The reported Cpk for a process with an average of 104 units, a spread of 18 units and upper and lower specification limits of 122 and 96 units would be?
The variance and the standard deviation are measures of the spread of the data around the mean. In this example they have given us the complete spread of the data (in other words the total variation of the process), and not the standard deviation. The spread of the process is calculated by USL - LSL, also known as the "tolerance". Here the total variation range is 18 units, and for a 6-sigma process we have 3-sigma on each side, therefore we will divide 18 by 2 to get the total variance on each side of the mean (mean of 104 units). 18 / 2 = 9. 9 is the total variation of data on each side of the mean.
If we divide 9 by 3-sigma, we get 3, which means, standard deviation per each sigma-level is 3. In the Cpk formula below we can see that the equation is divided by "3 x sd", 3 here is the 3-sigma and the sd is the calculated value of 3. So, 3 x 3 = 9. I hope you get the concept.
Cpk Formula = Minimum Of (USL - Mean / 3 x sd, Mean - LSL / 3 x sd)
Cpk formula Calculations:
=> (USL - Mean) / 3 (sigma) x sd = (122 - 104) / 3 x 3 = 18 / 9 => 2
=> (Mean - LSL) / 3 (sigma) x sd = (104 - 96) / 3 x 3 = 8 / 9 => 0.89
Since we have to pick the lowest value from the two calculations, .89 is our answer.
The minimum acceptable Cpk value must be at least 1, but here the value for LSL side is .89, whereas the value for the USL side is 2 (which is perfect). Our USL side is working at 6-sigma level (104 Mean + (6Sigma x 3sd) = 122), but our LSL side is working at a sigma level of below 3-sigma (104 Mean - (3Sigma x 3 sd) = 95). Since our LSL side is not at the 3-sigma level, the Cpk calculated was .89. .89 shows that we need to improve our process and reduce the standard deviation.
If the question had given us the standard deviation instead of the spread of 18, then we would have multiplied 3 with the standard deviation as shown in the Cpk formula, then we would not need to calculate the standard deviation with the help of the spread.
I made a typo in my previous comment, where I said:
"but our LSL side is working at a sigma level of below 3-sigma (104 Mean - (3Sigma x 3 sd) = 95)"
What I wanted to say was that the LSL value is 96, and when we multiply 3Sigma with 3sd and subtract with 104Mean, we get 95. This shows that our process is producing defective units with a value of 95 which is outside the LSL value of 96. Our LSL side is working at a sigma level of less than 3, and the 3rd sigma level is outside the LSL. We need to bring the 3rd sigma level inside the LSL value of 96, ideally, our process should be working at 5-sigma inside the LSL value of 96.
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