How to Calculate Cp and Cpk

Cp and Cpk are statistical tools that are used in quality management to ensure that a production process has met the specification limits defined for a particular process or products. Cp measures the process capability with respect to its specification using Upper Specification Limit (USL) and Lower Specification Limit (LSL), while Cpk measures the process variation with respect to its sample mean which is also considered to be the process mean. Process capability is determined by taking periodic samples from process under controlled conditions and calculating its standard deviation and sample mean. Standard deviation determines how far a sample is from the sample mean, while sample mean is the average of the samples taken under consideration.

### Calculate Sample Mean

Calculate the sample mean which is the sum total of individual variables divided by the total number of variables. It is also called average in most geometrical calculations. In quality control studies, sample mean(xbar) = (sum total of values of items considered/total number of items considered). It is generally denoted by xbar.

### Determine Standard Deviation

Determine the standard deviation of the samples. Calculate the square root of the co-variance. Covariance is the sum total of the square of the difference of individual item with its sample mean divided by the total number of variables.

Covariance = sum(x- xbar)^2/total number of items, where x is the individual item. It is generally denoted by Greek letter alpha.

Standard Deviation = square root of co-variance. It is denoted by Greek letter sigma.

### Define Fixed Upper Specification Limit

Define and keep fixed upper specification limit (USL) and lower specification limit (LSL) for a particular process or products. Using USL, LSL, sample mean and standard deviation, both Cp and Cpk indexes can be calculated.

### Determine Cp Index

To determine Cp index, subtract the LSL from USL (USL-LSL) and divide the value by six times of the standard deviation (6* standard deviation).

Cp = (USL-LSL)/6 x standard deviation. Cp index is the numerical value.

### Calculate Cpk Index

Determine Cpu and Cpl to calculate Cpk index. Cpu is the index of the total process variation with respect to upper specification limit (USL) and Cpl is the index of the total process variation associated with lower specification limit (LSL). After determining Cpu and Cpl, whichever results in the smallest value is the Cpk index.

Cpu = (USL-sample mean)/3x standard deviation.

Cpl = (LSL-sample mean)/3x standard deviation.

### Quality Management

In quality management, Cp is generally designed with higher value compared to Cpk with the consideration that the process is not centered within the specification limits. The Cpk index can change in the process by adjusting the centering of the process; however, Cp always remains the same unless any change is made on process itself. Centering a process involves well-defined specification limits and process variation having normal distribution.