HP Prime: Shewhart
X-Bar Chart (Quality Chart)
Introduction and Calculation
The program XCHART generates five variables that describe
the parameters of a Shewhart X-bar chart for quality control purposes. Two limits, upper and lower, are determined,
set 3 deviations from the mean, set the boundaries in which a process can vary
and still allow the process to be in control.
The samples are arranged in a matrix. Each row is a sample, with each column is a
data point of those samples. For
example, a 15 x 10 matrix represents 15 samples of with the sample size of 10.
The five parameters calculated are:
MM: Grand Mean. The mean of all the sample means.
Sample mean = Σx / n
RM: Range Mean. The mean of all ranges of all samples.
Range = max(sample) – min(sample)
MD: Mean
Deviation. The mean of all the standard
deviation of all samples.
Sample Deviation = √( (Σx^2 – (Σx)^2/n) / (n – 1)
L, U: Lower and Upper
limit of the x-bar chart. How it is
calculated is dependent on the sample size (number of columns).
If the size ≤ 25, then:
U = MM + a2 * RM
L = MM – a2 * RM
Where a2 is a constant.
The values of a2 for sample sizes 2 – 25 can be found here: http://onlinelibrary.wiley.com/doi/10.1002/9781119975328.app2/pdf
Some values (2 – 11):
|
Sample Size
|
a2
|
Sample Size
|
a2
|
|
2
|
1.880
|
7
|
0.419
|
|
3
|
1.023
|
8
|
0.373
|
|
4
|
0.729
|
9
|
0.337
|
|
5
|
0.577
|
10
|
0.308
|
|
6
|
0.483
|
11
|
0.285
|
HP Prime Program XCHART
EXPORT
XCHART(mat)
BEGIN
// 2017-08-25 EWS
// Quality control
// {grand mean, range mean,
// std.dev mean, LCL, UCL}
// Grand mean
LOCAL MM:=mean(mean(TRN(mat)));
// Average range
LOCAL RM,C,R,I,L,cl;
// sample size: columns
C:=colDim(mat);
// number of samples
R:=rowDim(mat);
FOR I FROM 1 TO R DO
cl:=row(mat,I);
RM:=RM+(MAX(cl)-MIN(cl));
END;
RM:=RM/R;
// Average deviation
LOCAL MD:=mean(stddev(TRN(mat)));
// Estimate LCL,UCL
LOCAL U,L;
IF C≤25 THEN
// a2 list for sample size 2-25
LOCAL a;
LOCAL a2:={1.88,1.023,.729,
.577,.483,.419,.373,.337,.308,
.285,.266,.249,.235,.223,
.212,.203,.194,.187,.18,.173,
.167,.162,.157,.153};
a:=a2(C-1);
L:=MM-a*RM;
U:=MM+a*RM;
ELSE
// c4 (C>25)
LOCAL c4:=√(2/(C-1))*((C/2-1)!/
((C-1)/2-1)!);
U:=MM+3*MD/(c4*√C);
L:=MM-3*MD/(c4*√C);
END;
// Results
RETURN {MM,RM,MD,L,U};
END;
Example:
Data Process 6 samples of 6 data points each:
|
9.48
|
10.07
|
10.98
|
8.9
|
11.53
|
11.05
|
|
10.04
|
8.78
|
10.8
|
9.45
|
11.34
|
8.9
|
|
10.75
|
10.48
|
10.08
|
9.36
|
10.61
|
9.92
|
|
10.05
|
8.99
|
10.00
|
8.33
|
9.3
|
9.77
|
|
9.65
|
9.41
|
10.08
|
8.91
|
11.43
|
9.08
|
|
9.74
|
9.08
|
11.17
|
11.21
|
9.94
|
8.89
|
Since the sample size is not greater than 25, the range is
used, with a2 = 0.483 (for the size of 6).
Results:
MM = 9.93194444444
RM = 2.19
MD = 0.785268047476
L = 8.87417444444
(lower limit)
U = 10.9897144444
(upper limit)
According to the results, the process needs further
investigation.
Sources:
Aczel, Amir D. and Sounderpandian, Jayavel. Complete Business Statistics
McGraw-Hill Irwin: Boston. 2006. ISBN
13: 978-0-07-286882-1
“6.3.2. What Are
Variables Control Charts?” Engineering
Statistics Handbook. http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc32.htm
Retrieved August 24, 2017
Six Sigma Improvement with Minitab “Appendix 2: Factors for control charts”
Published on line June 22, 2011. http://onlinelibrary.wiley.com/doi/10.1002/9781119975328.app2/pdf Retrieved August 24, 2017
Eddie
This blog is property of Edward Shore, 2017.
