01 process capability

Design for Manufacturing and Assembly

Process Capability Analysis

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PROCESS CAPABILITY ANALYSIS

PROCESS CAPABILITY (CP):

Process capability is the repeatability and consistency of a manufacturing

process relative to the customer requirements in terms of specification limits of a

product parameter. This measure is used to objectively measure the degree to which

your process is or is not meeting the requirements.

Process capability compares the output of an in-control process to the

specification limits by using capability indices. The comparison is made by forming the

ratio of the spread between the process specifications (the specification "width") to the

spread of the process values, as measured by 6 process standard deviation units (the

process "width").

Cp = (USL - LSL) / 6 sigma

Cp<1 means the process variation exceeds specification, and a significant number of

defects are being made.

Cp=1 means that the process is just meeting specifications. A minimum of .3% defects

will be made and more if the process is not centered.

Cp>1 means that the process variation is less than the specification, however, defects

might be made if the process is not centered on the target value.



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While Cp relates the spread of the process relative to the specification width, it does

not address how well the process average, X, is centered to the target value. Cp is

often referred to as process "potential".

We define process capability analysis as an engineering study to estimate process

capability. The estimate of process capability may be in the form of a probability

distribution having a specified shape, center (mean), and spread (standard deviation).

For example, we may determine that the process output is normally distributed with

mean 0.1=µ cm and standard deviation 001.0=σ cm. in this sense, a process capability

analysis may be performed without regard to specifications on the quality

characteristic.



3

PROCESS CAPABILITY INDICES (CPK):

A capable process is one where almost all the measurements fall inside the

specification limits. This can be represented pictorially by the plot below:

The Cp, Cpk, and Cpm statistics assume that the population of data values is normally

distributed. Assuming a two-sided specification, if and are the mean and standard

deviation, respectively, of the normal data and USL, LSL, and T are the upper and

lower specification limits and the target value, respectively, then the population

capability indices are defined as follows:

The estimator for Cpk can also be expressed as Cpk = Cp(1-k), where k is a scaled

distance between the midpoint of the specification range, m, and the process mean, .

Denote the midpoint of the specification range by m = (USL+LSL)/2. The distance

between the process mean, , and the optimum, which is m, is - m, where

. The scaled distance is



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(the absolute sign takes care of the case when ). To determine the

estimated value, , we estimate by . Note that .

The estimator for the Cp index, adjusted by the k factor, is

Since , it follows that .

To get an idea of the value of the Cp statistic for varying process widths, consider the

following plot

This can be expressed numerically by the table below:

where ppm = parts per million and ppb = parts per billion. Note that the reject figures

are based on the assumption that the distribution is centered at .Values of the

Process Capability Ratio (Cp) and Associated



5

We have discussed the situation with two spec. limits, the USL and LSL. This is known

as the bilateral or two-sided case. There are many cases where only the lower or

upper specifications are used. Using one spec limit is called unilateral or one-sided.

The corresponding capability indices are

where and are the process mean and standard deviation, respectively

Estimators of Cpu and Cpl are obtained by replacing and by and s, respectively.

The following relationship holds

Cp = (Cpu + Cpl) /2.

This can be represented pictorially by

Note that we also can write:

Cpk = min {Cpl, Cpu}.



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CAPABILITY INDEX EXAMPLE:

For a certain process the USL = 20 and the LSL = 8. The observed process average,

= 16, and the standard deviation, s = 2. From this we obtain

This means that the process is capable as long as it is located at the midpoint,

m = (USL + LSL)/2 = 14.

But it doesn't, since = 16. The factor is found by

and

We would like to have at least 1.0, so this is not a good process. If possible,

reduce the variability or/and center the process. We can compute the and

From this we see that the , which is the smallest of the above indices, is 0.6667.

Note that the formula is the algebraic equivalent of the min{ ,

} definition.

PROCESS CAPABILITY METRICS:

For processes that are in statistical control and that are normally distributed, we

can do a Process Capability Analysis. Last month's e-zine contained an explanation of

process capability and introduced one metric (Cp) to measure process capability. This

month's e-zine introduces a second metric to measure process capability - Cpk.



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Cpk

EXAMPLE FOR CPK :



8

Cp values are not the best indicators of process capability. As shown last month, Cp is

the ratio of the engineering tolerance (USL - LSL) to the natural tolerance (6s). The

value of Cp does not take into account where the process is centered. Just knowing

that a process is capable (Cp > 1.0) does not ensure that all the product or service

being received is within the specifications. A process can have a Cp > 1.0 and

produce no product or service within specifications. In addition, Cp values can't be

calculated for one- sided specifications. A better measure of process capability is Cpk.

