c Heldermann Verlag
ISSN 0940-5151
Economic Quality Control
Vol 17 (2002), No. 1, 81 – 98
Prediction Intervals, Tolerance Intervals and
Standards in Quality Control - Part 1
Elart von Collani and Karl Baur
Abstract: Industrial quality control is necessary at least for two reasons:
• Monitoring product or process quality in order to perform interventions for assuring a
desired quality level.
• Monitoring product or process quality in order to meet certain requirements (imposed
by law or customers) for documentation.
Especially in the second case legal requirements or consumers demand that “professional”
methods are used. Therefore, industry relies on the methods offered by “professional” bodies
like the International Organization for Standardization (ISO) or its national counterparts for
example the German DIN Deutsches Institut für Normung, because they are widely acknowledged as professional.
One of the standards which are in use in German industries is DIN 55303 (Teil 5): “Statistische
Auswertung von Daten - Bestimmung eines statistischen Anteilsbereichs” of February 1987
which corresponds to ISO 3207-1975 “Statistical interpretation of data: determination of a
statistical tolerance interval.” This paper introduces in detail statistical tolerance intervals
and subsequently examines the standard critical. This first part is devoted to the case that
the normal approximation is used and results in the recommendation not to use the methods
offered in the standard. A second part will investigate the case that the normal approximation
is not made.
1
Introduction
Any meaningful statistical method is necessarily based on a stochastic model and a prediction procedure. Therefore, models and predictions constitute the core of theory and
application of stochastics, which was established about 300 years ago by Jakob Bernoulli
as the Art of Conjecturing. Any prediction refers to a random variable and, as a matter of
fact, predictions are often used as auxiliary tools for making statements on deterministic
quantities, like proportions, totals, average values, etc. Calling these quantities deterministic variables, one observes that any situation is described by a deterministic variable, a
82
Elart von Collani and Karl Baur
random variable and a probability measure which represents the connection between the
two variables and thus between past and future.
Let D denote the deterministic variable and assume that there is a random variable with
probability distribution depending on the actual value d of D. Accordingly, the random
variable is denoted by X|d with probability distribution PX|d .
If the actual value d of the deterministic variable is known, one can predict the future
event with respect to X|d, and if d is unknown, one can use the outcome x of X|d to
draw inference on the actual value of the deterministic variable. Thus, before solving a
problem by statistical methods there should be a clear picture about the deterministic
variable, the random variable, the connecting probability measure and the nature of the
problem by distinguishing between predicting a future event of the random variable or
determining the actual value of the deterministic variable.
In 1931 tolerance limits were introduced by Shewhart in his pioneering book “ Economic
Control of Quality of Manufactured Product.” Shewhart notes:
As a perfectly general case, let us assume that the quality X upon which we
wish to set tolerance limits depends upon the qualities X1 , X2 , . . . , Xi , . . . , Xm
of m different piece-parts or kind of raw material. Interpreted from the viewpoint of control, this means that we wish to set two limits on X which will
include a certain fraction P of the product in the long run.
The main purpose of Shewhart’s tolerance limits was to determine engineering tolerance
limits, i.e., specifications for complex product which would assure operativeness with a
given probability taking into account the economic situation.
In 1941 Shewhart’s ideas were taken up by Samuel S. Wilks [13] who, assuming a continuous model, obtained tolerance limits as order statistics. However, he formulated the
problem of setting tolerance limits as follows:
The problem now arises as to how we should calculate a tolerance range
(L1 , L2 ) for x from a sample, and how large the sample should be in order
for the tolerance range to have a given degree of stability. More specifically,
for a given method of calculating tolerance limits, how large should our sample
be in order that the proportion P of the universe included between L1 and
L2 have an average value a, and will be such that the probability is at least p
that P will lie between two given numbers, say b and c.
Wilk’s tolerance regions were called statistical tolerance regions in contrast to Shewhart’s
engineering tolerance or specification regions. Nowadays, statistical tolerance regions
constitute a frequently used tool particularly in industrial quality control for judging the
quality of product. To this end there are ISO and national standards which offer a number
of methods for different situations.
Some questions arise with respect to tolerance regions in general and to those offered in
the industrial standards:
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
83
1. Are tolerance regions used for making predictions about the quality of product to
be produced, i.e., about a future event, or are they used for determining the quality
of given product, i.e., the outcome of a past event?
