## Key Points

- p-Charts are a control chart used to see nonconforming data in a set of data.
- They can be used to pinpoint issues for further process improvement.
- Proper usage allows you to see the proportion of data falling outside of conformance.

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 an LCL should not exist, since rates of nonconforming products outside the LCL are a good thing; we WANT low rates of nonconforming products. However, if we treat these LCL violations as simply another search for an assignable cause, we may learn from 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.

## Why Use a p-Chart?

You’ll end up with quite a few data points throughout a project. As such, seeing where those points stack up is a handy way of optimizing your production. This is where the p-chart shines. By allowing you to see where points are exceeding the UCL or LCL, you can make adjustments as needed.

**Steps in Constructing a p-Chart**

- 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

- Determine the rate of nonconformities in each subgroup by using:

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

- Find pbar; there are k subgroups.

- Estimate sigma-p if needed and determine the UCL and LCL:

- Plot the centerline, pbar, the LCL and UCL, and the process measurements, the phat’s.
- Interpret the data to determine if the process is in control.

**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:

## Other Useful Tools and Concepts

Now that we’ve finished up with the calculations, you might be hungry for a little more. Having an out-of-control action plan is a handy way of keeping everything in check even when things go pear-shaped. Learning how OCAPs can impact your business is one of the most important steps in Six Sigma.

Additionally, you’ll be dealing with variations of all sorts when going through statistical analysis. Learning the differences between common cause and special cause variation is going to save a lot of time and energy on your part. You can learn all about it in our exhaustive guide.