I recently saw a question on the open forum that asked about process capability numbers (Cp/Cpk) and Sigma Levels. I thought it was a good topic for us to discuss here. Are they related and if so, how do we translate from one to the other?

Suresh,

The capability of the process is calculated using the Sigma.

Cp= USL - LSL/ 6Sigma.

Process capability is defined as the ability of the process to sustain within the required specifications. Sigma refers to variation of the process.

Cp is closely releated to Sigma (Variation).

but Cpk tells about the process shift.

So Cp is directly proportional to Process Sigma.

Cpk has no relation with Process sigma.

Regards,

Ramabadran

Dear SJ,

Cp is correctly defined as below;

Cp = Specification width / Process width = (USL ~ LSL ) / (UCL ~ LCL)

[ N.B: Often,people use the formula (USL-LSL) / (UCL-LCL) which might lead to confusion sometimes if understanding not clear]

Now,

Case 1:

USL = + 3 SD ; LSL = - 3 SD

Process width is mostly kept at 6 SD for all the cases.

Cp = ( 6 SD / 6 SD ) = 1.0

A Cp of 1 corresponds to a Sigma Level of 3 as the area occupied between the USL and LSL is 99.73% ( assuming Short term capability and a Normally distributed process) which in turn equals to a yield of 99.73% which in turn equals to a DPMO of 2700 [( 100% -99.73%)x1000000 = 2700] which equals to a Sigma Level of 3.

Hence ,

Cp of 1 equals to Sigma Level of 3

Proceeding on the same lines,

Cp of 1.33 equals to Sigma Level of 4

Cp of 1.66 equals to Sigma Level of 5

Cp of 2.0 equals to Sigma Level of 6

Readers may please note that all the above Sigma Levels are Short term Sigma levels.

Hope this helps.

Regards,

Ari

When people talk about Sigma levels, they usually refer to Z (Bench) where defects on both sides (greater than USL, and less than LSL) are included. In the above formulation, we are only talking about Sigma level on one side.

So, for a Cp = 1.0, the Sigma level for defects greater than USL is 3.0 and Sigma level for defects less than LSL is 3.0. The total Sigma level (bench) is 2.78.

SJ

Dear Ari/Ramabadran,

You have rightly pointed out the Cpk cannot be translated directly into Sigma levels. Here are some additional points regarding that topic.

Cpk (unlike Cp) includes both variation and shift to calculate the process capability. The traditional formula for Cpk (for a normally distributed process data) is given by:

Cpk = min [ (USL - Xbar)/(3*S), (Xbar - LSL)/(3*S)]

Where, LSL and USL are the Lower and Upper Specification Limits (as determined from the customer), Xbar is the average of the process data, and S is the sample standard deviation.

When, we look at the formula, we see that we compute the minimum. Which means, we are looking at the defects that are greater than USL on one side and the defects that are less than the LSL on the other side and only picking the worst case (A smaller process capability number relates to larger number of defects).

So, if a process had 10,000 PPM defects to the left of LSL and 20,000 PPM defects to the right of USL, then Cpk will look at both of them and then compute the process capability number related to 20,000.

As pointed out earlier, Sigma Level (Bench) looks at defects on both sides and adds them up. So, it would not be directly possible to translate Cpk numbers to Sigma Levels. However, assuming the worst case defects occurs on both sides, it would be possible to estimate a conservative estimate of Sigma level if we so desire.

Example:

Let's say LSL = 11.5, USL = 18, Xbar = 15, S = 1.

Cpk = min[(18-15)/(3*1), (15-11.5)/(3*1)] = min[1, 1.167] = 1.0

CpU (only looking at USL) = 1.0

CpL (only looking at LSL) = 1.167

PPM_U (only looking at USL) = 1350

PPM_L (only looking at LSL) = 232

PPM_Total (both sides) = 1582

Sigma_U = 3.0

Sigma_L = 3.5

Sigma_Bench = 2.95

Conservative estimate for Sigma_Bench based on Cpk = 1.0 (assume both sides have PPM = 1350 - centered process!). Hence, Total PPM = 2700 => Sigma_Bench = 2.78.

CONCLUSION:

Cp * 3 gives you sigma level on one side (Z_LSL and Z_USL).

Cpk cannot be directly translated into sigma level for bilateral tolerances as it only looks at the worst side. Of course, if it is a unilateral tolerance, then we can predict Sigma levels from Cpk numbers.

SJ has explained this in detail.

Two more lines to summarise - In a two sided specification limit, Cpk multiplied by three is the smaller of the two sigma levels (sigma level upper and sigma level lower)

In the example above Cpk = 1 and Cpk multiplied by three is Sigma Leval (Upper).