1. Tolerance Limits Statistical Analysis & Specification
SMU
EMIS 7364
NTU
TO-570-N
Tolerance Limits Statistical Analysis & Specification
Updated: 2/14/02
Statistical Quality Control
Dr. Jerrell T. Stracener, SAE Fellow

2. (Ideal level for use in product)
Product Specification
Lower
Specification
Limit
Nominal
Upper
Target
(Ideal level for use in product)
Tolerance x
(Product
characteristic)
(Maximum range of variation of the product
characteristic that will still work in the product.)

3. Traditional US Approach to Quality
(Make it to specifications)
good T
USL
LSL
Loss ($)
No-Good x

5. Setting Specification Limits on Discrete Components

6. Don’t just conform to specifications
Variability Reduction
Variability reduction is a modern concept of design
and manufacturing excellence
Reducing variability around the target value leads
to better performing, more uniform, defect-free
product
Virtually eliminates rework and waste
Consistent with continuous improvement concept
Don’t just conform to specifications
Reduce variability
around the target
accept
reject
reject
target

7. True Impact of Product Variability
Sources of loss
- scrap
- rework
- warranty obligations
- decline of reputation
- forfeiture of market share
Loss function - dollar loss due to deviation of
product from ideal characteristic
Loss characteristic is continuous - not a step
function.

8. Representative Loss Function Characteristicsx
Loss $
X nominal is best
L = k (x - T)2
X smaller is better
L = k (x2)
X larger is better
L = k (1/x2) T

9. Variability-Loss Relationship
LSL
USL
Target
Loss
$ savings
due to
reduced
variability
Maximum
$ loss
per item

10. Loss Computation for Total Product Population
X nominal is best
L = k (x - T)2 x
Loss $ T

11. Statistical Tolerancing - Convention
Normal Probability
Distribution
LTL
Nominal
UTL
0.9973
+3
-3 

12. Statistical Tolerancing - Concept
LTL
UTL
Nominal x

13. Caution
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:
% outside the natural tolerances sounds
small, but this corresponds to 2700 nonconforming
parts per million.
2. If the distribution of process output is non
normal, then the percentage of output falling
outside   3 may differ considerably from 0.27%.

14. Normal Distribution
Probability Density Function:
 < x < 
where  =
e =

15. Normal Distribution
Mean or expected value of X
Mean = E(X) = 
Median value of X
X0.5 = 
Standard deviation

16. Normal Distribution
Standard Normal Distribution
If X ~ N(, ) and if , then Z ~ N(0, 1).
A normal distribution with  = 0 and  = 1, is called
the standard normal distribution.

17. Normal Distribution - example
The diameter of a metal shaft used in a disk-drive unit
is normally distributed with mean inches and
standard deviation inches. The specifications
on the shaft have been established as 
inches. We wish to determine what fraction of
the shafts produced conform to specifications.

18. Normal Distribution - example solution
0.2508
0.2515
USL
0.2485
LSL
0.2500
f(x) x
nominal

19. Normal Distribution - example solution
Thus, we would expect the process yield to be
approximately 91.92%; that is, about 91.92% of
the shafts produced conform to specifications. Note
that almost all of the nonconforming shafts are too
large, because the process mean is located very
near to the upper specification limit. Suppose we
can recenter the manufacturing process, perhaps
by adjusting the machine, so that the process mean
is exactly equal to the nominal value of
Then we have

20. Normal Distribution - example solution
0.2500
0.2515
USL
0.2485
LSL
f(x) x
nominal

21. What is the magnitude of the difference between sigma levels?
Sigma Area Spelling Time Distance
One Floor of typos/page 31 years/century earth to moon
Astrodome in a book
Two Large supermarket 25 typos/page 4 years/century 1.5 times around
in a book the earth
Three small hardware 1.5 typos/page 3 months/century CA to NY
store in a book
Four Typical living 1 typo/30 pages 2 days/century Dallas to Fort Worth
room ~(1 chapter)
Five Size of the bottom 1 typo in a set of 30 minutes/century SMU to 75 Central
of your telephone encyclopedias
Six Size of a typical 1 typo in a 6 seconds/century four steps
diamond small library
Seven Point of a sewing 1 typo in several 1 eye-blink/century 1/8 inch
needle large libraries

22. Linear Combination of Tolerances
Xi = part characteristic for ith part, i = 1, 2, ... , n
Xi ~ N(i, i)
X1, X2, ..., Xn are independent

23. Linear Combination of Tolerances
Y = assembly characteristic
If , where the a1, ..., an are
constants,
then
Y ~ N(Y, Y),
where
and

24. Concept x1 x2 . xn y

25. Statistical Tolerancing - Concept
0.2500
0.2515
USL
0.2485
LSL
f(x) x
nominal

26. Tolerance Analysis - example
The mean external diameter of a shaft is S = 1.048
inches and the standard deviation is S =
inches. The mean inside diameter of the mating
bearing is b = inches and the standard
deviation is b = inches. Assume that both
diameters are normally and independently distributed.
(a) What is the required clearance, C, such that the
probability of an assembly having a clearance less than C is 1/1000?
(b) What is the probability of interference?