2. What are the differences among the tolerance regions offered in the standards and
in which situation should the one or the other be used?
3. What are the quality characteristics which can be monitored or controlled by means
of tolerance regions?
There are two types of statistical tolerance regions proposed in literature: the tolerance
region, as defined by Wilks [13], and another type called β-expectation region. The
relevant standards do not deal with β-expectation region and are restricted to the onedimensional and unimodal case implying that the tolerance regions reduce to intervals.
2
Prediction Intervals
Let X denote an one-dimensional random variable with range of variability X = (a, b)
where the cases a = −∞ and b = ∞ are admitted. In most cases a meaningful prediction
(p)
is given by an interval AX ⊂ X where p denotes the confidence level, i.e. a lower bound
for the probability that the predicted event will actually occur which determines the
procedure’s reliability. Any prediction interval with reliability p may be specified by an
upper and a lower quantile.
(p)
()
(u)
(1)
AX = qX (1 + p − γ) , qX (γ)
()
(u)
where qX (a) denotes the lower and qX (a) the upper quantile of order a of the random
variable X and γ ∈ [p, 1]. These quantiles are defined by:
(u)
(2)
qX (a) = inf x ∈ IR PX (−∞, x] ≥ p
()
(3)
qX (a) = sup x ∈ IR PX ([x, ∞) ≥ p
(p)
The probability of the predicted event AX is given by:
(p)
(u)
()
PX AX
=
PX (−∞, qX (γ)] − PX (−∞, qX (1 + p − γ)
=
γ − [1 − (1 + p − γ)] = p
if X is of continuous type
≥
γ − [1 − (1 + p − γ)] = p
if X is of discrete type
(4)
If the distribution PX is known, the two quantiles defined by (1) may easily be calculated.
The question of selecting a suitable value for γ has to be answered according to the given
situation.
84
Elart von Collani and Karl Baur
• If the prediction should be of the form “X will be at most . . .”, then the prediction
(p)
interval AX should cover the lower tail and γ = p is selected resulting in a prediction
of the form
(p)
(u)
AX = a, qX (p)
(5)
• If the prediction should be of the form “X will be at least . . .”, then the prediction
interval should cover the upper tail and γ = 1 is selected resulting in a prediction
of the form
(p)
()
(6)
AX = qX (p), b
• If the most accurate prediction is desired, the following optimization problem has
to be solved for the optimal γ ∗ :
(u)
()
∗
(7)
γ = arg min qX (γ) − qX (1 + p − γ)
γ∈[p,1]
In the special case of a symmetric probability distribution PX , the optimization
problem is solved by setting
γ∗ =
1+p
2
(8)
resulting in the prediction interval
(p)
AX
=
() 1
qX (
+ p (u) 1 + p ), qX (
)
2
2
(9)
Clearly, for a symmetric distribution the lower tail
()
a, qX
1+p not covered by the
(u) ,b .
prediction interval has the same probability as the uncovered upper tail qX 1+p
2
2
Any prediction interval with confidence level p represents a future event with respect to
X which occurs with probability of at least p. However, the prediction interval itself is
completely determined by the probability distribution and the aim of prediction and does
not depend on random implying that it is a deterministic quantity.
3
Prediction Intervals in Quality Control
Two cases have to be distinguished with respect to the application of prediction intervals
in industrial practice. In the first situation the aim is to make a statement about a lot of
given manufactured items and in the second situation a statement shall be made about
the future product of a manufacturing process.
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
85
• Case 1:
In the first case a lot of N items is given represented by the set of N values of an
items’ quality characteristic:
(10)
x1 , x2 , . . . , x N
The set (10) has a discrete frequency distribution, which determines the quality of
the lot. The problem is that the frequency distribution of the lot and, thus, its
quality is not known.
Introducing the random variable X which refers to drawing randomly one item from
the lot, we note that X has a discrete probability distribution coinciding with the
(p)
frequency distribution of the lot. It follows that any prediction interval AX for X
()
(u)
given by the quantiles qX (1 + p − γ) and qX (γ) for γ ∈ [p, 1] covers at least a
proportion of 100p% of the lot.
