Process Capability Index 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 the 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.
What is Process Capability Index
Process Capability is a calculation tool used by many to estimate the product output. After the industrial revolution that introduced machine lines for production, the method of manufacturing turned into a controlled process. Later the investors tried to make calculations based on the observed speed of factors of production. They wanted to estimate the possible output they are going to have to calculate and manage the resources for maximum performance.
The inputs for the process capability index are usually the normal statistical analysis and distribution of data obtained. It is somewhat similar to 'mean' or 'average value' and the 'standard deviation' that is taught in arithmetics. But it differs in the terms of using a control chart analysis to check the statistical control of the data. Usually, three measured output data is used. The data of production conditions, materials and people involved in the process is taken into consideration. This method is utilized by both established production units and startups. The outputs are obtained in the plus and minus deviations. The Whole process is responsible for understanding and meeting customer requirements, engineering tolerance and specifications of manufacture of produced goods.
When this process is expressed in a single number by the use of the Process Capability Index (Cpk) it gets easier for users to meet the specifications. All in all, it provides important information about the stability of the process that is expected to be uncertain with ever-changing variables.
The Process Capability Index comes in handy while solving the estimation of products in a real-life situation.
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, the 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 = \[\frac {(Mean - LSL)}{3 \times SD}\]
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 = \[\frac {(USL - Mean)}{3 \times SD}\]
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 \[\frac {(30-27)}{7}\] or \[\frac {(27 - 20)}{7}\]
= min of \[\frac {3}{7}\] or \[\frac {7}{7}\]
= 0.43 or 1
Here, the Cpk calculation is 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.
FAQs on Cpk Formulas
1. State Some Important Points About Cpk.
Cpk is a short-term process index that describes the potential capability of a process.
It’s an option along with a z-score in statistics.
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.
The Cp can perform the best process if that process is centered on its midpoint.
The minimum value of “k” is 0.0 and the maximum is 1.0.
A perfect centered process will have Cp = Cpk.
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.
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.
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
If Cpk is less than zero, the process mean has crossed one of the specification limits.
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.
If Cpk is greater than one, the process mean is perfectly centered and is within the specification limits.
3. What are the benefits of using the Process Capability Index?
With the growing time the production system is getting more vast and complex day by day. So the task of controlling and managing the production units is getting daunting. In such situations, we can use Mathematics as our way out. Mathematicians and economists have evolved many logical and empirical formulas for making the work of management boards or quality analysers easier. It is a very good way to have a precise estimation of the future gains any investor is going to have. The process can also help in better management of the resource allocation and the whole business in the long term.
4. Do I need to study all about the 'Process Capability' before learning about the 'Process Capability Index'?
'Process Capability is the basic understanding of what the concept stands for and gives you the complete understanding of its requirement and application for what it is meant for. Statistical data analysis and study is the prior thing needed for the application of the many theories in the real world. This duplication of calculations for the same value of the variable is what gave rise to the 'Process Capability Index'. It is pre-prepared Process Capability values for various sets of conditions that anyone uses and compares to get rid of the calculations.
5. What is the best method to learn Process Capability index?
The chapter on Process Capability is an important part of the statistical studies syllabus. The books are available for the Commerce stream after the 10th board exam. The best way to go through this chapter is to first learn all the theories to get clear about the concepts and by heart the formulas if necessary. Once you are capable of solving all the problems in the book then you are ready to refer to the index available for various values of factors. You can also practice solving more questions from various other sources.
6. Where can I get all the formulas for the 'Process Capability Index'?
'Formulas' are very essential for the chapter on Process Capability during solving the problems and while using the Process Capability Index. These formulas are mentioned once in the chapter with the explanation and content. The formulas are also available on the Vedantu website. If you register yourself on the portal and get enrolled you will be able to access all the free study materials available on the website. You can also get the hard copies by downloading and printing the pdfs.