Cpk Formulas

Cpk Statistics Formula

CP stands for Process Capability (Cp) , it is a technique to find out the measurable property of a process to a specification.And Cpk stands for Process Capability Index (Cpk). It measures the producer's capability to produce a product within customer’s tolerance range. Cpk is used to calculate how close you are to a given target and how consistent you are to your average performance. Cpk gives you the best-case scenario for the existing process. Let us study the basics of cpk and cpk formulas.

Capability Index Formula is Given as:

Cp = (USL - LSL/ 6 x σ)

Cpk = min(USL−mean / 3σ, mean−LSL/3σ)

Cpk is a standard index to estimate the capability of one process, the higher the Cpk value the better the process is. For example, Machine 1 has a Cpk of 1.5 and machine 2 has a Cpk of 1.2. From the Cpk value, one can say that machine 1 is better than machine 2.

Important Terms Used in the Cpk Formulas

Cp and Cpk both are used to define the ability of a system to make a product that meets requirements. Let us understand the product specifications.

  • Cpk, or Cpk, stands for Process Capability Index. 

  • Cp, or Cp, stands for Process Capability. 

  • Product specifications define the requirements a product must meet for it to be accepted by the customer. Specifications are defined in terms of nominal tolerances or ranges, using "+/-" to specify what the range is. A specification for the width of a component, for example, might be 27 mm +/- 0.02 mm.

  • The upper specification limit (USL) represents the target rise of your process. In the example of 28 mm +/- 0.02 mm, the USL = 28.02 mm.

  • The lower specification limit (LSL) represents the target fall of your process. In the example of 28 mm +/- 0.02 mm, the LSL = 27.98 mm. 

  • The target specification is the ideal specification. In this example, that would be 27 mm. Because the target specification is in the middle between the USL and LSL, it's often referred to as the center.

  • Standard deviation (SD) measures the variability in the production process. The root mean square (RMS) deviation from the mean average tells you how much the process will likely vary from that average. 

How to Calculate Cpk

Cpk is a measurement of process capability. Process capability index measures your process to see if it is capable of making an item within the given specifications. On the other hand, process capability index lets you know where your process is in relation to the center of those specifications.The Cp for the upper limit is represented as the Cpu, while the Cp for the lower limit is represented as the Cpl. The Cpk equations for each are nearly identical.

Calculate Cpl

To calculate the Cpl, subtract the lower specification limit from the mean, then divide this by 3 times the standard deviation (SD):

Cpl = (Mean – LSL) / (3 * SD)

Calculate Cpu

To calculate the Cpu, subtract the upper specification limit from the mean, then divide this by 3 times the standard deviation (SD):

Cpu = (USL – Mean) / (3 * SD)

Determine the Cpk

Once you have the Cpl and Cpu calculated, you can put this into the Cpk formula 

The Formula for Cpk = Min (Cpl, Cpu)

The standard deviation is multiplied by three because six standard deviations (or six sigmas), account for just about every eventuality in a process using a normal distribution curve. Six divided by two (for the upper and lower limits) is three.

Cpk Calculation Formula

The value of Cp index is not sufficient by itself to analyze a process capability. What would happen if the nominal production output value shifts towards either the upper or lower limits and some of the production falls outside the design specifications? This is when a Cpk calculation is needed.

The Cpk Equation is:

Cpk = Min ((Upper specification limit - Nominal value)/3 Sigma spread or (Nominal value - Lower specification limit)/3 Sigma spread))

For example, consider the manufacture of water bottles. The target size is 25 ounces. The specifications require that the manufacturing process produce bottles ranging in size from an upper limit of 30 ounces to a lower limit of 20 ounces.Suppose the mean shifts to the right to 27 ounces. 

The Calculations for Cpk are as Follows:

Cpk = Min ((30 - 27)/7 or (27 - 20)/7)

 = min of 3/7 or 7/7 

= 0.43 or 1

Here, the Cpk calculation is the lesser or 0.43. Since this value is less than one, this process is not acceptable because a large portion of the production falls outside the upper specification and is considered defective.

FAQ (Frequently Asked Questions)

1. State Some Important Points About Cpk.

Answer:

  1. Cpk is a short term process index that describes the potential capability of a process.

  2. It’s an option along with z-score in statistics.

  3. Time-series and SPC charts are used to determine process control. If the process is out of control, then assessing the current process is unlikely to reflect the long term performance.

  4. The Cp can perform the best process if that process is centered on its midpoint.

  5. The minimum value of “k” is 0.0 and the maximum is 1.0. 

  6. A perfect centered process will have Cp = Cpk.

  7. An estimate for Cpk = Cp(1-k). Since the max value for k is 1.0, so the value for Cpk will always be less or equal to Cp.

  8. Input is required from the customer regarding the lower specification limit (LSL) and the upper specification limit (USL).

2. What is the Interpretation of Cpk Value.

Answer: 

  1. If Cp value is equal to Cpk value, then the process is operating at borderline conditions. The production capability falls within the design specifications for Six Sigma standards and is acceptable

  2. If Cpk is less than zero, the process mean has crossed one of the specification limits.

  3. If Cpk is greater than zero but less than one, the process mean is within the specification limits, but some part of the production output has crossed the specification limits.

  4. If Cpk is greater than one, the process mean is perfectly centered and is within the specification limits.