Cpk takes into account where the process is centered. The value of Cpk is the

minimum of two process capability indices. One process capability is Cpu, which is the

process capability based on the upper specification limit. The other is Cpl, which the

process capability is based on the lower specification limit. Algebraically, Cpk is

defined as shown in the figure.

Both Cpu and Cpl take into account where the process is centered. The value of Cpk

is the difference between the process average and the nearest specification limit

divided by three times the standard deviation. It should be noted that the standard

deviation is the standard deviation based on a R or s chart - not the standard deviation

of the individual measurements.

Cpk values above 1.0 are desired. This means that essentially no product or service is

being produced above USL or below LSL. The figure above shows how the Cpk

values are calculated. If Cpk is less than 1.0, this means that there is some product

being produced out of specification. If there is only one specification, the value of Cpk

is either Cpu or Cpl, whichever is appropriate for the specification.

A bagging operation is designed to place 50 pounds of sand into each bag. The

specifications for the operation are a minimum of 49.5 pounds and a maximum of 50.5

pounds. So, the lower specification limit (LSL) is 49.5. The upper specification limit

(USL) is 50.5 pounds.



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The operation is being monitored using a Xbar-R chart with a subgroup size of 4. Each

hour, four consecutive bags are weighed. The subgroup average and range are

calculated and plotted on an Xbar-R chart. The control chart is shown in the figure and

is in statistical control. The standard deviation (from the range chart data) is 0.212.

The average bag weight is 50.05. The calculations for Cpk are shown below. Since

Cpk is the minimum of Cpu and Cpl, Cpk = 0.71.



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Definitions Mean ( x ): The arithmetic mean of a set of ‘n’ numbers is the sum of the numbers divided by ‘n’. Mean is expressed algebrically,

,.........321

n

XXXXx n++++=

Where the symbol x represents the arithmetic mean.

nXXXX ,.....,, 321 , are the n values of the variate X

i.e., n

xx∑=

If X1 occurs f1 times, X2 occurs f2 times, etc and finally Xn occurs fn times, then,

nffffn ..........321 +++=

Then n

nn

ffff

XfXfXfXfx

++++++++

=.........

.........

321

332211

The mean is used to report average size, average yield, average percent defective etc.

Median: When all the observations are arranged in ascending or descending order, then the median is the magnitude of the middle case.

If n is odd, 2

1+=n

Median

If n is even, Median is average of

2

nth and

+12

nth value.

Where n = No of observations.

Mode: The mode of a set of data is the value which occurs most frequently.

Range(R): In the control chart, the range is difference between the largest observed value and the smallest observed value.

Variance ( 2σ ): It is defined as the sum of the squares of the deviations from the arithmetic mean divided by the number of observations ‘n’.

Variance (2σ ) =

( ) ( ) ( )n

xxxxxx n

22

2

2

1 −+−+−

Sample problems



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Example 1: The no of orders received for a particular item on each day for five days are as follows. Calculate the mode and variance. 1, 2, 0, 3, 2

Solution: Mode = 2 (it occurs more than the other values) This can be put more succinctly using the summation notation as;

Variance 2σ = ( )21

xxn

−∑

It is possible to rearrange this formula in a way which makes the calculation of the variance much easier in general.

Variance =2σ ( )221xx

n−∑ , To calculate x ,we use

Mean 6.15

23021=

+++++=x

Using this formula, the variance for the data used above is calculated as follows:

Variance 2σ =

5

1 (12 + 22 + 02 + 32 + 22 ) - 1.62

= 5

18 - 2.56

=2σ 1.04

Standard deviation = ( )2σ

Standard deviation = )04.1( = 1.02

Example 2: Calculate mean, variance, standard deviation for the given order size data.

Order size range

Class mark(x)

Frequency (f)

fx fx2

1-10 5.5 1 5.5 30.25

11-20 15.5 2 31.0 480.50

21-30 25.5 4 102.0 2601.00

31-40 35.5 12 426.0 15123.00

41-50 45.5 13 591.5 26913.25

51-60 55.5 8 444.0 24642.00

61-70 65.5 8 524.0 34322.00

71-80 75.5 1 75.5 5700.25

81-90 85.5 1 85.5 7310.25

50=∑ f 0.2285=∑ fx 50.1171222 =∑ fx



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Figure 1 Skewed curves

Mean x = 50

2285

= 45.7

Variance =2σ 50

5.117122 - (45.7) 2

= 253.96

Standard deviation = 96.253

= 15.9

Skewness: The curve, which does not follow the shape of the normal curve. These generally represent a purely temporary process condition, and serve as a guide to detecting the presence of some unusual factor like defective material, or abnormal machining conditions. (e.g.) tool chatter, tool vibration, etc. These curves are like normal curves in that the frequencies decrease continuously from the centre to extreme values, but unlike the normal curve they are not symmetrical. Their extreme values occur more frequently in one direction from the centre than in the other. They appear like “disturbed normal” curves and hence are called “skewed curves”. The normal distribution is the most commonly occurring symmetrical frequency distribution. Positive skewness is also quite common, as for instance the shape of the distribution of personal incomes. Another example is the distribution of the time intervals between randomly occurring events, such as the arrival of customers at the ends of a queue. Negative skewness is less common, but occurs, for instance, in the distribution of times to failure of certain types of equipment.

