# Calculating Lower and Upper 3 Sigma Control Limits

D

#### dmk_miz

I've been stuck on this problem for far too long. I can get numbers close to E, but I am not sure if what I am doing is correct. If somebody has the time to give me a walk through with this I would very much appreciate the help! Thank you!

A manufacturer of fax machines checks samples of fifty units for functionality at the conclusion of the assembly process. The results of ten samples are shown in the table below. Calculate the lower and upper 3-sigma control limits.

Sample # 1-2-3-4-5-6-7-8-9-10

A. (–0.01, 0.09)
B. (–0.01, 0.29)
C. (–0.01, 0.09)
D. (0.00, 0.09)
E. (0.00, 0.29)

#### Steve Prevette

##### Deming Disciple
Super Moderator
I've been stuck on this problem for far too long. I can get numbers close to E, but I am not sure if what I am doing is correct. If somebody has the time to give me a walk through with this I would very much appreciate the help! Thank you!

Take your data, divide each by 50. Determine the average of the data (since the sample size is always 50, this makes it easier.

Then use the p-chart calculation for control limits which is the average plus and minus 3 times the square root of (avg x (1-avg) / 50)

#### David-D

##### Involved In Discussions
Good explanation although I'd add that since the calculated lower control limit is negative; you set it equal to zero (since you can't get negative defects).

Since clearly this is an exam type question there is a much faster way to get the right answer by elimination if you understand the underlying concepts. You need to first find the centerline of your control chart (to be able to +/- 3 SD). As you'll find that to be .144 (72 defects /500 samples) you can immediately eliminate choices A, C, and D as all three have both the upper and lower control limits on the same side of the centerline. Then looking at B and E you can immediately also eliminate B since the lower control limit can't be negative. Thus, if you understand the fundental concepts, you can quickly come to the correct answer with very little calculation.

Of course, in actual practice on a production line this shortcut/approach isn't useful for determining actual control limits, but it is very useful as a sanity check to make sure you didnt screw them up. In my experience with the CQE exam, being able to harness 'tricks' to quickly get the answer will allow more time to focus on other questions. Think of it as a time bonus for understanding the concepts above just brute-forcing you're way through the math.

Good luck,

David