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

K

#### 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).]

R

#### 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

How to control when control limit is out of Specs limit Statistical Analysis Tools, Techniques and SPC 11
P-Chart Control Limit Calculation help Statistical Analysis Tools, Techniques and SPC 1
R How to calculate UCL (Upper COntrol Limit) and LCL (Lower Control Limit) Statistical Analysis Tools, Techniques and SPC 5
A CQE exam primer Control Limit question Statistical Analysis Tools, Techniques and SPC 2
S SPC Control Limit Calculation Statistical Analysis Tools, Techniques and SPC 2
Setting Lower Control Limit on Attributes Charts Statistical Analysis Tools, Techniques and SPC 9
I Shewhart Constants vs Central Limit Theorem in calculating Control Limits Statistical Analysis Tools, Techniques and SPC 18
A Change Ratio [Guideline in Control Limit Update] Statistical Analysis Tools, Techniques and SPC 3
D Why a Lower Control Limit on Range and Sigma Statistical Analysis Tools, Techniques and SPC 7
S S-Type Tolerances: How to calculate Cs? Control Limit(s)? Statistical Analysis Tools, Techniques and SPC 8
K Calculating UCL (Upper Control Limit) and LCL (Lower Control Limit) Statistical Analysis Tools, Techniques and SPC 14
K How to establish critical limit for Critical Control Points (CCPs) Food Safety - ISO 22000, HACCP (21 CFR 120) 7
J What Control Chart should be used for Microbial Limit (NMT 100 cfu/m3) Statistical Analysis Tools, Techniques and SPC 12
L When really deserve to compress the Control Limit as an Improvement? Statistical Analysis Tools, Techniques and SPC 9
J Xbar - R Control Chart taking into account the Probability Limit Using Minitab Software 8
B Analytical Lab Environmental Condition (Temperature & Humidity Control Limit) General Measurement Device and Calibration Topics 5
Q UCL (Upper Control Limit) Calculations - Sample Problem Statistical Analysis Tools, Techniques and SPC 11
L Why the difference in control limit value when using different calculation methods? Statistical Analysis Tools, Techniques and SPC 2
I X-MR Control Limit calculation question Statistical Analysis Tools, Techniques and SPC 4
D Control limit calculation (Xbar R charts) using Minitab Statistical Analysis Tools, Techniques and SPC 14
R ISO9001:2000 "Control of Documents"- What's the limit? ISO 9000, ISO 9001, and ISO 9004 Quality Management Systems Standards 21
2 What type of Control Chart? USL of 3 and a lower spec limit of 1 Statistical Analysis Tools, Techniques and SPC 1
H When shall I recalculate control limit? Capability, Accuracy and Stability - Processes, Machines, etc. 5
J Control and Handling of Limit or Boundary Sample - Molding company and we get flash Quality Tools, Improvement and Analysis 6
C Designing Control Charts - Variable with Small Variation and Tight Control Limit Statistical Analysis Tools, Techniques and SPC 20
Changing LCL (Lower Control Limit) on a Minitab Control Chart Using Minitab Software 13
M Subgroup Size for Control Limit Calculations - SPC Statistical Analysis Tools, Techniques and SPC 5
M How to Calculate UCL (Upper Control Limit) & LCL (Lower Control Limit) & CL Statistical Analysis Tools, Techniques and SPC 73
A Point Outside of Control Limit for R Chart - This is not SPC...It's GR&R study Gage R&R (GR&R) and MSA (Measurement Systems Analysis) 2
M Bias - Upper and Lower control Limit - Bias Study page 88 of the MSA 3rd Edition Gage R&R (GR&R) and MSA (Measurement Systems Analysis) 1
S Control chart limit calculation confusion concerning SPC control limit calculation Statistical Analysis Tools, Techniques and SPC 30
K SPC for One-sided Specification - Setting the Lower Control Limit Statistical Analysis Tools, Techniques and SPC 4
B How do I calculate Cpk with a single Control Limit? Capability, Accuracy and Stability - Processes, Machines, etc. 15
COPLANARITY: Composite profile tolerance on multiple surfaces- what does" lower dimensional reference frame tolerance" control? Inspection, Prints (Drawings), Testing, Sampling and Related Topics 4
Medical Device Marketing Material - Control of Social Media 21 CFR Part 820 - US FDA Quality System Regulations (QSR) 2
Does 4.5 - Alternative RISK CONTROL apply to the Particular Standards? IEC 60601 - Medical Electrical Equipment Safety Standards Series 3
Control of Change i.e. process and WI IATF 16949 - Automotive Quality Systems Standard 9
AIAG VDA PFMEA and Control Plan training FMEA and Control Plans 6
Quality Control in Electrical Calibration ISO 17025 related Discussions 0
Why Control Limits are not the same depending on type of exclusion of data points Using Minitab Software 7
Engineering Change order vs. Change Control ISO 13485:2016 - Medical Device Quality Management Systems 3
Computer access, password control ISO 13485:2016 - Medical Device Quality Management Systems 6
How to control distribution list? Document Control Systems, Procedures, Forms and Templates 2
Control plan for IATF - example? FMEA and Control Plans 1
ISO 9001:2015 Clause 8.5.1 Control of Production and service provision - Help with Work Instruction Access ISO 9000, ISO 9001, and ISO 9004 Quality Management Systems Standards 13
Final quality control corrective actions ISO 9000, ISO 9001, and ISO 9004 Quality Management Systems Standards 10
Training on the importance of Document Control Document Control Systems, Procedures, Forms and Templates 3
How to implement H&S and Quality Control Requirements in Contract for Potential Supplier? Contract Review Process 6
GAMP CAT 5 Change Control Qualification and Validation (including 21 CFR Part 11) 9
Use Error Risk Controls and Control Verification ISO 14971 - Medical Device Risk Management 6