Coefficient of Skewness= σModeMean −

Several measures of skewness have been proposed, but are rarely used in practice. The simplest way of describing skewness is to quote the mean, the median, and,



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where possible the mode. For symmetrical distributions, these three measures will approximately coincide. For positively skewed distributions, the mode will be less than the median, which will in turn be less than the mean. This is very noticeable for the distribution of personal incomes.

For negatively skewed distributions, these three measures will be in the reverse order. The differences between the measures give some indication of the extent of the skewness. When the distribution is moderately, there is an approximate relationship between the three measures, expressed as Mean-Mode=3(Mean-Median).

Measure Of Skewness:

(A) Absolute Skeweness

(a) Absolute Sk = Mean-Mode (when mode is not ill-defined) (b) Absolute Sk = 3(Mean-Median)

(B) Relative Skewness

(a) Karl Pearson’s coefficient of Skewness.

Coefficient of Skewness= σModeMean −

When mode is ill-defined

Coefficient of Skewness= σ

)(3 ModeMean −

(b) Measure of Skewness based on moments:

With the help of moments Skewness can be determined, Karl Pearson suggested

1β as Measure of Skewness.

3

2

2

3

1µ

µβ =

For a symmetrical distribution 1β = 0.

Moments (i) Moments about Mean

N

XX )(1

−=∑µ

N

XX∑ −=

2

2

)(µ =

2σ or σ = 2µ



14

N

XX∑ −=

3

3

)(µ

N

XX∑ −=

4

4

)(µ

In case of frequency distribution N

XXf )(1

−=∑µ = 0

N

XXf∑ −=

2

2

)(µ =

2σ , etc.

(ii) Moments about arbitrary origin A

=1

1µ ( )N

AX∑ −

=1

2µ ( )N

AX∑ − 2

=1

3µ ( )N

AX∑ − 3

=1

4µ ( )N

AX∑ − 4

In a frequency distribution the moments about an arbitrary origin will be calculated as follows:

iN

fd×=∑1

1µ or ( )

iN

AXf×

−∑

2

2

1

2 iN

fd×=∑µ or

( )2

2

iN

AXf×

−∑

3

3

1

3 iN

fd×=∑µ or

( )3

3

iN

AXf×

−∑

4

4

1

4 iN

fd×=∑µ or

( )4

4

iN

AXf×

−∑

Where ‘ i ’ is the class interval and l=

−i

AX.

Order to simplify calculations the moments are first calculated about an origin A. They can then be converted with the help of the following relationships to obtain moments about mean.

µ1= µ11-µ1

1 =0

µ2 = µ21-( µ1

1)2

µ3 = µ31-(3µ1

1 µ21)+2 ( µ1

1)3

µ4 = µ41-(4µ1

1 µ31)+6 ( µ1

1)2 µ21 -3( µ1

1)4



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Example 3:

Calculate any measure of skewness from the following data:

X 0 1 2 3 4 5 6 7

f 12 27 29 19 8 4 1 0

Solution: Since the question is to calculate any measure of skewness, we should prefer Karl Pearson’s coefficient of skewness because it is considered to be the best measure for calculating skewness. The formula is:

Coefficient of Skewness=σModeMean −

Hence for calculating skewness we have to determine the values of mean, mode and standard deviation.

Calculation of Coefficient of Skewness

X f x-2 d

fd fd2

0 12 -2 -24 48

1 27 -1 -27 27

2 29 0 0 0

3 19 +1 +19 12

4 8 +2 +16 32

5 4 +3 +12 36

6 1 +4 +4 16

7 0 +5 0 0

N=100 ∑ = 0fd 1782 =∑ fd

x = A+ 2)100/0(2 =+=∑N

fd

Standard deviation

∑−

∑=

22

N

fd

N

fdσ

334.178.1

100

0

100

1782

==

−

=σ



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Mode: Since the highest frequency is 29, by inspection the mode is the value corresponding to the frequency 29 i.e. 2.

x=2 , Mo =2, σ =1.334

Substituting these values in the formula,

Coefficient of Skewness =σModeMean −

= 334.1

22 − = 0.