• Case 2:
In the second case a production process is considered and the random variable X
refers to the quality characteristic of the product in question. Any lot of size N to
be produced is represented by the sequence of random variables
X1 , X2 , . . . , XN
(11)
where Xi , i = 1, . . . , N , is the quality characteristic of the ith item to be produced.
Assume that the sequence consists of independent and identically like X distributed
(p)
random variables of continuous type, and let AX be a prediction interval for X with
reliability p, then the following statements on the quality of the future lot (11) can
be made:
1. Assertion 2.1: The random number M among the N items to be produced
(p)
with quality characteristic X falling into the prediction interval AX is binomially distributed with parameters N and p.
2. Assertion 2.2: An expected proportion of 90% of the lot to be produced is
(p)
covered by AX .
As to Assertion 2.1, it is interesting to note that the probability that at least pN items
(p)
(M ≥ pN ) fall into a prediction interval AX is only about 0.5. However, at least pN
(p+e)
items fall almost with certainty into the prediction interval AX , where e depends on
N , but is a rather small number even for medium sized lots. For N = 1000 and p = 0.9,
this relation is illustrated in Figure 1.
86
Elart von Collani and Karl Baur
inclusion probability
1
0.8
0.6
0.4
0.2
-0.04
-0.02
0.02
0.04
Fig. 1: Probability (inclusion probability) that at least 90 % of the 1000 items fall into
(0.9+e)
AX
for −0.05 ≤ e ≤ 0.05.
In the example represented in Figure 1, the small probability overshoot of e = 0.02 over
p = 0.9 leads to an inclusion probability of 0.98985 which is close to certainty.
(p)
In view of Assertion 2.2, the prediction interval AX for X is called a p-expectation interval
for the future lot.
Note, that in either case there is a random variable X representing the quality characteristic and reflecting uncertainty of the drawing or production procedure, respectively.
The predictions with respect to X are used either for making a statement about the
determinate quality of a given lot or about the future quality of a lot to be manufactured.
If the probability distribution PX is known, there are no problems at all to determine
(p)
any desired prediction interval AX by means of the corresponding quantiles. However,
generally PX is completely or at least partially unknown and, thus, it is not possible to
calculate the necessary quantiles which, therefore, have to be estimated on the basis of a
random sample (X1 , . . . , Xn ), implying that the corresponding prediction intervals are not
anymore deterministic, but random. In contrast to the deterministic prediction intervals
these random prediction intervals are called tolerance intervals, indexed by the reliability
of the estimation procedure.
4
Specifications and Tolerance Intervals
As to the requirements with respect to the quality characteristic X given by specifications,
there are three cases to be distinguished:
1. item conforming ⇔ x ≤ U (onesided upper specification)
87
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
2. item conforming ⇔ x ≥ L (onesided lower specification)
3. item conforming ⇔ L ≤ x ≥ U (two-sided specification)
(u)
In the first case the left tail of the distribution and, thus, the upper quantile qX (p) of
X are of interest. Analogously, in the second case the deterministic variable of interest is
given by the lower quantile q () (p).
The third case is generally more complicated. However, assuming that the quality charac(u)
()
) and qX ( 1+p
) determine
teristic X is symmetrically distributed, the two quantiles qX ( 1+p
2
2
a shortest prediction interval, which covers the center of the distribution symmetrically
around the expectations E[X].
We conclude that according to the product specifications tolerance intervals for the tails
) are used in quality control.
(γ = p and γ = 1) and for the central region (γ = 1+p
2
Tolerance regions are obtained by an estimation procedure based on a sample
= (X1 , . . . , Xn )T and an appropriate sample function T (X).
It is assumed here that
X
the sample consists of independent and identically distributed random variables. This
assumption is met sufficiently well in Case 1, if the lot size N for the given lot is large
compared with the sample size n and in Case 2, if the production conditions are more or
less stable and the outcome for an item does not affect the outcome of the subsequently
produced items.
Assuming that the distribution of X may be approximated by a continuous one, there
T2 (X)
and T3 (X)
to be applied in the
is the need for three sample functions T1 (X),
three situations with respect to the specifications. To be meaningful the reliability of
the estimation procedures must be specified by a confidence level β and the following
requirements:
1. For onesided, upper specification U the condition
(u)
PT1 {t1 t1 ≥ qX (p)} = β
(12)
has to be met by T1 (X).
