# Control Chart Constants and Confidence Interval

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#### rwhitney

I've read several threads on the calculation of the control chart constants. Concensus is that the constant * variation estimator (e.g. Asub2 * Rbar) estimates sigma * 3 to yield the factor added or subtracted from the central value to net UCL or LCL. Is this estimation quantified by a confidence interval? 90%, 95%, 99%? Maybe buried deep in the calculation or concept?

#### Steve Prevette

##### Deming Disciple
Super Moderator
No. the control chart limits are "not an exercise in probability" (Dr. Deming).

The control limits are set at 3 standard deviations from the average, a value determined by Dr. Shewhart in 1930 as the best compromise between failure to detect and false alarms across many different data distributions.

Dr. Shewhart invoked the Tchebychev Inequality as theoretical basis to use one number across several distributions, and he tested the results against the Normal distribution, a square distribution and a triangular distribution.

Dr. Wheeler has followed up with several simulation studies, and concluded the level of significance for SPC control limits run from 3% to 5% worst case.

R

Thank you Steve!

#### bobdoering

Trusted Information Resource
Dr. Shewhart invoked the Tchebychev Inequality as theoretical basis to use one number across several distributions, and he tested the results against the Normal distribution, a square distribution and a triangular distribution.

Key to note that those distributions need to be from random and independent variation process output. Otherwise, Shewhart's charts do not not work well. Tchebychev Inequality requires random and independent variation to be valid.

#### Bev D

##### Heretical Statistician
Super Moderator
I'll wade in cautiously here. buried deep in the concept and the calculations of the constants is the core concept that also results in a 'confidence interval'. However, as Steve pointed out, the control chart limits are not confidence intervals. Confidence intervals and control limits can be thought of as cousins. and there is certainly no attempt or ability or need to have a precise probability of any particular result with a control limit as Steve also pointed out.

certainly any particular result has some large or small probability of occurring If the process hasn't changed. The same can be said when comparing two point estimates to each other...but confidence intervals are intended for comparing a very small number of point estimates, typically not differentiated by time but be a different source of change like different vendors, equipment, materials or processes. control limits are used to compare often large numbers of samples that differentiate themselves in time sequence. These different comparisons while similar in some aspects are really very different types of comparisons...

#### Statistical Steven

##### Statistician
Super Moderator
I've read several threads on the calculation of the control chart constants. Concensus is that the constant * variation estimator (e.g. Asub2 * Rbar) estimates sigma * 3 to yield the factor added or subtracted from the central value to net UCL or LCL. Is this estimation quantified by a confidence interval? 90%, 95%, 99%? Maybe buried deep in the calculation or concept?

I will dip my feet in the water to give you the answer your are expecting. The "probability" is that 3 out of 1000 will be outside the control limits by chance alone. So in the essence the control limit are like 99.73% limits on the distribution of the data.

#### Steve Prevette

##### Deming Disciple
Super Moderator
The answer of 3 in 1000 is only valid for the Normal Distribution . . .

#### Statistical Steven

##### Statistician
Super Moderator
The answer of 3 in 1000 is only valid for the Normal Distribution . . .

That is correct, and if you use Xbar charts then the distribution of the means is normal (or approximately normal) regardless of the distribution of the individuals. Of course, if you are a purest, I am probably understating the reality, but I am a realist so 3 out of 1000 is a good estimate.

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