Example: 4 Calculate Karl Pearson’s co-efficient of skewness from the following data:

Size 1 2 3 4 5 6 7

Frequency 10 18 30 25 12 3 2

Solution:

Calculation of Karl Pearson’s Coefficient of Skewness

Size x

Frequency f

x-4 d

fd fd2

1 10 -3 -30 90

2 18 -2 -36 72

3 30 -1 -30 30

4 25 0 0 0

5 12 +1 +12 12

6 3 +2 +6 12

7 2 +3 +6 18

N=100 ∑ −= 72fd 2342 =∑ fd

Coefficient of Skewness=

σModeMean −

Mean: x = A+ N

fd∑ = 4 - (72/100) =3.28

Standard deviation

∑−

∑=

22

N

fd

N

fdσ

= 1.152 Mode: Since the maximum frequency is 30, by inspection the mode is the value corresponding to the frequency 30 i.e. 3.

28856.2100

72

100

2322

=

−−

=σ



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x=3.28 , Mo =3, σ =1.518

Substituting these values in the formula,

Coefficient of Skewness =σModeMean −

=

518.1

328.3 − = 0.184

Kurtosis: The fourth momentum will provide a numerical value associated with the peakedness or flatness of the data as it is a distributed about the mean also known as “kurtosis”. The following equation incorporates the fourth moment about the mean and the fourth power of the samples standard deviation to measure kurtosis.

( )41

4

1xx

ni

n

i

−= ∑=

µ

Kurtosis =4

4

s

µ

The following equation is commonly used to calculate the zero based kurtosis in statistical analysis computer programming.

Zero-based kurtosis= 31

1

4

−

−∑=

n

i

i

s

xx

n

Note that the value of 3 is subtracted from the kurtosis value. This force the value to be zero based, as opposed to be centered around the number 3. The common approach to quantity kurtosis is that the normal peak distribution is centered about the value 3. As the kurtosis deviates above or below 3. The peakedness or flatness begins to take a numerical significance as described below.

Mesokurtic: They are three general distributions types used to define nature of kurtosis. The first is mesokurtic distribution as shown in the Figure 2. In it the data is normal

distributed about the mean the kurtosis will be equal to 3.

Figure 2 Mesokurtic distribution



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Platykurtic: The second is platykurtic distribution, shown in figure 3. In it the data is dispersed

bout the mean in a manner that is flat in nature: the kurtosis will be less than 3.

Leptokurtic: The third is leptokurtic distribution, shown in figure 4. In the data is dispersed

about the mean in a manner that is very peaked in nature; the kurtosis will be greater than 3.

Figure 3 Platykurtic distribution

Figure 4 Leptokurtic distribution



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Measures Of Kurtosis: Kurtosis are measured by the coefficient

2

2

4

2µ

µβ =

or 322 −= βγ

For normal distribution 2β =3. If 2β is more than 3 the curve is leptokurtic and if it

less than 3 the curve is platykurtic.

Example: 5 Calculate first four moments from the following data:

Also calculate the values of β 1 and β 2 and comment on the nature of the

distribution X 0 1 2 3 4 5 6 7 8 Y 5 10 15 20 25 20 15 10 5 Solution:

Calculation of moments X f fX (X-4) f(X-4) f(x-4)2 f(X-4)3 f(X-4)4

0 5 0 -4 -20 80 -320 1280 1 10 10 -3 -30 90 -270 810 2 15 30 -2 -30 60 -120 240 3 20 60 -1 -20 20 -20 20 4 25 100 0 0 0 0 0 5 20 100 +1 +20 20 +20 2 6 15 90 +2 +30 60 +120 40 7 10 70 +3 +30 90 +270 10 8 5 40 +4 +20 80 +320 280 N=125 ∑fX ∑f(X-4) ∑ f(x-4)2 ∑ f(X-4)3 ∑ f(X-4)4

= 500 = 0 =500 =0 =4700

4125

500===∑

N

fXx

N

XXf )(1

−=∑µ



20

125,0)( ==−∑ NXXf

0125

01 ==µ

4125

500)( 2

2 ==−

=∑

N

XXfµ

0125

0)( 3

3 ==−

=∑N

XXfµ

6.37125

4700)( 4

4 ==−

=∑N

XXfµ

04

03

2

3

2

2

3

1 ===µ

µβ

Since β 1 is zero, the distribution is symmetrical

35.216

6.372

2

4

2 ===µ

µβ

Since β 2 is less than 3, the distribution is platykurtic.

