How to Calculate Cp and Cpk

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Cp and Cpk are two different ways of measuring the capability, or accuracy, of a process. They're typically used in manufacturing to help ensure that products will meet the specifications required by customers. They originate from the Six Sigma system for quality.

Even the most accurate equipment, the most skilled workers and the highest quality assurance methodologies won't always meet the exact target specification. If you buy a 100g chocolate bar and put it on an extremely sensitive scale, it's unlikely to weigh in at 100.000 grams. One could weigh 100.01 grams and another might weigh 99.92 grams.

The question Cp and Cpk are designed to answer is how far from being 100 grams are the chocolate bars going to be.

Terms You Need to Know

Both Cp and Cpk are used to define the ability of a system to make a product that meets requirements. Before you can fully understand Cp and Cpk, you should first have an understanding of product specifications.

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

  • Cp, or Cp, stands for Process Capability. Just to confuse things, Cp is also sometimes called the Cp Index.

  • 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 how far above the target your process can be. In the example of 27 mm +/- 0.02 mm, the USL = 27.02 mm.

  • The lower specification limit (LSL) represents how far below the target the process can be. In the example of 27 mm +/- 0.02 mm, the LSL = 26.98 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. 

Cp vs. Cpk Indices

There's a fundamental difference between Cp and Cpk that takes into account the mean average. To calculate Cp, you need to know the specification limits and the standard deviation. It simply tells you whether or not the production process is capable of making products within the specifications.

To calculate Cpk, you need the specification limits, the standard deviation and the mean. Cpk not only tells you whether or not the process can make products within specifications, but it also tells you if it's capable of meeting the target specification.

Things You Need

Before you can reliably determine Cp and Cpk, you first need to:

  • Ensure the sample size is large enough.
  • Test the collected data for normality.
  • Ensure the process is under statistical control.

For both the mean and the standard deviation to be stable over time, your production process must be under statistical control.

Cpk is used more often than Cp because it accounts for both the standard deviation and the mean in its calculation. A difference between Cp and Cpk values indicates how far the average production will be from the target specification. The closer production is to the target specification, the closer Cpk will be to Cp. However, the value for Cpk can never exceed the value for Cp.

How to Calculate Cpk

Cpk is a measurement of how many standard deviations the specification limits are from the target specification or center. The Cp for the upper limit is the Cpu, while the Cp for the lower limit is the Cpl. The equation 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 use the smaller of these two values as the Cpk or put this into the Cpk formula:

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.

Calculating Cpk Using a Z Value

Another way of figuring out the Cpk index is to use Z scores, provided the upper and lower specification limits are the same distance from the center.

First, you need to calculate a Z score for the upper and lower specification limits. These are called the Z USL and the Z LSL. A Z score is the same as a standard score, representing the number of standard deviations above the mean average.

Once you have a Z score, simply divide it by three to determine the Cpk:

Cpk = Z / 3

Understanding Cpk Values

If the Cpk is a negative number, your process will routinely produce products outside of the specification limits. The higher the Cpk value is, when it's positive, the better your process will be at hitting the target specification without going higher or lower than that value. In other words, your process is well centered.

A typical requirement is that Cpk should be at least 1.33. Anything above this number would be considered excellent, such as a Cpk 1.67, meaning that your production process is extremely reliable.

If the Cpk is less than one, it will often miss the target but won't necessarily exceed the specification limits.

How to Calculate Cp Values

To calculate the Cp index, you need to know the upper and lower specification limits (USL and LSL), as well as the standard deviation. You can then use the following formula:

Cp = (USL-LSL) / 6 x SD

A Cp of one means that your process matches the width of the specification limits. If the Cp is less than one, your process spread exceeds that of the specifications, which means some of the products will be rejected. If the Cp is greater than one, the process spread is inside the specification limits.

Understanding Cp Values

Always keep in mind that the Cp index doesn't include the mean average of your process, so a high Cp value in itself doesn't mean that your process is going to produce acceptable goods.

To illustrate this, suppose your process was to drill 5 mm holes in 10 mm metal washers using a drill press. Even if the drill makes the right sized hole every time, this won't mean a thing if the drill press is misaligned and is putting the holes 7 mm left of center. Instead of washers, you would have c-shaped scraps of metal.