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Appendix 13: SPCHART
SPSS creates nine types of Shewhart control charts. In this appendix, the charts are grouped into five sections:
- X-Bar and R Charts
- X-Bar and s Charts
- Individual and Moving Range Charts
- p and np Charts
- u and c Charts
For each type of control chart, the process, the center line, and the control limits (upper and lower) are described.
Notation
The following notation is used throughout this appendix unless otherwise stated:
σ Population standard deviation for measurements X
A Number of sigmas specified by the user, 0 ≤ A≤ 9 K Number of subgroups ni Number of units (samples) for subgroup i N (^) Total sample size, equal to n 1 +K +n (^) K x (^) ij Measurement (observation) for the jth unit (sample) of subgroup i
x (^) i (^) Mean of measurements for subgroup i, x (^) i x (^) ij n i
n i
i
=
∑ 1
Si Sample standard deviation for subgroup i, Si x (^) ij x (^) i n i
n i
i 2 2
1
=
∑^3 8 /^1
Ri (^) Sample range for subgroup i, R x x x x i =^ max^4 i 1 , K,^ in (^) i 9 −min^4 i 1 ,^ K, ini 9 LCL Lower Control Limit UCL Upper Control Limit
Weight
Weights can be used when the data organization is Cases are units.
- Each value for weight must be a positive integer.
- Cases with either non-positive or fractional weights are dropped.
- When weight is in effect, ni is a weighted sum for all the units in subgroup i and x (^) i and x are weighted means.
X-Bar and R Charts
When X-Bar and R charts are paired, the sample range statistic R is used to construct the control limits for the X-Bar chart. Note: Subgroups whose sample sizes are less than the specified minimum value are dropped.
Equal Sample Sizes
Assume that n (^) i = n for i =1, K, K. The process for the X-Bar chart is ; x (^) i :i = 1 , K,K@. The center line for an X-Bar chart is the grand mean statistic x :
x K
x (^) i i
K
=
∑
1
and the control limits are
LCL
UCL
x AR d n n
x AR d n n
2
2
3 0 5 8
3 0 5 8
and the control limits for subgroup i are
LCL
UCL
x A n
x A n
i
i
σ
σ
The center line for an R chart for subgroup i is Ri = σ$^ d (^2) 1 6n (^) i for i = 1 , K, Kwhere
σ =^ $^ /
∑
2 1
K
R (^) i d ni i
K 2 1 6 7
and the control limits for subgroup i are
LCL
UCL
max $^ , $
R A d n
R A d n
i i
i i
σ
σ
3
3
2 1 6^07
1 6
X-Bar and s Charts
When X-Bar and s charts are paired, the sample sandard deviation is used to construct the control limits for the X-Bar chart.
Equal Sample Sizes
Assume n (^) i = n. The process for the X-Bar chart is (^) ;x (^) i :i = 1 , K,K@. The center line for an X-Bar chart is x and the control limits are
LCL
UCL
x AS c n n
x AS c n n
4
4
3 0 5 8
3 0 5 8
The process for an s chart is (^) ;S (^) i :i = 1 , K,K@. The center line for an s chart is
S
K
Si i
K
=
∑
1
and the control limits are
LCL
UCL
max / ,
S A c n c n
S A c n c n
4
2 4
4
2 4
4 1 0 5^69 0 5
4 1 0 5^69 0 5
The auxiliary function is
c n n
n (^4) n
0 5 0 5 10 5 6
where Γ(.) is the complete Gamma function.
Note: When n ≥ 25, c 4 0 5n n can be approximated by n − 0 5. ,
(^1 ) 2 4 −^1 c^ 0 5n^69 /^ c 4 0 5n can be approximated by^1 /^2 n^ −^ 2 5. , and^ c^4 0 5n^ can be
approximated by 04 n − 5 5 0/ 4 n− 35.
