Cpk Formula

Process capability index i.e. Cpk is an important statistical tool. It is used to measure the ability of a process to produce desired output within the customer’s specification limits. In other words, it measures the producer’s capability to give a product within the tolerance range. Also, it is used to estimate how close we are to a given target as well as about the consistency around the average performance. We will learn the basic concept of cpk as well as Cpk formulas with examples. Let us learn it!

Cpk Formulas

Concept of Cpk Formulas

Cp and Cpk both are used for Process Capability measurement. Generally, we may use this when a process is under statistical control. This generally happens with a mature process that will be around for a while. Process capability will use the process of sigma value determined from either the Moving Range, Range or Sigma control charts.

Therefore, Process capability index i.e. cpk is the measurement of process capability. It shows how closely a process is able to produce the required output in comparison to its overall specifications. It also decides about the consistency around the average performance.

Cpk gives us the best-case scenario for the existing process. It may also estimate the future process performance, assuming performance is consistent over time. In Six Sigma we have to describe the quality of processes in terms of sigma. This is because it gives us an easy way to talk about how capable different processes are with a common mathematical model.

Important Points about Cpk

1. Cpk is a short term process index that describes the potential capability of a process assuming it was analyzed and stays in the control.
2. It’s an option along with z-score in statistics.
3. We may use time-series and SPC charts 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 is the best a process can perform if that process is centered on its midpoint.
5. The addition of “k” in Cpk quantifies the amount of which a distribution is centered. A perfectly cantered process where the mean is the same as the midpoint will have a “k” value of 0.
6. The minimum value of “k” is 0.0 and the maximum is 1.0. A proper 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).

The Formula for Cpk

Where,

Calculating Cp & Cpk

Cp and Cpk are process capability indices used in statistical process control to measure the ability of a process to meet customer specifications.

Cp measures the capability of the process to produce within the specification limits, while Cpk measures the capability of the process to produce within the specification limits and account for the variability of the process.

Here are the steps to calculate Cp and Cpk:

1. Determine the upper specification limit (USL) and lower specification limit (LSL) for the process. These are the customer requirements or target values.
2. Collect a sample of data from the process, and calculate the sample mean (x-bar) and sample standard deviation (s). You should collect at least 30 data points to ensure the sample is representative.
3. Calculate Cp using the formula:
4. Cp = (USL – LSL) / (6 x s)
5. Calculate Cpk using the formula:
6. Cpk = min[(USL – x-bar) / (3 x s), (x-bar – LSL) / (3 x s)]
7. where min[] means the minimum value of the two options in brackets.
8. Interpret the results:

• Cp > 1 indicates the process is capable of meeting customer specifications
• Cpk > 1 indicates the process is capable of meeting customer specifications and accounts for variability
• Cp and Cpk values closer to 1 indicate a greater risk of producing nonconforming products

It’s important to note that process capability indices are just one tool in process control and improvement. Other statistical tools, such as control charts and root cause analysis, are also necessary for effective process management.

What is the Difference between Cp, Cpk and Pp, PPk?

Cp and Cpk are called Process Capability. Pp and Ppk are called Process Performance. In both cases, we want to try to verify if the process can meet Customer CTQs (requirements).

Cp and Cpk are used for Process Capability. Generally, you use this when a process is under statistical control. This often happens with a mature process that has been around for a while. Process capability uses the process sigma value determined from either the Moving Range, Range, or Sigma control charts.

Pp and PPk are used for Process Performance. Generally, you use this when a process is too new to determine if it is under statistical control. Ex. you are piloting a new process or testing a short pre-production run. Because there is not a lot of historical data, we take large samples from the process to account for variation. Process Performance generally uses sample sigma in its calculation.

In theory, Cpk will always be greater than or equal to Ppk. There are anomalies seen when the sample size is small, and the data represents a short amount of time where estimating using R will overstate standard deviation and make Cpk smaller than Ppk. It is not real; there can never be less variation in the long term since the long term is using all of the data, not just two pieces of data from every subgroup.

Evaluating process capability with Cp & Cpk mirrors what is done (and why it is done) when following the Pp & Ppk approach. The main difference is that you use Cp & Cpk after a process has reached stability or statistical control.

Cpk vs. Ppk

P_pk tells us how a process has performed in the past, and you cannot use it to predict the future because the process is not in a state of control.

If a process is under statistical control;

The values for C_pk and P_pk will converge to almost the same value because the sigma and the sample standard deviation will be identical (use an F test to determine).

In other words, if Cpk == Ppk, the process is likely in statistical control.