2. For onesided, lower specification L the condition
()
PT2 {t2 t2 ≤ qX (p)} = β
(13)
has to be met by T2 (X).
3. For the two-sided specification (L, U ) the condition
() 1 + p
(u) 1 + p
P(T3,1 ,T3,2 ) {(t3,1 , t3,2 ) t3,1 ≤ qX (
), qX (
) ≤ t3,1 = β
2
2
= T3,1 (X),
T3,2 (X)
. ,
has to be met by T3 (X)
(14)
88
Elart von Collani and Karl Baur
determines a onesided confidence interval with confidence level
The sample function T1 (X)
(u)
does not fall below q (u) (p) with probability β implying
β for qX (p) implying that T1 (X)
X
that
(u)
PT1 [qX (p), ∞) ≥ β
(15)
holds or equivalently we have with probability β
(u)
(a, qX (p)] ⊂ a, T1 (X)
(16)
covers with probability of
In other words, in Case 1 the tolerance interval (a, T1 (X)]
at least β a proportion of 100p % of the items of a given lot. In Case 2, an expected
proportion of 100p % of a lot to be produced is covered by the tolerance interval with
probability β.
If the event {x | T1 (x) ≤ U } occurs, it is concluded that
• in Case 1, a proportion of at least 100p % of the items in the given lot meet the
specification, or
• in Case 2, an expected proportion of at least 100p % of the items in the lot to be
produced will meet the specification.
determines analogously a onesided confidence interval with
The sample function T2 (X)
()
confidence level β for qX (p) and, therefore, if the event {x | T2 (x) ≥ L} occurs, the same
conclusions are drawn.
Finally, the sample function T3 (X) = T3,1 (X), T3,2 (X) determines a confidence interval
with confidence level β for the central prediction interval, i.e., with probability β we have:
1+p
1+p
()
(u)
(17)
qX
, qX
⊂ T3,1 (X), T3,2 (X)
2
2
If the event {x | L ≤ T3,1 (x) and T3,2 ≤ U } occurs, again the same conclusions are drawn
as in the two previous situations.
5
Approximations of PX
some assumptions have to be
In order to derive appropriate sample functions Tk (X)
made concerning the random variable X in order to allow the use of approximations of
the unknown probability measure PX . Generally, two types of situation are distinguished:
1. The distribution PX of the quality characteristic of interest X may be approximated
by the normal model.
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
89
2. The distribution PX of the quality characteristic X is of continuous type, but may
not be approximated by the normal model.
In each of the situation a continuous distribution is assumed. For continuous distributions
the following relation holds:
()
(u)
qX (1 + p − γ) = qX (γ − p)
(18)
and, therefore, the problem reduces
procedures for estimating the upper quan
to provide
(u)
(u)
(u) (u) 1+p and
q
.
tiles qX (p), qX (1 − p), qX 1−p
X
2
2
5.1
Procedures Based on the Normal Model
In case that the normal approximation N (µ, σ 2 ) can be used, the distribution is completely
determined by the actual value µ of the expectation E[X] and the actual value σ 2 of the
variance V [X]. One may think of four different situations with respect to prior knowledge
about the distribution parameters.
1. The values µ and σ 2 are known.
2. The value µ is known and the value σ 2 is unknown.
3. The value σ 2 is known and the value µ is unknown.
4. Neither of the values µ is σ 2 is known.
From the viewpoint of industrial practice the first case and the third case seem to be less
important, compared with the second and especially the forth case.
5.2
Known Expectation and Variance
If the µ and σ 2 are known, there is no estimation problem, as the necessary quantiles are
readily available in almost any textbook on statistics. Therefore, the prediction intervals
can be determined and there is no need for tolerance intervals. For the cases mentioned
above, we obtain
•
•
•
(−∞, µ + λ(p)σ]
[µ − λ(p)σ, +∞)
1+p
1−p
)σ, µ + λ(
)σ]
[µ + λ(
2
2
(19)
(20)
(21)
where λ(a) denotes the upper quantile of order a of the standardized normal distribution.