Example: 6 Using moments, calculate a measure of relative skewness and a measure of relative kurtosis for the following distribution and comment on the result obtained: Weekly Wages No. of Workers (Rs) 70 but below 90 8 90 “ 110 11 110 “ 130 18 130 “ 150 9 150 “ 170 4



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Solution:

Weekly wages

(Rs) f m.p d fd fd2 fd3 fd4

70-90 8 80 -2 -16 32 -64 128

90-110 11 100 -1 -11 11 -11 11

110-130 18 120 0 0 0 0 0

130-150 9 140 1 9 9 9 9

150-170 4 160 2 8 16 32 64

N=50 ∑fd =-10 ∑fd2=68 ∑ fd3=-34 ∑fd4=212

42050

101

1 −=×−

=×=∑ iN

fdµ

54440050

682

2

1

2 =×=×= ∑ iN

fdµ

5440800050

343

3

1

3 −=×−

=×=∑ iN

fdµ

67840016000050

2124

4

1

4 =×=×=∑ iN

fdµ

Moment about Mean

µ2 = µ21-( µ1

1)2 =544-(-4)2=528

µ3 = µ31-3(µ1

1 µ21)+2 ( µ1

1)3 = -5440-3(-4) (544) +2(-4)3

= 960.

µ4 = µ41-4(µ1

1 µ31)+6 ( µ1

1)2 µ21 -3( µ1

1)4

= 674800-4(-4) (-5440) +6(-4)2(544)-3(-4)4

= 642816



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Skewness= 08.0528

9603

2

3

2

2

3

1 ===µ

µβ

306.2278784

6428162

2

4

2 ===µ

µβ

Since β 2 is less than 3, the distributions platykurtic.

Example: 7 Calculate coefficient of skewness by Karl Pearson’s method and the values of β 1

and β 2 from the following data:

Profits(Rs. Lakhs) 10-20 0

20-30 30-40 40-50 50-60

No. of companies 18

20 30 22 10

Solution:

Calculation of Karl Pearson’s Coefficient of Skewness β 1 and β 2

Profits (Rs.

Lakhs)

No. of cos f

m.p m

(m-45)/10

d

fd

fd2

fd3

fd4

10-20 18 15 -2 -36 72 -144 288

20-30 20 25 -1 -20 20 -20 20

30-40 30 35 0 0 0 0 0

40-50 22 45 +1 +22 22 +22 22

50-60 10 55 +2 +20 40 +80 160

N=100 ∑ −= 14fd ∑ =1542fd ∑ −= 623fd ∑ = 4904fd

Karl Pearson’s Coefficient of Skewness=

σModeMean −

Mean: x = A+ iN

fd×∑

A=35, ∑ fd =-14 , N=100, i=10

x = 35+ 6.334.13510100

14=−=×

Mode: Mode= iL ×∆−∆

∆+

21

1

By inspection mode lies in the class 30-40



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L=30 10,82230,102030 21 ==−=∆=−=∆ i

Mode= 30+ 10810

10×

+= 30 + 5.56 = 35.56

Standard deviation

×

∑−

∑= i

N

fd

N

fd22

σ

Karl Pearson’s Coefficient of Skewness=33.12

56.356.33 −=-0.159

Calculation of 3

2

2

3

1µ

µβ =

We will have to calculate moments

4.110100

141

1 −=×−

=×=∑ iN

fdµ

154100100

1542

2

1

2 =×=×=∑ iN

fdµ

6201000110

623

3

1

3 −=×−

=×=∑ iN

fdµ

4900010000100

4904

4

1

4 =×=×=∑ iN

fdµ

µ2 = µ2

1-( µ11)2 =154-(-1.4)2=152.04

µ3 = µ3

1-3(µ11 µ2

1)+2 ( µ11)3

= -620-3(-1.4) (154) +2(-1.4)3

= 20.32

µ4 = µ41-4(µ1

1 µ31)+6 ( µ1

1)2 µ21 -3( µ1

1)4

= 49000-{4(-1.4) (-620)} +{6(-1.4)2(154)-{3(-1.4)4} = 47327.516

33.1210100

14

100

1542

=

×

−−

=σ



24

( )( )

( )047.2

04.152

516.47327

00013.01.3514581

54.454

04.152

32.21

22

2

4

2

2

2

2

2

2

3

1

===

====

µ

µβ

µ

µβ

Process Capability Analysis: Statistical techniques can be helpful throughout the product cycle, including development activities prior to manufacturing, in quantifying process variability, in analyzing this variability relative to product requirements or specifications, and in assisting development and manufacturing in eliminating or greatly reducing this variability. This general activity is called process capability analysis. Product capability refers to the uniformity of the process. Obviously, the variability in the process is a measure of the uniformity of output. There are two ways to think of this variability:

