1. Page 1
Tolerance stackup
analysis
Winson Mao
August 22, 2012

2. Page 2
Tolerance stackup analysis
Performed to determine the variation of a single untoleranced
dimension or distance on a component or in an assembly.

3. Page 3
Tolerancing

4. Page 4
Methods
Arithmetic method
Worst case method (WC)
Statistical method
Root sum square method (RSS)
Modified root sum square method (MRSS)
Six sigma method (6S)

5. Page 5
Formulas
WC RSS MRSS 6S
Mean
dimension
Standard
deviation
N/A
Bilateral
tolerance
Close
dimension
∑∑ =
−
=
−=
n
ki
i
k
i
i µµµ
1
1
∑=
=
n
i
iTTWC
1
WCTD ±= µ
∑∑ =
−
=
−=
n
ki
i
k
i
i µµµ
1
1
∑∑ =
−
=
−=
n
ki
i
k
i
i µµµ
1
1
σ3=RSST
RSSTD ±= µ
CT f
MRSS ×= σ3
MRSSTD ±= µ
∑∑ =
−
=
−=
n
ki
i
k
i
i µµµ
1
1
∑=
=
n
i
Ti
1
2
)
3
(σ
σ×= aT t eS6
TD S6
±= µ
∑ 





=
=
n
i
Ti
1
2
3
σ ∑ 







=
=
n
i
e
C
T
pki
i
1
2
3
σ

6. Page 6
Process capability
σ6
LSLUSL
Cp
−
= ( )CCC puplpk
,min=
σ
µ
3
LSL
Cpl
−
=
σ
µ
3
−
=
USL
Cpu
where
,
,

7. Page 7
Coefficients
( )
( ) 1
1
5.0
+
−
−
=
nT
TTC
RSS
RSSWC
f
• MRSS coefficient,
• Limit size coefficient,
Cf
6~3=at
at

8. Page 8
Coefficients
( )
( ) 1
1
5.0
+
−
−
=
nT
TTC
RSS
RSSWC
f
• MRSS coefficient,
• Limit size coefficient,
Cf
6~3=at
at
Limit sizes Process yield
(%)
Number of
rejects per
million
component
produced
µ±1σ 68.2 317310
µ±2σ 95.4 45500
µ±3σ 99.73 2700
µ±3.5σ 99.95 465
µ±4σ 99.994 63
µ±4.5σ 99.9993 6.8
µ±5σ 99.99994 0.6
µ±6σ 99.9999998 0.002

9. Page 9
Arithmetic vs. statistical
• Number of tolerances in the tolerance stackup.
• The quantity of parts to be manufactures.
• Manufacturing process controls.
• Past company practices.
• Willingness to accept risk.

10. Page 10
Step-by-step procedure
Equal bilateral
tolerancing
Define direction
of dimensions
Ite
m
+
Dim
-
Dim
Tol
1 30 ±0.2
2 45.5 ±0.5
3 13.5 ±0.5
Total 45.5 43.5 ±1.2
25.435.45 =−=µ 2.1±=Tol
Fill in table