Steps in Constructing a p-Chart

Attribute control charts arise when items are compared with some standard and then are classified as to whether they meet the standard or not. The control chart is used to determine if the rate of nonconforming product is stable and detect when a deviation from stability has occurred. The argument can be made that a LCL should not exist, since rates of nonconforming product outside the LCL is in fact a good thing; we WANT low rates of nonconforming product. However, if we treat these LCL violations as simply another search for an assignable cause, we may learn for the drop in nonconformities rate and be able to permanently improve the process.

p Charts can be used when the subgroups are not of equal size. The np chart is used in the more limited case of equal subgroups.

Steps in Constructing a p Chart

1. Determine the size of the subgroups needed. The size, n(i), has to be sufficiently large to have defects present in the subgroup most of the time. If we have some idea as to what the historical rate of nonconformance, p, is we can use the following formula to estimate the subgroup size:

n=3/p

2. Determine the rate of nonconformities in each subgroup by using:
3. phat(i)=x(i)/n(i)

where:

phat(i)=the rate of nonconformities in subgroup i

x(i)=the number of nonconformities in subgroup i

n(i)= the size of subgroup i

4. Find pbar; there are k subgroups.
5. Estimate sigma-p if needed and determine the UCL and LCL:
6. Plot the centerline, pbar, the LCL and UCL, and the process measurements, the phat’s.
7. Interpret the data to determine if the process is in control.
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Example:

```Farnum Example:
data is from Farnum (1994):
Modern Statistical Quality Control and Improvement, p. 242
Number
Day	Rejects	Tested	Proportion
1	14	286	0.0490
2	22	281	0.0783
3	9	310	0.0290
4	19	313	0.0607
5	21	293	0.0717
6	18	305	0.0590
7	16	322	0.0497
8	16	316	0.0506
9	21	293	0.0717
10	14	287	0.0488
11	15	307	0.0489
12	16	328	0.0488
13	21	296	0.0709
14	9	296	0.0304
15	25	317	0.0789
16	15	297	0.0505
17	14	283	0.0495
18	13	321	0.0405
19	10	317	0.0315
20	21	307	0.0684
21	19	317	0.0599
22	23	323	0.0712
23	15	304	0.0493
24	12	304	0.0395
25	19	324	0.0586
26	17	289	0.0588
27	15	299	0.0502
28	13	318	0.0409
29	19	313	0.0607
30	12	289	0.0415
Calculations:
PBAR =	0.0539
UCL =	pbar + 3*sqrt(pbar*(1-pbar)/n(i))
LCL =	pbar - 3*sqrt(pbar*(1-pbar)/n(i))
Day	CL	UCL		LCL		Proportion
1	0.0539	0.093892049	0.013808661	0.0490
2	0.0539	0.094246721	0.013453989	0.0783
3	0.0539	0.092310827	0.015389883	0.0290
4	0.0539	0.092126068	0.015574642	0.0607
5	0.0539	0.093410843	0.014289867	0.0717
6	0.0539	0.092624795	0.015075915	0.0590
7	0.0539	0.091587368	0.016113342	0.0497
8	0.0539	0.091943946	0.015756764	0.0506
9	0.0539	0.093410843	0.014289867	0.0717
10	0.0539	0.093822229	0.013878481	0.0488
11	0.0539	0.092498288	0.015202422	0.0489
12	0.0539	0.091240619	0.016460091	0.0488
13	0.0539	0.093209857	0.014490853	0.0709
14	0.0539	0.093209857	0.014490853	0.0304
15	0.0539	0.091883814	0.015816896	0.0789
16	0.0539	0.09314354	0.01455717	0.0505
17	0.0539	0.094103724	0.013596986	0.0495
18	0.0539	0.091646103	0.016054607	0.0405
19	0.0539	0.091883814	0.015816896	0.0315
20	0.0539	0.092498288	0.015202422	0.0684
21	0.0539	0.091883814	0.015816896	0.0599
22	0.0539	0.091528906	0.016171804	0.0712
23	0.0539	0.092688517	0.015012193	0.0493
24	0.0539	0.092688517	0.015012193	0.0395
25	0.0539	0.091470715	0.016229995	0.0586
26	0.0539	0.093683678	0.014017032	0.0588
27	0.0539	0.093011904	0.014688806	0.0502
28	0.0539	0.091823966	0.015876744	0.0409
29	0.0539	0.092126068	0.015574642	0.0607
30	0.0539	0.093683678	0.014017032	0.0415
p - Chart:

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