Design Margin 
Statistical Tolerance Part 2
for use with DFSS
 Design for Six Sigma
Piece part design margins, using statistical tolerancing
along with a six sigma agenda can indeed reduce the number of
failures in a manufacturing process. However, legacy
parts and processes may make this impossible. Retooling and/or
redesigning can be cost prohibitive. But it may be possible to
start with new processes, to include updated processes. This
should be a big part of any designing for six sigma
methodologies. Of course, new designs and processes lend
themselves to be shaped into six sigma or more designs
margins.
One way to do this is by the addition of statistical
tolerances. If you have 3 parts that fit together, for
example, and you can get a six sigma design margin on two of
them but only a 4.5 sigma design margin on the third, you may
still get so close to six sigma that in the end it will not
really matter.
Whenever more than one assembly is put together, new
dimensions are created as well as new distributions. Because
of this, we need to look at the addition of distributions. The
distributions that exist on the first distribution are added
to the second piece and so on till the building of the
component is finished. Because the addition of distribution is
statistical in nature, we need to know some rudimentary
statistical laws so that we may use this information to
predict the best and most reasonable resolution for
production.
There are four laws that govern the addition of
distribution.
1. Law Of Additions Averages. If parts are
assembled in such a way that one dimension is added to another,
the average dimension of the entire assembly will be equal to
the sum of the average dimensions of the parts.
Let 
_{A} 
= 
The Mean of Part A 

_{B} 
= 
The Mean of Part B 

_{C} 
= 
The Mean of Part C 
Average dimension of assembly 
= 
_{A} 
+ 
_{B} 
+ 
_{C} 
,etc. 
See Fig. 1 
2. Law Of Differences. If parts are assembled in
such a way that one dimension is subtracted from another, the
average dimension of the entire assembly will be the difference
between the average dimensions of the parts.
Let 
_{D} 
= 
The Mean of Part D 

_{E} 
= 
The Mean of Part E 
Average dimension of assembly 
=( 
_{D} 
 
_{E} 
)or( 
_{E} 
 
_{D} 
)as the case may be. 
See Fig. 2 
3. Law Of Sums and Differences. If parts are
assembled in such a way that some dimensions are added to each
other and some dimensions are subtracted from another, the
average dimension of the entire assembly will be the algebraic
sum of the average dimensions of the parts.
Average dimension of assembly 
= 
_{A} 
+ 
_{B} 
+ 
_{C} 
 
_{D} 
+ 
_{E} 
,etc 
4. Law Of The Addition and Standard Deviations or
Variances. If the parts are assembled at random, Standard
Deviation, (Sigma), of the assembly WILL NOT BE the simple sum
standard deviations of the parts, but rather, it will be the
value obtained by squaring each of the component standard
deviations, totaling the squares, and then taking the square
root of the total.^{*}
Let 
_{A} 
= 
The Standard Deviation of Part A 

_{B} 
= 
The Standard Deviation of Part B 
Standard Deviation
of the assembly = 


(_{A})^{2}
+ (_{B})^{2} 
The forth law should be carefully examined because the
statistical addition gives a different result from the one which
he/she would be likely to get naturally.
One should note that the squares of the standard
deviations are always added regardless of whether the average
dimension is gotten by sums of differences. DO NOT attempt to
subtract one standard deviation from another as may be done in
the case of averages.
The forth law can also be expressed using variances
instead of sigma, (standard deviations). Standard deviation is
the square root of variance (^{2}).
If (_{A})^{2}
is the variance of Part A and (_{B})^{2}
is the variance of Part B, the
variance of the assembly will be (_{A})^{2}
+ (_{B})^{2}
.
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