[Maths Class Notes] on Cpk Formulas Pdf for Exam

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.