90
Elart von Collani and Karl Baur
5.3
Known Expectation and Unknown Variance
As the value µ is known, it makes sense to select the following sample functions:
= µ + k1 S(X)
•
T1 (X)
•
•
(22)
= µ + k2 S(X)
T2 (X)
= µ − k3 S(X)
= µ + k3 S(X)
and T3,2 (X)
T3,1 (X)
(23)
(24)
with
1 (Xi − X)2
S (X1 , . . . , Xn ) =
n − 1 i=1
n
2
(25)
1
Xi
n i=1
n
X=
(26)
and to determine ki , i = 1, 2, 3 so as to meet the conditions (12), (13) and (14), respectively.
The normal assumption for X implies:
(u)
qX (γ) = µ + λ(γ) · σ
(27)
and, therefore, the conditions (12), (13) and (14) take the following form:
PT1
t1 t1 ≥ µ + λ(p)σ
=β
t2 t2 ≤ µ − λ(p)σ
PT2
=β
1+p
1+p
PT3
(t3,1 , t3,2 ) t3,1 ≤ µ − λ(
)σ, t3,2 ≥ µ + λ(
)σ
=β
2
2
(28)
(29)
(30)
These conditions can be reformulated by means of the random variable (n − 1) S(σX)
2 .
2
λ(p)
P(n−1) S(X)
s s ≥ (n − 1)
=β
(31)
k1
σ2
2 λ(p)
s s ≥ (n − 1)
=β
(32)
P(n−1) S(X)
k2
σ2

2 
 
λ 1+p
2

=β
P(n−1) S(X)
s s ≥ (n − 1)
(33)


k3
σ2
Evidently, the lower bounds of the events given in (31), (32) und (33) are the lower
quantiles of order β of the sample function (n − 1) S(σX)
2 , which is distributed according to
a χ2 -distribution with (n − 1) degrees of freedom. The required lower quantiles may be
expressed by means of upper quantiles, which are available in statistical tables.
()
(u)
qχ2 (n−1) (β) = qχ2 (n−1) (1 − β)
(34)
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
Thus, the constants ki meeting the above conditions may easily be calculated.
n−1
k1 = k2 =
· λ(p)
(u)
qχ2 (n−1) (1 − β)
n−1
1+p
·λ
k3 =
(u)
2
qχ2 (n−1) (1 − β)
5.4
91
(35)
(36)
Unknown Expectation and Known Variance
This situation seems to be rather weird as it is difficult to imagine that the variance σ 2
is known, but there are no information about the expectation µ, which essentially enters
σ2.
by
Proceeding analogously as in the previous case means to exchange µ by X and S(X)
σ yielding the following sample functions:
= X + k1 σ
•
T1 (X)
(37)
= X + k2 σ
•
T2 (X)
(38)
= T3,1 (X),
T3,2 (X)
= X − k3 σ, X + k3 σ
•
T3 (X)
(39)
√
n it is easy to see that the conditions (12), (13)
Introducing the random variable X−µ
σ
and (14) are equivalent to the following conditions.
√ =β
(40)
P X−µ √n y y ≤ (k1 − λ(p)) n
σ
√
P X−µ √n y y ≤ (k2 − λ(p)) n
=β
(41)
σ
√ n
=β
(42)
P X−µ √n y y ≤ k3 − λ 1+p
σ
2
√
The random variable X−µ
n is distributed according to the standardized normal distriσ
bution. In order to meet the reliability requirement given by the confidence level β, we
obtain:
λ(β)
(43)
k1 = k2 = λ(p) + √
n
1+p
λ(β)
+ √
k3 = λ
(44)
2
n
5.5
Unknown Expectation and Unknown Variance
This case seems most important for industrial practice, because generally there are no
information about the mean or variance in the lot. In this case the following sample
functions are obtained.
