The Relationship Between Cp/Cpk and Sigma Level

Cpk = minimum (USL-mean, mean – LSL)/(3*std)

Therefore, in order for Cpk to be negative your mean will have to either be less than the LSL or greater than the USL.

Sigma = minimum (USL-mean, mean – LSL)/std

Therefore, if the mean is greater than USL then (USL – mean) will be the smaller value and it will be negative which means Sigma will be negative.

If the mean is less than the LSL then the minimum value will be (mean – LSL) which will also be negative which means your Sigma will be negative once again.

Thus, the results that give you a negative Cpk cannot result in a positive Sigma

Look at the total ppm defective.  Translate between Cpk and sigma level…for that one side of the curve.

Hello –

If you have a negative CPk, it means you have at least 50% Failures, assuming the formula is correct.

The lower the negative CPk, the more Percent failures.

As an Example;  Negative 1 CPk would be about 99.7% Failures, to the Specification

Then look at the CP Value.  If CP is positive, you probably only need to adjust the mean (X-Bar), to center the process.

It is possible to have a negative CPk, with a positive sigma (Std. Dev.) as the Std. Dev. will always be a positive number.

The CPk value is the Real Calculated Value

The CP Value is a Theoretical Value, Assuming the Process is perfectly centered around the Nominal Dimension.

Hope this helps

@[email protected] – One of my pet gripes is the fact that when the folks were developing their terminology for six sigma the took the word “sigma” and applied it to the calculation indicated in my first post.  That formula is the formula for the sigma level which has nothing to do with the sigma of standard deviation fame.  Unfortunately, a number of programs and a lot of verbiage, both posted and published in the professional journals, insist on shortening the term “sigma level” to just “sigma.”

With regard to this confusion – on the less than worrisome end are posts like our exchange – what can really get interesting is when someone makes a statement about sigma being negative (what they mean is the sigma level is negative) and when someone else, who knows that sigma (as in standard deviation) is the square root of the variance, jumps in and calls the first poster an idiot for trying to claim a negative value for a square root….and so it goes

If you go by the formulas, Cp cannot be negative , Cpk can be negative when mean is lower than lower spec or higher than upper spec . In that case,  Sigma level is also negative. example, for a torque spec limit of 16 to 20 N on a bolt,

Z = ( USL- x bar ) / s

for mean of 23, and usl of 20, std dev of 3,

Z = ( 20-23)/3 = -1

cpku = ( usl – x bar ) / 3s =  -3 / 3 ( 3 )     = – 0.33

cpkl = ( x bar – lsl ) / 3s = ( 23-20) / 3 ( 3 ) = 0.33

Cp = ( usl – lsl ) 6s = ( 20 – 16 ) / 6 ( 3 ) = 4/18 = 0.2