Unequal Sample Sizes
The processes for X-Bar and s charts are the same as the processes in the section “Equal Sample Sizes” above. The center line for an X-Bar chart is x (as defined in equation (1)) and the control limits are
LCL
UCL
x A n
x A n
i
i
σ
σ or LCL
UCL
x AS c n n
x AS c n n
i i i
i i i
4
4
3 1 6 8
3 1 6 8
The average moving range is
R
N m
Ri m
N
− + ∑
0 15
The center line for an Individual chart is x and the control limits for an Individual chart are
LCL
UCL
x AR d m x AR d m
2 2
0 5 0 5
The process for a moving range chart is {Ri, i = m,..., N}. The center line for a moving range chart is R. The control limits for a moving range chart are
LCL
UCL
max / ,
/
R Ad m d m
R Ad m d m
3 2
3 2
3 1 0 5^ 0 5^68
1 0 5^ 0 5 6
p and np Charts
The data for p and np charts are attribute data. Each measurement x (^) ij is either 0 or 1, where 1 indicates a non-conforming measurement. Therefore,
x (^) i xij j
ni
=
= (^) ∑ 1
is the count of non-conforming units for subgroup i. When a weight variable is specified, x^ i+ is^ a^ weighted^ sum^ of^ non-conforming^ units.^ If^ the^ data^ are aggregated and the value of the count variable is greater than the total number of units for any subgroup, this subgroup is dropped.
Equal Sample Sizes
Assume n (^) i = nThe process for a p chart is ; p (^) i : i = 1 , K,K@where p (^) i = x (^) i+ / n. The center line for a p chart is
p K
pi i
K
=
∑
1
and the control limits are
LCL
UCL
= − −
= + −
max / ,
min / ,
p A p p n
p A p p n
2 0 57
2 0 57
The process for an np chart is ;x (^) i+ : i = 1 , K,K@. The center line for an np chart is
x K
x (^) i i
K = (^) + =
∑
1
and the control limits are
LCL
UCL
max ,
min ,
x A np p
x A np p n
4 0 5 9
4 0 5 9
Unequal Sample Sizes
The process for a p chart is (^) ; p (^) i :i = 1 , K,K@ where p (^) i = x (^) i + / ni. The center line for a p chart is
p N
x N i n p i
K i i i
K = (^) += = =
∑ ∑
1 1
Equal Sample Sizes
Assume ni = n. The process for a u chart is ;u (^) i :i = 1 , K,K@ where u (^) i = x (^) i+ / n. The center line for a u chart is
u K
u (^) i i
K
=
∑
1
and the control limits are
LCL
UCL
max / ,
/
u A u n
u A u n
3 08
The process for a c chart is ;x (^) i+ :i = 1 , K,K@. The center line for a c chart is
c K
x (^) i i
K = (^) + =
∑
1
and the control limits for a c chart are
LCL
UCL
max c A c,
c A c
3 08
Unequal Sample Sizes
The process for a u chart is ;u (^) i : i = 1 , K,K@ where u (^) i = x (^) i + / niui = xi+ / ni. The center line for a u chart is
u N
x (^) i i
K = (^) + =
∑
1
and the control limits are
LCL
UCL
max / ,
/
u A u n
u A u n
i
i
3 08
The process for a c chart is ; x (^) i+ : i = 1 , K,K@. The center line for subgroup i is n ui and the control limits are
LCL
UCL
max n u A n u,
n u A n u
i i
i i
3 08
Note: A warning message is issued when a c chart is requested for subgroups of unequal sample sizes.
References
ASTM Special Technical Publication (STP) 15D 1990. Manual on presentation of data and control chart analysis. 6th ed. Prepared by Committee E-11 on Quality and Statistics, American Society for Testing and Materials. Philadelphia, Penn.
Grant, E. L., and Leavenworth, R. S. 1980. Statistical quality control: 5th ed. New York: McGraw-Hill.
Harter, H. L. 1969. Order statistics and their use in testing and estimation, Volumes 1 and 2. Aerospace Research Laboratories, United States Air Force. Washington, D.C.: U.S. Government Printing Office.
