C_{p} and C_{pk} are statistical tools to measure process capability. They are statistical tools to understand variations from the targeted nominal dimensions of parts and are employed widely in the manufacturing world. They also form one of the most essential study when it comes to Quality check and analysis.

To calculate both C_{p} and C_{pk}, you need data which can only be obtained from samples.

**C _{p} is called Process Capability**

“C

_{p}is an index (a simple number) which measures how close a process is running to its specification limits, relative to the natural variability of the process. The larger the index, the less likely it is that any item will be outside the specs.” Neil PolChemus

*C _{p} = (USL-LSL)/6σ*

**σ** is the **standard deviation** of your observations

**USL** is the upper limit of your dimension (ie nominal dimension + upper level of specified tolerance)

**LSL** is the lower limit of your dimension (ie nominal dimension – lower level of specified tolerance)

*6σ is called the natural tolerance of the system(click here to know more)
*

Looking at the equation of C_{p}, we can understand certain things. *σ* is the natural tolerance of the system. Natural tolerance is determined by the observed variability of the data collected from sample parts. The higher the variation among parts, the higher the natural tolerance would be.

(USL-LSL) just signifies the overall variation according to specification. If you have a drawing with a dimension specified as 20±0.5mm, the (USL-LSL) would be (20.5-19.5) which is equal to 1mm.

Considering the above, we can clearly tell that the lesser the variation, the higher the C_{p} is. A higher C_{p} would mean that the process is more capable.

**C _{pk} is called Process Capability Index**

“If you hunt our shoot targets with bow, darts, or gun try this analogy. If your shots are falling in the same spot forming a good group this is a high

C_{p}, and when the sighting is adjusted so this tight group of shots is landing on the bullseye, you now have a highC_{pk}.”Tommy

*C _{pk} = Min {C_{p}.upper, C_{p}.lower}*

*C _{p}.upper = (USL-μ)/3σ*

*C _{p}.lower = (μ-LSL)/3σ*

**σ** is the **standard deviation** of your observations

**μ** is the **mean** of your observations

**USL** is the upper limit of your dimension (ie nominal dimension + upper level of specified tolerance)

**LSL** is the lower limit of your dimension (ie nominal dimension – lower level of specified tolerance)

C_{pk} brings centricity around the nominal dimension into the equation. This is why it is divided into two halves (the mid-point of which is the nominal dimension) and considers the minimum value out of the lower side and upper side. If C_{p}.upper is lower, this would mean that the observed data actually leans to the upper half side (and vice versa).

C

_{pk}measures how close you are to your target and how consistent you are to around your average performance. A person may be performing with minimum variation, but he can be away from his target towards one of the specification limit, which indicates lowerC_{pk}, whereasC_{p}will be high. On the other hand, a person may be on average exactly at the target, but the variation in performance is high (but still lower than the tolerance band (i.e., specification interval). In such case alsoC_{pk}will be lower, butC_{p}will be high.C_{pk}will be higher only when you are meeting the target consistently with minimum variation.

“The value itself can be thought of as the amount the process (car) can widen before hitting the nearest spec limit (garage door edge). Taken from here)

*C*_{pk }=1/2 means you’ve crunched nearest the door edge (ouch!)

*C*_{pk }=1 means you’re just touching the nearest edge

*C*_{pk }=2 means your width can grow 2 times before touching

*C*_{pk }=3 means your width can grow 3 times before touching”

**How many observations do I need to take to calculate C _{p} and C_{pk}?**

It completely depends on the person or the company. Definitely, the higher the number of observations, the better calculations are going to be. But take a look at the graph below.

Degrees of freedom is actually the number of data in hand minus 1.

As you can see, the graph starts to flatten out after 30. For this reason, mathematically speaking, 30 observations are considered sufficient to estimate Cp and Cpk levels with a relatively good accuracy.

At 30 observations, your uncertainty is at 13.1%

To reduce the uncertainty further by half, you would need to reach 120 observations (a fourfold increase).

The time and the effort spent to collect data is usually not worth it.