CP, and CPK, PP, and PPK

Cp = (USL – LSL)/ 6σ

CPU = (USL – X)/ 3 σ

CPL = (LSL – X)/ 3 σ

CPK = minimum of CPU and CPL

PP , Cp = Capability Index ( spread of data/process) PPK , CPK = Performance index ( centralization of data/process)

For short term data For long term data

Standard Deviation

( σ )= ∑ (X-X )2/(n-1) Standard Deviation

( σ )= R/dn

In both cases all formulas stay same, only formula for standard deviation changed, and notification changed from C to P.

Pp = (USL – LSL)/ 6σ PPU = (USL – X )/ 3 σPPL = (LSL – X )/ 3 σPPK = minimum of PPU and PPL

All of 1st understand CP and CPK Graphically

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0 1 2 3 4 5 6 7 8 9 10111213141516171819

02468101214

Cent

er li

neAxis Title

Axis Title

US

L

LSL

CP

CPL CPU

All Data looks to be inside of this yellow curve, mean good spread of data. Mean good CP. But all data is not centralized, it is towards LSL , So we can get high number of rejections, mean – CPK is not good.

Upper specified limitLower specified limit

Desired location of data, centralized.

good Cp K

To study, Cp, CpK, Pp , PpK you must know HISTOGRAM .

If you don’t know , then 1st study it, and next 4-slides will explain it.

And if you already know, then that’s great, just skip, next 4 -slides.

3 4 5 6 7 8 9 10 112 3 4 5 6 7 8 9 10

0

2

4

6

8

10HISTOGRAM

Lower limit

Upper limit

Trend line

It is used to observe that , how is the process going. Or we can say, use to predict future performance of a process. Any change in process. It is simply a bar chart, from which we get, info of the process- how its going, it is in limits or not.

HISTOGRAM

How to build HISTOGRAM in Excel.(with example)

1.) 1st we have to Study/Collect specifications like -diameter (Data) for 24 products.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24D DIA 6 5 7 10 9 8 4 7 5 6 7 7 8 6 8 7 5 7 8 6 7 7 8 8

2.) Calculate RANGE.RANGE= Maximum value - Minimum valueSo here , maximum value= 10, Minimum value = 4So RANGE = 10- 4 =6

3.) Now decide the NUMBER OF CELLS. Data Points Number of Cells

20 -50 6 51-100 7101-200 8201-500 9501-1000 10Over 1000 11-20

We have 24 data points , and it fall in 1st group ,

so- No of cells = 6

4.) Calculate the approximately cell width.Cell width= RANGE/ NO OF CELLS = 6/6 =1

5.) Round Off the cell width.If cell width come in a complicated manner, like 0.34, 0.89 or else , then round off it to , one you want, like : 0.50 or 1 or else.

Go to yellow block, type, =frequency( D1:D24, B1:B10), and press Enter. Then select yellow block and all sky blue blocks, press F2, and press CTRL , SHIFT, ENTER. ( frequency formula will get implement in all sky blue blocks as in yellow block )And you will get frequency of data values in each group, As in group (7 – 8) , frequency is 6.

6.) Now construct the Cell Groups with keep in mind cell width( cell width=1)

2 33 44 55 66 77 88 99 10

10 1111 12

Cell width=1

Cell width same for all cell groups =1

7.) Now find number of data values/ Frequencies in each Cell Group.You can do this manually , by counting itself or by using formula .(frequency formula) Mean how many values fall in each group.

A B Cfrequency

1 2 3 02 3 4 13 4 5 34 5 6 45 6 7 86 7 8 67 8 9 18 9 10 19 10 11 010 11 12 0

cell groups (D1:D24) values are on previous page)

8.) Now we got frequency data in each group, now we can build Histogram.

frequency data is our final data.

now select this data a build a bar chart. That’s it.

3 4 5 6 7 8 9 10 112 3 4 5 6 7 8 9 10

0

2

4

6

8

10

Dia (mm)

Freq

uenc

y

Group 4 - 5, show values from 4.1 to 5Group 5 - 6, show values from 5.1 to 6

So this rule for all groups.

Trend line – also give an visual idea of moving

process.

sample lot 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

sample 1 5.5 5 5.4 5.2 5.1 5.1 5.4 5.5 5 5 5.2 5.7 5.4 5.2 5.2sample 2 5.1 5 5 5.2 5 5.2 5 5.1 5 5.1 5.4 5 5 5.2 5sample 3 5 5 5.1 5.4 5 5 5 5 5.5 5.1 5 5.2 5.4 5 5sample 4 5.2 5.5 5.3 5.5 5.1 5.2 5.5 5.4 5.1 5 5.3 5.5 5.2 5 5.1sample 5 5.4 5.1 5.4 5.4 5.1 5.5 5 5.2 5.2 5 5.1 5.4 5.4 5.1 5.4

Range(R ) 0.5 0.5 0.4 0.3 0.1 0.5 0.5 0.5 0.5 0.1 0.4 0.7 0.4 0.2 0.4 0.4average

dn = 2.326 ( for 5 samples, from table 1.1, last slide)

R

σ = R/dn = 0.17196

LETS FIND , CP, AND CPK FOR SHORT TERM DATA.