27. Tolerance Analysis - example solution
Bearing
Shaft Xb XS
fb(x)
fs(x)

28. Tolerance Analysis - example solution
Intersection Region
fb(x)
fs(x)

29. The Normal Model - example solution
D = xb-xs = Clearance of bearing inside diameter minus shaft outside diameter
D = b - S = 0.011
D = (b2 + S2)1/2 =
so D~N(0.011,0.0036)
fD(x)
d=xb-xs

30. The Normal Model - example solution
(a) Find c such that P(D < c) =
so that
From the normal table (found in the resource section of the website), the Z = -3.09
and

31. The Normal Model - example solution
Since c < 0, there is no value of c for which the probability is equal to 0.001
(b) Find the probability of interference, i.e.,
From the normal table (found in the resource section of the website), the Z of -3.1 =

32. Tolerance Analysis - example
Using Monte Carlo Simulation (n=1000):
(a) What is the required clearance, C, such that the
probability of an assembly having a clearance less
than C is 1/1000?
(b) What is the probability of interference?

33. Tolerance Analysis - example
Using Monte Carlo Simulation
First generate random samples from (I used n=1000)
Xbi~N(mb, sb) = N(1.059, )
and
Xsi~N(ms, ss) = N(1.048, )

34. Tolerance Analysis - example
Then calculate the differences
Estimate
Estimate ms by taking the mean. (You can use the AVERAGE() function.)
Estimate ss by calculating the standard deviation. (You can use the STDEV() function.)

35. Tolerance Analysis - example=
and =
(a)
This is close to c =

36. Tolerance Analysis - example
(b)
This can be compared to P(I) =

37. Statistical Tolerance Analysis Process
Assembly consists of K components
Specifications Assembly:
Specifications Component:
Assembly Nominal
where ai = 1 or -1 as appropriate

38. Statistical Tolerance Analysis Process
Assembly tolerance
If dimension
with parameters and , then
where
and

39. Statistical Tolerance Analysis Process
is specified
is determined during design
is calculated
Case 1: if probability is too small, then
(1) component tolerance(s) must be reduced
or (2) tA must be increased

40. Statistical Tolerance Analysis Process
Case 2: if probability is too large, then some or all components tolerances must be increased.
Note: Do not perform a worst-case tolerance analysis

41. Estimating the Natural Tolerance Limits of a Process

42. Tolerance Limits Based on the Normal Distribution
Suppose a random variable x is distributed with mean m and variance s2, both unknown. From a random sample of n observations, the sample mean and sample variance S2 may be computed. A logical procedure for estimating the natural tolerance limits m ± Za/2 s is to replace m by and s by S, yielding.

43. Tolerance Intervals - Two-Sided
Since and S are only estimates and not the true parameters values, we cannot say that the above interval always contains 100(1 - a)% of the distribution. However, one may determine a constant K, such that in a large number of samples a faction

44. Tolerance Limits Based on the Normal Distribution

45. Tolerance Intervals - Two-Sided
If X1, X2, …, Xn is a random sample of size n from
a normal distribution with unknown mean  and
unknown standard deviation , then a two-sided tolerance interval is (LTL,UTL), i.e., an interval that contains at least the proportion P of the population, with g 100% confidence is:
and
is a function of n, P, and g and may be obtained from the table Factors for Two-Sided Tolerance Limits for Normal Distributions (Located in the resource section on the website).

46. Tolerance Intervals - One-Sided
If X1, X2, …, Xn is a random sample of size n from
a normal distribution with unknown mean  and
unknown standard deviation , then a one-sided lower (upper) tolerance interval is defined by the lower tolerance limit LTL (upper tolerance limit UTL), the value for which at least the proportion P of the population lies above (below) LTL (UTL) with g100% confidence where
is a function of n, P, and g and may be obtained from the table Factors for Once-Sided Tolerance Limits for Normal Distributions (Located in the resource section on the website).

47. Tolerance Intervals - Two-Sided Example
Ten washers are selected at random from a population that can be described by a normal distribution. The measured thicknesses, in inches, are:
Establish an interval that contains at least 90% of the population of washer thicknesses with 95% confidence.

48. Tolerance Intervals - Two-Sided Example Solution
From the sample data
and
The K value can be found on Tolerance Limits Table- Two-Sided with gamma 95 and 99 and n=2 to 27 (Located in the resource section on the website).

49. Tolerance Intervals - Two-Sided Example Solution
so that
Therefore, with 95% confidence at least 90% of the population of washer thicknesses, in inches, will be contained in the interval (0.116,0.136).