1. The natural or inherent variability at a specified time; that is, “Instantaneous” variability. 2. The variability over time. ,

We present methods for investigating and assessing both aspects of process capability. It is customary to take the 6-sigma spread in the distribution of the product quality characteristic as a measure of process capability. Figure 5 shows a process for which the quality characteristic has a normal distribution with mean µ and standard

deviationσ . The upper and lower “natural tolerance limits” (UNTL & LNTL) of the

process fall at σµ 3+ and0 σµ 3− , respectively. That is,

σµ 3+=UNTL

σµ 3−=LNTL

For a normal distribution, the natural tolerance limits include 99.73% of the variable, or put another way, only 0.27% of the process output will fall outside the natural tolerance limits. Two points should be remembered:

1.0.27% outside the natural tolerances sounds small, but this corresponds to 2700 nonconforming parts per million. 2. If the distribution of process output is nonnormal, then the percentage of

output falling outside σµ 3± may differ considerably from 0.27%.

We define process capability analysis as an engineering study to estimate process capability. The estimate of process capability may be in the form of a probability distribution having a specified shape, center (mean), and spread (standard deviation). For example, we may determine that the process output is normally distributed with mean 0.1=µ cm and standard deviation 001.0=σ cm. in this sense, a process

capability analysis may be performed without regard to specifications on the quality characteristic. Alternatively, we may express process capability as a percentage outside of specifications. However, specifications are not necessary to perform a process capability analysis.



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A process capability study usually measures functional parameters on the product, not the process itself. When the analyst can directly observe the process and can control or monitor the data-collection activity, the study is a true process capability study, because by controlling the data collection and knowing the time sequence of the data, interferences can be made about the stability of the process over time. However, when we have available only sample units of products, perhaps supplied by the vendor or obtained via receiving inspection, and there is no direct observation of the process or time history of production, then the study is more properly called product characterization. In a characteristic or the process yield (fraction conforming to specifications); we can say nothing about the dynamic behavior of the process or its state of statistical control. Process capability analysis is a vital part of an overall quality-improvement program. Among the major uses of data from a process capability analysis are the following:

1. Predicting how well the process will hold the tolerances.

2. Assisting product developers/designers in selecting or modifying a process. 3. Assisting in establishing an interval between sampling for process monitoring. 4. Specifying performance requirements for few equipment. 5. Selecting between competing vendors. 6. Planning the sequence of production process when there is an interactive effect of process on tolerances. 7. Reducing the variability in a manufacturing process.

Thus, process capability analysis is a technique that has application in many segments of the product cycle, including product and process design, vendor sourcing, production or manufacturing planning, and manufacturing. Three primary techniques are used in process capability analysis: histograms or probability plots, control charts, and designed experiments.

Figure 5 Upper and Lower natural tolerance limits in the normal distribution.



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Six-Sigma:

Sigma (σ ) is a character of the Greek alphabet which is used in mathematical

statistics to define standard deviation. The standard deviation indicates how tightly all the various examples are clustered around the mean in a set of data.

Six Sigma is a business method for improving quality by removing defects and their causes in business process activities. It concentrates on those outputs which are important to customers. The method uses various statistical tools to measure business processes. In technical terms, Six Sigma means that there are 3.4 defects per million events. The main goal is continuous improvement.

Six Sigma is carried out as projects. Most common type is the DMAIC method (Define, Measure, Analyze, Improve, and Control). First, the project and the process to be improved are defined after which the performance of the process is measured. The data is then analyzed and bottle-necks and problems identified. After analysis, improvement program is defined and defects removed. This development program is controlled by a management group. After DMAIC circle it is time to define a new project.

Example: 7 This sample example at GE, illustrates how the concept of Six Sigma affects different people. Average Vs Variation Customer expectations: 8 day order to Delay Cycle.

Internal Look

Existing Process After conventional Delay Cycle (days) improvements (days) 20 17 15 2 30 5 10 12 5 4 16 Days (Average) 8Days (Average) “Internal Calibration “= 16 – 8 = 8 Therefore improvement is 50%



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CUSTOMER LOOK

CUSTOMER LOOK

GE employees claimed that they had achieved Six Sigma capability after improving the delivery time for a medical product by 50% (brining it from an average of 16 days to average of 8 days).But this effect was not reflected on the customer’s side as they were still getting their products delivered at random as seen from the Figure 6 From the above Figure Customer feels no change And once the feedback from the customer was heard, they modified the process to reflect Six Sigma delivery for the customer which resulted in the following:

6Sigma Internal Process

7 9 9 8 7 8 Days (Average) Here the internal look is same. But the customer feels Six Sigma (Figure 7).

