSL
% Total Defects = (defects100)/(Total points)

Parts Per Million (PPM) and Expected PPM

PPM = % Total Defects 10000
Expected PPM = (Normsdist(Zlower) + (1-Normsdist(Zupper)))1,000,000

Sigma

If observed PPM > 0, it uses the observed value to calculate Sigma.
If observed PPM = 0, it uses Expected PPM to calculate Sigma.
If both are zero, it defaults to 6 sigma.
If there are no USL/LSL, it leaves it blank.