# Control limit - How is table for x-bar & R control chart derived?

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#### Kleeve

Does anyone know how is table for x-bar & R ctrl chart derived?
Well the problem is, i try to calculate the ctrl limit:UCL and LCL. Thru' understanding the limit are +/-3sigma thus i obtain the sigma value,then applied x_bar+/-3sigma to cal the ctrl limit.
But had no idea why the result varies from using the formula (x_bar+/-A2R_bar)? Very much greater... ...

R

#### Rick Goodson

Kleeve,

It is not easy to express formula in the forums but I will give it a try. If it is difficult to follow post your email address and I will send you the derivation in a Word document.

The control limits are equal to X-bar plus and minus 3 sigma sub x-bar.
Sigma sub x-bar is equal to sigma prime divided by the square root of n.
Sigma prime is estimated from R-bar divided by d2 (the factor for subgroup size).

Hope this helps.

Rick

A

#### Al Dyer

Right on Rick, formulas are tough!

Just a shot in the dark,

But could the difference be that A2 is a constant based on the groups sample size, but sigma in not based on sample size?

I think the only deviation (no pun intended) in sigma is that for Cpk values we use the sigma of the samples (however many), and for Ppk values we use the sigma based on the entire population.

Am I getting close?

ASD...

[This message has been edited by Al Dyer (edited 27 June 2001).]

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#### Rick Goodson

Al,

You are absolutely right. The standard deviation of the universe and the standard deviation of the samples drawn from the universe are related but obviously different. Sigma of the X-bar's (standard deviation of the samples) is equal to sigma prime (standard deviation of the universe) divided by the square root of the sample size. At a sample size of 5 the standard deviation of the samples is equal to 0.45 sigma of the universe. At a sample size of 4 it is 0.50 sigma of the universe. Of course the standard deviation of the universe is an estimate based on the average range divided by the d2 factor which further confuses the issue.

Rick

G

#### Graeme

Originally posted by Kleeve:
Does anyone know how is table for x-bar & R ctrl chart derived?
Well the problem is, i try to calculate the ctrl limit:UCL and LCL. Thru' understanding the limit are +/-3sigma thus i obtain the sigma value,then applied x_bar+/-3sigma to cal the ctrl limit.
But had no idea why the result varies from using the formula (x_bar+/-A2R_bar)? Very much greater... ...

The basic mathematics for the control charts was developed in the 1920's and 1930's ... most people now simply refer to either ASTM Manual 7 "Manual on Presentation of Data and Control Chart Analysis" or ANSI/ASQC B1-1996 through B3-1996 "Quality Control chart Methodologies".

Some key points are:
• As stated by Rick Goodson, values in the control chart tables are more closely related to (3*sigma) divided by (square-root ). It is actually more complex than that for small sample sizes, because factors such as Student's t are involved as well.
• If your process is not in a state of statistical control, you should expect differences in the values from different calculation formulas. That is because equations using sigma assume a normal distribution, but the presence of assignable causes is a statement that the distribution is not normal.

ASQ - http://www.asq.org
ASTM - http://www.astm.org

------------------
Graeme C. Payne
ASQ Certified Quality Engineer

K

#### Ken K.

From Montgomery's "Introduction to Statistical Quality Control" (great book), pages 183 & 184:

sigma-hat = R-bar / d2

A2 = 3 / [d2 * SQRT ]

Remember that we're talking about the 3 sigma limits on the distribution of X-bar, not just X, and that the standard deviation of X-bar is simga/SQRT(n. So . . .

X-bar +/- 3*sigma_of_X-bar

which equals:

X-bar +/- 3*[sigma/SQRT ]

which using sigma-hat = R-bar / d2 equals:

X-bar +/- 3*[R-bar / d2)/SQRT ]

which equals:

X-bar +/- R-bar*[3 / (d2 * SQRT ]

which equals:

X-bar +/- R-bar*A2

S

#### student without holiday

Simple and short question:

what is the formula behind d2 (or dn, hartley's conversion constant) and is this constant limited to a sample size of 12 ?

greetings

R

#### Rick Goodson

student without holiday

Short, yes. Simple, no.

The d2 factor is not limited to a sample of 12. Tables for d2 factors run from samples of 2 to over 100 (reference Statistical Quality Control, seventh edition, Grant and Leavenworth, ISBN 0-07-024162-7, Table C, page 717).

I do not know the derivation of the d2 factor however it is the expected value of R bar divided by sigma of the universe at different sample sizes. For a discussion of the d2 factor see the text mentioned above or Quality Control and Industrial Statistics By Acheson J. Duncan. My edition is pretty old but I am sure it is discussed in newer revisions of the text.

Regards,

Rick