Figure 6

Figure 7



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Process Capability Ratios:

Use and interpretation of Cp It is frequently convenient to have a sample, quantitative way to express process capability.

σ6LSLUSL

Cp−

= ----------------------------------------- (2-1)

Where USL and LSL are the upper and lower specification limits, respectively. Usually, the process standard deviation σ is unknown and must be replaced by an

estimateσ . To estimate σ we typically use either the sample standard deviation S or

R /d2 (when variables control charts are used in the capability study). This results in an estimate of the Cp, say

σ6LSLUSL

Cp−

= --------------------------------------------- (2-2)

To illustrate the calculation of the Cp, The specifications on piston-ring diameter are

USL=74.05mm and LSL=73.95mm, and σ =0.0099.

Thus, our estimate of the Cp is

σ6LSLUSL

Cp−

=

)0099.0(6

95.7305.74 −=

68.1=

We assumed that piston-ring diameter is approximately normally distributed and the cumulative normal distribution table in the appendix was used to estimate that the process produces approximately 20PPM (Parts Per Million) defective. The Cp in equation (2-1) has a useful practical interpretation, namely

1001

=

CpP ---------------------------------------------- (2-3)

Is the percentage of specification band used up by the process. The piston-ring process uses

10068.1

1

=P

5.59= percent of the specification band.

Equation (2-1) and (2-2) assume that the process has both upper and lower specification limits. For one-sided specifications, we define the Cp as follows.

σµ

3

−=USL

C pu (upper specification only)---

-------------- (2-4)



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σµ

3

LSLC pl

−= (lower specification only) --

--------------- (2-5)

Estimate UCp and LCp would be obtained by replacing µ and σ in equation (2-4)

and (2-5) by estimate µ and σ , respectively.

The process capability ratio is a measure of the ability of the process to manufacture

product that means specification table 2.1 shows several values of Cp along with the associated values of process fallout, expressed in defective are non-conforming parts per million. This process fallout were calculated assuming a normal distribution of the quality characteristics, and the case of two sides specification, assuming the process mean is centered between the upper and lower specification limits. These assumptions are essential to the accuracy of the reported numbers, and if they are not true,

Table 2.1 Values of the Process Capability Ratio (Cp) and Associated Process Fallout for a normal distribution process (in defective PPM)

Cp

A process fallout(in defective PPM)

One side Specifications Two-sided specifications

0.25 226,628 453,255

050 66,807 133,614

0.60 35,931 71,861

0.70 17,865 35,729

0.80 8,198 16,395

0.90 3,467 6,934

1.00 1,350 2,700

1.10 484 967

1.20 159 318

1.30 48 96

1.40 14 27

1.50 4 7

1.60 1 2

1.70 0.17 0.34

1.80 0.03 0.06

2.00 0.0009 0.0018

Then the table is invalid. To illustrate the use of table, notice, that the Cp of one implies a fallout rate of 2700 PPM of two sides specifications, while the Cp of 1.5 implies fallout rate of 4 PPM of one side specification.



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Table2.2 Recommended Minimum Values of the Process Capability Ratio (Cp)

Two-sided specifications One-sided specifications

Existing processes 1.33 1.25

New processes 1.50 1.45

Safety, strength, or critical parameter, existing process

1.50 1.45

Safety, strength, or critical parameter, new process

1.67 1.60

Table2.2 represents some recommended guidelines for minimum values of Cp the bottle strength characteristics a parameter closely related to the safety of product, bottles with inadequate pressure strength may fail and injury customers. This implies that the Cp should be atleast 1.45 perhaps one way the Cp could be improved would be increasing the mean strength of the bottles, say by pouring more glass in the mould. We point out that the values in the table 2.2 are only recommended minimum. In recent years, many companies have adopted criteria for evaluating their processes that include process capability objectives that are more stringent that those of table2.2. For example, Motorola’s “six-sigma” program essentially requires that when the process mean in control, it will not be closer that six standard deviations from the nearest specification limit. This, in effect, requires that the process capability ratio will be least 2.0. Within Motorola, this has become a corporate quality objective. Many other organizations, including their suppliers and customers, have adopted similar criteria.

Process Capability Ratio For An Off-Center Process The process capability ratio (Cp) does not take into account where the process mean is located relative to the specifications. Cp simply measures the spread of the specifications relative to the 6-sigma spread in the process. For example the top two normal distributions in figure 2.8 both have Cp =2.0, but the process in panel (b) of the figure clearly has lower capability than the process in panel (a) because it is not operating at the midpoint of the interval between the specifications.



