Process Capability Analysis

In this tutorial, you will learn all about process capability analysis, when to use it, how to calculate it and how to interpret the results.

Process Capability Example

In order to understand the concept of process capability, we will start with a simple example. Let‘s say we work for a company producing muffins, which are sold to supermarkets. Our label on the muffin states that the muffin weighs 90 grams. So of course we must make sure that we produce muffins around 90 grams.

Underweight muffins can violate labeling laws and trigger recalls. Overweight muffins waste ingredients. So our goal is muffin weight control. We want to produce muffins within a defined range. For example, a muffin should weigh a minimum of 85 g and a maximum of 95 g.

These are our specification limits. 85g is the so-called ower specification limit (lsl) and 95 g is our so called upper specification limit (usl).

But how can we make sure that we produce muffins within specifications? That’s where process capability analysis comes in.

What is Process Capability Analysis?

Process capability analysis is a statistical method that determines if a process can consistently meet specifications by comparing process variation with the specification limits. For sure, each process has a certain variation. The muffins we produce will vary to some extent.

Now, process capability analysis compares this variation with the specification limits. And it tells you whether the process can consistently meet those specifications.

Process Capability Analysis checks how well a stable process can meet spec limits—typically summarized by Cp/Cpk (short-term) or Pp/Ppk (long-term)—by comparing the process spread and centering to the allowed tolerance.

How is a Process Capability Analysis calculated?

We want to measure the weight of our muffins, so we must make sure the scale we use is measuring correctly. To do this, we can perform a measurement system analysis to ensure the scale isn’t adding false variation. If you need more information about measurement system analysis, you can have a look into our tutorials or training videos.

Next, we can collect data. Let’s take a sample of four muffins every hour and record their weights. Now we can calculate the so called process capability indices Cp and CpK and further the process performance indices Pp and Ppk.

What are the Process Capability Indices (CP/CpK)?

In our example, we have our lower specification limit (lsl) with 85g and our upper specification limit (usl) with 95g. That gives us a specification width or tolerance of 10g.

And of course, we have a certain variation of the process. If the process variation is large compared to the tolerance, the process won’t be capable of reliably producing muffins within specifications.

And this is what Cp tells us: it’s the ratio of the tolerance to the process width. Ok, the tolerance is clear it is the upper specification limit minus the lower specification limit.

What is the formula for the Cp?

Cp is calculated using the following formula:

The process width is estimated with 6 times σ. If the process variation is large compared to the specification width, Cp is less than 1. If the variation is small compared to the tolerance, Cp is greater than 1. And if 6sigma is equal to the tolerance, Cp is 1.

So, which CP value is good, and which is no longer good? There isn’t one universal “official” table, but these widely used rules of thumb cover most industries. Below one is "not capable", 1 is the Borderline and above 1.33 is considered as "capable".

Now, there’s one thing to consider: If we plot weight on the y-axis, mark the specification limits, and show the process variation, we’ll get the same Cp value no matter where our process is centered. So, process can be centered here, or there, in all cases we get the same CP value. Because the cp value just depends on the process width and the tolerance.

Now the question is ‘What good is Cp if I can get a great number while producing completely out of specification?’ The answer: a high Cp tells you the process has low variation compared to the tolerance—and that’s great.

It’s usually much easier to re-center a process than it is to reduce variation. So, it means in general the process is capable, it just needs to be recentered. So Cp compares the tolerance with the process spread. But it doesn’t tell us where that spread sits. That’s where Cpk comes in.

How do we calculate Cpk?

Cpk is calculated with this formula. It looks a bit complicated, but let’s make it simple.

As we now know Cp reflects the potential capability under ideal centering. So we have the same distance from the mean to the upper limit and lower limit.

In contrast to that, Cpk shows the actual capability, accounting for any mean shift. For example here the critical limit would be the upper specification limit, because the mean is closer to this limit, so this distance is used to calculate the cpk.

If we go back to the formula, this distance is mu minus the lower specification limit and this distance is the upper specification minus mu. And we use the minimum of both to calculate Cpk.

So, as long as the process mean lies between the lower and upper specification limits, we get a positive Cpk value. If the process mean is outside the limits, either on the right or the left side, we get a negative Cpk.

We then divide the distance from the mean to the nearest specification limit by three times the standard deviation (3σ), which is half of the 6σ process width. So the bigger the process width, the smaller the cpk value. If the process is perfectly centered, so we have the same distance left and right, Cp and Cpk are the same.

So with Cp we have a value that reflects the potential capability under ideal centering, and with Cpk we have a value that shows the actual capability, accounting for any shift from the center.

We can use the same Cp table to categorize Cpk values.

How are the process performance indices Pp and Ppk calculated?

We alreday know that sigma is the within-group standard deviation, so let’s denote it as 𝜎_𝑤. What is the within-group standard deviation?

We measured the weight of four muffins at each time point. Now the within-group standard deviation captures the variation within each subgroup. Simply speaking, it calculates the deviation in these groups and averages all deviations.

So we get a standard deviation that tells us how much the muffins vary within the groups. But this means we don’t account for any deviations between groups.

Even if the measurements look like this, we would still see a small within-group variation, because the variation within each group stays the same.

But for sure, the total variation also affects capability. This is where Pp and Ppk come in.

Basically, the formulas for Cp and Pp and for Cpk and Ppk are the same. The only difference is which standard deviation we use. Cp and Cpk use the within-group, or short-term, standard deviation. Pp and Ppk use the total, or long-term, standard deviation. So in case of Pp and Ppk all individual observations across time are used, ignoring subgrouping.

How can we calculate a process capability analysis online with numiqo?

If you like you can load this example data with the link above or you can copy your own data into the Process Capability Calculator. Now we just click on Process capability and then on weigth and on subgroub. Here we can enter the Lower specification limit 85 and the upper specification limit 95.

Then we can see the results here. There we just see the process data. Here we have the Within the groups data with Cp and Cpk and here we have the overall data with Pp and Ppk. But lets just briefly discuss the results. So our Cp value is 1.5, which—according to this table—means the process is capable.

What about Cpl and Cpu? Looking at the formula for Cpk, Cpl is this term and Cpu is that term. To calculate Cpk, we take the minimum of these two values.

So Cpk is 0.82—the smaller of the two. A Cpk of 0.82 is not acceptable, so we need to recenter the process.

PPM means parts per million. So a value of 6869 means, about 6,869 out of 1,000,000 units are expected to be out of specification.

And here are the Overall results:

As we know, “Within” considers within-subgroup variation, while “Overall” uses the total (long-term) variation.

A Pp of 1.46 indicates that the long-term potential capability is good. Pp measures the process’s potential capability based on long-term variation only, it ignores centering.

A Ppk of 0.82 indicates that the long-term capability is not good. Ppk tells you the process’s actual (long-term) capability to meet both specifications and taking centering into account.

And if you like, you can also just click on interpretation, then you will get a summary in words of this results table.

numiqo is therefore an alternative to Minitab. If you like, just give it a try!