If a process is NOT in statistical control;

Cpk and Ppk values will differ distinctly, perhaps by a very wide margin.

What is the Difference Between Cp and Cpk?

Cp vs. Cpk

Cp and Cpk measure your consistency compared to your average performance.

The ‘k’ stands for ‘centralizing factor.’ The index considers the fact that your data may not be centered.

C_pk tells us what a process can do in the future, assuming it remains in a state of statistical control.

What is Cpk?

The Parking a Car in the Garage Analogy

Think of the walls of your garage – where you have to fit your car – they become the customer specification limits. If you go past those limits, you will crash, and the customer will not be happy!

When your process has a lot of variation, the process average is all over the place. Not good for parking a car or any other process. To give your parking process the best chance of success, you should reduce variation and centering.

If the car is too wide for the garage, nothing you do to center the process will help. You have to change the dispersion of the process (make the car smaller.)

If the car is a lot smaller than the garage, it doesn’t matter if you park it exactly in the middle; it will fit, and you have plenty of room on either side. That’s one of the reasons the Six Sigma philosophy focuses on removing variation in a process.

If you have a process that is in control and with little variation, you should be able to park the car easily within the garage and thus meet customer requirements. C_pk tells you the relationship between the car’s size, the garage’s size, and how far away from the middle of the garage you parked the car.”

How to Calculate Cpk

Cpk is a measure to show how many standard deviations the specification limits are from the center of the process. On some processes, you can do this visually. Others require an equation.

To find Cpk you need to calculate a Z score for the upper specification limit (called Z USL) and a Z score for the lower specification limit (called Z LSL).

Since we are trying to measure how many standard deviations fit between the center line and the specification limit, you should not be surprised that the value of those limits, the process mean, and the standard deviation are all components of the Z calculation.

Cp is an abbreviation. There are really two parts, the upper and the lower denoted Cpu and Cpl, respectively. Their equations are:

C_pl = (Process Mean – LSL)/(3*Standard Deviation)
C_pu = (USL – Process Mean)/(3*Standard Deviation)

Cpk is merely the smallest value of the Cpl or Cpu denoted:  C_pk= Min (C_pl, C_pu)

Why are we dividing by 3 to find Cpk?

We know that any specification limit has an upper and lower bound. Because you know that 6 sigmas (or six standard deviations account for nearly all eventualities in a process (assuming normal distribution)), you shouldn’t be surprised to see the “/ 3” because we are looking at only one side of the distribution.

Calculating Cpk using a Z ValueIf you have a Z value, the equation is very easy;Cpk can be determined by dividing the Z score by three.A z score is the same as a standard score; the number of standard deviations above the mean.

Z = x – mean of the population / standard deviation.

Notes and Characteristics of Cpk

Cpk and Centered Processes

If a process is perfectly centered, it has a Cp of 1. That would indicate that the mean was 3 standard deviations away from the upper limit and the lower limit.

A perfectly centered process is a process that has a mean exactly in between the 2 specification limits (meaning halfway between the two will have a Cpk of 1. How is this possible? Let’s check the math.

If a process is perfectly centered, then we know that the (USL – Process mean) equals the same thing as the (Process Mean – LSL). Let’s call that A.

Z USL = USL – Process Mean / Standard Deviation. then becomes Z USL = A/ Standard Deviation

Z LSL = Process Mean – LSL / Standard Deviation then becomes Z LSL = A / Standard Deviation.

The exact same thing.

Notes on Cpk

• C_pk measures how close a process is performing compared to its specification limits and accounting for the natural variability of the process.
• Larger is better. The larger Cpk is, the less likely it is that any item will be outside the specification limits.
• When Cpk is negative, it means that a process will produce output that is outside the customer specification limits.
• When the process’s mean is outside the customer specification limits, the value of Cpk will be negative.
• To satisfy most customers, we generally want a Cpk of at least 1.33 [4 sigmas] or higher.
• Cpk can have an upper and lower value reported.
  • If the upper value is 2 and the lower is 1, we say it has been shifted to the left.
  • This tells us nothing about whether the process is stable or not.
  • We must report the lower of the 2 values.

What are Good Values for Cpk?

Remember the Car parking in the garage analogy?

• C_pk = Negative number: Your process will regularly crash the car into the wall.
• C_pk =0.5: You have a good chance of hitting the wall on entry.
• C_pk =1:  Your car may be just touching the nearest edge of the entry.
• C_pk =2: Great! You have great clearance. You could double the width of your car before you hit the side of the garage.
• C_pk =3: Excellent!  You have excellent clearance. You could triple the width of your car before you hit the side of the garage.