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This situation may be more accurately reflected by defining a new process capability ratio that takes process centering into account. This quantity is

),min( plpupk CCC = ---------------------------------------------- (2-6)

Notice that Cpk is just the one-sided Cp for the specification limit nearest to the process average. For the process shown in figure 8a, we would have

),min( plpupk CCC =

)3

,3

min(σ

µσ

µ LSLC

USLC plpu

−=

−==

)5.2)2(3

3853,5.1

)2(3

5362min( =

−==

−= plpu CC

Generally, if Cp= Cpk, the process is centered at the midpoint of the specifications, and when Cpk < Cp the process is off-center. The magnitude of Cpk relative to Cp is a direct measure of how off-center the process is operating. Several commonly encountered cases are illustrated in figure 2.8. Note in panel (c) of figure 2.8 that Cpk =1.0 while Cp=2.0. One can use table 2.1 to get a

Figure 8 Relationship of Cp and Cpk



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quick estimate of potential improvement that would be possible by centering the process. If we take Cp =1.0 in table 2.1 and read the fallout from the one-sided specifications column, we can estimate the estimate the actual fallout as 1350 PPM. However, if we can center the process, then Cp=2.0 can be achieved, and table 2.1 (using Cp=2.0 and two sided specifications) suggests that the potential fallout is 0.0018 PPM, an improvement of several orders of magnitude in process performance. Thus, we usually say that PCR measures potential capability in the process, while Cpk measures actual capability.

Panel (d) of figure 8 illustrates the case in which the process mean is exactly equal to one of the specification limits, leading to Cpk = 0. As panel (e) illustrates, when Cpk < 0 the implication is that the process mean lies outside the specifications. Clearly, if Cpk < -1, the entire process lies outside the specification limits. Some authors define Cpk to be nonnegative, so that values less than zero are defined as zero.

Many quality engineering authorities have advised against the routine use of process capability ratios such as Cp and Cpk (or the others discussed later in this section) on the grounds that they are an oversimplification of a complex phenomenon. Certainly, any statistic that combines information about both location (the mean and process centering) and variability, and which requires the assumption of normality for its meaning full interpolation is likely to be misused (or abused). Furthermore, as we will see, point estimates of process capability ratios are virtually useless if they are computed from small samples. Clearly, these ratios need to be used and interpreted very carefully.

Manufacturing Process Capability Metrics: Tolerances are always related to manufacturing processes or to materials used in the manufacture of a product, and they must be designed in conjunction with the application of the specific manufacturing process. If a tolerance band is determined without considering a manufacturing process, there is great risk in having a mismatch between the required tolerance and the capability of a given process- when the engineer finally gets around to selecting one. It is a fundamental precept in concurrent engineering to develop technology concepts or product-design concepts in simultaneously with the necessary manufacturing processes to support the timely and economic commercialization of the desired product. Often during technology development it is necessary to invent and co develop the manufacturing technology required to make the product. It is unwise to wait until the tolerance design phase of a product-commercialization process to select or optimize a manufacturing process. Capable manufacturing processes must be aligned with the product concept as early as possible. Only in this way will there be enough time develops necessary relationship between tolerances and manufacturing processes. It is also essential to perform manufacturing-process parameter optimization just as one would for design-component parameters. The design engineer and the manufacturing engineer have to define a common metric that quantifies the relationship that exists between the nominal design specifications, their tolerances, and the variability associated with the measurable output from the manufacturing process. The manufacturing engineer must also provide tolerances on the manufacturing process parameters and raw materials to help the team stay well within the tolerances assigned to the component being



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manufactured. The focus on manufacturing process set-point tolerances should be directed at keeping the component specifications as close to on-target as possible. The manufacturing process capability index, typically expressed as Cp, Cpk, Cp(upper limit), or Cp(lower limit), is the ratio of design tolerance boundaries to the measured variability of the manufacturing process output response. Cp is defined arithmetically as follows:

Cp = ( )

σ6LSLUSL −

Where, USL=Upper specification limit. LSL= Lower specification limit. 6σ stands for six times the short-term sample standard deviation of the production

measure of part quality in engineering units; the use of σ is really a misapplication

of the population parameter for a standard deviation. The true measure of variability most often used in the alteration of cp is 6s(where σ is the sample standard

deviation).

References: 1. “Statistical Quality Control”by Eugene L.Grant, Richard S.Leavenworth, 7th

edition Mcgraw hill book co., 2. “Advanced Practical Statistics” by S.P.Guptha. 3. “Statistical Sources and Techniques” by F.J.Rendall and D.M.Wolf. 4. “Tolerance Design” by Creveling C.M. , Addison Wesley Publication Company.

01 process capability

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