USL = 6.2, and LSL = 4.2 ( GIVEN)Data Points Number of Cells 20 -50 651-100 7101-200 8201-500 9501-1000 10Over 1000 11-20

USL = 6.2, and LSL = 4.2

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.63.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4

0

5

10

15

20

25

30

LSL

US

L

CELL GROUP frequency

3.8 4 0

4 4.2 0

4.2 4.4 0

4.4 4.6 0

4.6 4.8 0

4.8 5 24

5 5.2 27

5.2 5.4 15

5.4 5.6 8

5.6 5.8 1

5.8 6 0

6 6.2 0

6.2 6.4 0

6.4 6.6 0

• Cp = (USL – LSL)/ 6σ = 1.938333333

• CPU = (USL – X)/ 3 σ = 1.957716667

• CPL = (LSL – X)/ 3 σ = 1.91895

• CPK = minimum of CPU and CPL = 1.9188

RANGE = (max-min ) = 0.7NUMBER OF CELLS = 7CELL WIDTH = 0.7/7 = 0.1ROUND OFF = 0.2

standard deviation

σ = R/dn = 0.171969046

Lets find Pp and PpK for long term data

X (X - X ) (X - X )21 5.5 -0.26 0.070222 5.1 0.135 0.018233 5 0.235 0.055234 5.2 0.035 0.001235 5.4 -0.16 0.027226 5 0.235 0.055237 5 0.235 0.055238 5 0.235 0.055239 5.5 -0.26 0.07022

10 5.1 0.135 0.0182311 5.4 -0.16 0.0272212 5 0.235 0.0552313 5.1 0.135 0.0182314 5.3 -0.06 0.0042215 5.4 -0.16 0.0272216 5.2 0.035 0.0012317 5.2 0.035 0.0012318 5.4 -0.16 0.0272219 5.5 -0.26 0.0702220 5.4 -0.16 0.02722

SUM 104.7 0.6855AVERAGE 5.235

Standard Deviation

( σ )= ∑ (X-X )2/(n-1) = 0.6855/(20-1)= 0.1899

∑(X-X )2

Data Points Number of Cells 20 -50 651-100 7101-200 8201-500 9501-1000 10Over 1000 11-20

4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.64 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4

0

1

2

3

4

5

6

7

LSL

US

L

Standard Deviation

( σ )= ∑ (X-X )2/(n-1) Pp = (USL – LSL)/ 6σ = 1.75PPU = (USL – X )/ 3 σ = 1.69PPL = (LSL – X )/ 3 σ = 1.81PPK = minimum of PPU and PPL = 1.81

= 0.6855/(20-1)= 0.1899

frequency4 4.2 0

4.2 4.4 04.4 4.6 04.6 4.8 04.8 5 55 5.2 6

5.2 5.4 65.4 5.6 35.6 5.8 05.8 6 06 6.2 0

6.2 6.4 06.4 6.6 0

cell group

Range = (max – min ) = 0.5Number of cells = 6Cell width = 0.5/6 = 0.08Round off cell width = 0.2

USL= 6.2 , LSL= 4.2 (GIVEN)

PPM – Parts rejected Per Million

CPK PPMPpk PPM

0 500,0000.1 382,0890.2 274,2530.3 184,0600.4 115,0700.5 66,8070.6 35,9300.7 17,8640.8 8,1980.9 3,4671 1,350

1.1 4831.2 1591.3 481.4 131.5 31.6 11.7 0.1701.8 0.0331.9 0.0062 0.001

PPM and Ppk(Assumes all PPM out on same side)

0

100,000

200,000

300,000

400,000

500,000

600,000

0 0.5 1 1.5 2 2.5

Ppk

PPM

0.5 133,614

0.75 24,449

1 2,700

1.1 967

1.2 318

1.3 96

1.4 27

1.5 7

1.6 2

1.7 0.34

1.8 0.067

1.9 0.012

2 0.0018

CP PPM

Sample LCL UPL LCL UCL

Size = N A2 A3 dn D3 D4 B3 B4

2 1.88 2.659 1.128 0 3.267 0 3.267

3 1.023 1.954 1.693 0 2.574 0 2.568

4 0.729 1.628 2.059 0 2.282 0 2.266

5 0.577 1.427 2.326 0 2.114 0 2.0896 0.483 1.287 2.534 0 2.004 0.03 1.97

7 0.419 1.182 2.704 0.076 1.924 0.118 1.882

8 0.373 1.099 2.847 0.136 1.864 0.185 1.815

9 0.337 1.032 2.97 0.184 1.816 0.239 1.761

10 0.308 0.975 3.078 0.223 1.777 0.284 1.716

11 0.285 0.927 3.173 0.256 1.744 0.321 1.679

12 0.266 0.886 3.258 0.283 1.717 0.354 1.646

13 0.249 0.85 3.336 0.307 1.693 0.382 1.618

14 0.235 0.817 3.407 0.328 1.672 0.406 1.594

15 0.223 0.789 3.472 0.347 1.653 0.428 1.572

16 0.212 0.763 3.532 0.363 1.637 0.448 1.552

17 0.203 0.739 3.588 0.378 1.622 0.466 1.534

18 0.194 0.718 3.64 0.391 1.608 0.482 1.518

19 0.187 0.698 3.689 0.403 1.597 0.497 1.503

20 0.18 0.68 3.735 0.415 1.585 0.51 1.49

21 0.173 0.663 3.778 0.425 1.575 0.523 1.477

22 0.167 0.647 3.819 0.434 1.566 0.534 1.466

23 0.162 0.633 3.858 0.443 1.557 0.545 1.455

24 0.157 0.619 3.895 0.451 1.548 0.555 1.445

25 0.153 0.606 3.931 0.459 1.541 0.565 1.435

X-bar Chart for sigma R Chart Constants S Chart Constants

Table 1.1

• That’s it .

• I Hope you got it.

• Have any question, please let me know.