92
•
•
•
Elart von Collani and Karl Baur
= X + k1 S(X)
T1 (X)
= X + k2 S(X)
T2 (X)
T3 (X) = T3,1 (X), T3,2 (X) = X − k3 S(X), X + k3 S(X)
The random variables
√
X−µ √
n + λp n
σ
and
S
X−µ √
n
σ
√
+ λ 1+p n
σ
2
S
σ
P X−µ √n+λ
σ
S
σ
√
1+p n
2
√ z z ≤ k3 n
=β
The random variables
√
X−µ √
n+λ(p) n
σ
S
σ
and
(46)
(47)
(48)
S
σ
are considered and the following events are deduced from (12), (13) and (14):
√ z z ≤ ki n
=β
for i = 1, 2
P X−µ √n+λp √n
σ
(45)
(49)
(50)
X−µ √
n+λ 1+p
σ
2
S
σ
(
√
)
n
are distributed according to the
noncentral
t-distribution
√ with (n − 1) degrees of freedom and non-centrality parameter
√
n, respectively. In order to meet the reliability requirement given
λ(p) n and λ 1+p
2
by the confidence level β, we obtain:
1
k1 = k2 = √ t(n−1,λ(p)√n) (β)
for i = 1, 2
(51)
n
1
k3 = √ t(n−1,λ( 1+p )√n) (β)
(52)
2
n
where tn,c (a) denotes the upper quantile of the noncentral t-distribution with n degrees
of freedom, non-centrality parameter c and order a.
6
Tolerance Intervals of DIN 55303 (Part 5)
The standard DIN 55303 (Part 5) contains “statistical tolerance intervals” for making
statements on the proportion of items of a given lot falling into the center or into one
of the tails of the frequency distribution. The length and location of a tolerance interval
covering the center or the tails can be compared with the specification interval and, thus,
used for evaluating the quality of a given lot.
The standard DIN 55303 (Part 5) contains a brief description of how to determine statistical tolerance intervals for two types of situation with respect to the used approximation:
1. normal model, and
2. non-normal but continuous model.
93
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
In the following the procedures offered in the standard are compared with those derived
in the previous section in case the normal approximation is used.
6.1
Known Expectation and Known Varianz (DIN 55303)
In the case of complete knowledge about the distribution, the tolerance intervals become
simple prediction intervals and, of course, the standard gives the same results (19), (20)
and (21) as derived above. This case has been discussed here for the reason of clarification
and illustration and not because it is of any importance for practical application. In
contrast to an academic paper, an industrial standard aims at presenting solutions for
problems arising in practical operation.
The solutions offered in DIN 55303 (Part 5) refer to a given lot {x1 , x2 , . . . , xN } and
N
serve to evaluate its unknown quality. It is difficult to imagine that x̄ =
xi and
i=1
s2 =
1
N
N
(xi − x̄)2 are known for a given lot, but still there is the demand for a quality
i=1
evaluation by means of tolerance intervals of a normally distributed random variable.
6.2
Known Expectation and Known Varianz (DIN 55303)
In the previous section the case of known expectation and unknown variance has been
considered, because it may in fact arise in industrial practice in situations like the following:
Let the quality characteristic be the weight of the elements and assume that
the weight of the whole lot can be measured, then the average mean weight of
the elements which is identical to the expectation of X is exactly known.
However, this case is omitted in the standard.
6.3
Unknown Expectation and Known Variance (DIN 55303)
The second case considered in the standard refers to an unknown expectation and a known
variance. As mentioned earlier this case seems to be rather out of touch with reality. In
fact it is difficult to imagine a situation that for a given lot the value of s2 is known, but
the value of x̄ not.
The solutions given in the standard for determining tolerance intervals for the tails of the
distribution are the same as derived here and given by (43).
the solution
of a
However,
(u) 1−p (u) 1+p , qX
tolerance interval covering the central prediction interval qX
given in
2
2
the standard differs from (43). In fact, the method described in the standard leads to
shorter tolerance intervals and, therefore, seems to be superior.
The approach used in the previous section starts by selecting the desired prediction interval by means of the value of γ and proceeds by deriving a procedure meeting the reliability
94
Elart von Collani and Karl Baur
requirement given by the confidence level β as a lower bound for the probability that the
procedure yields an interval which covers the corresponding prediction interval.
The method used in the standard in the case of a central prediction interval proceeds in
almost exactly the opposite way. The value of γ and, thus, the desired
prediction interval
are not specified and, therefore, a result T3 (x) = (T3,1 (x), T3,2 (x) is considered correct,
if it covers a proportion of 100p % of the items in the lot, no matter where the proportion
is located. Thus, a result (t3,1 , t3,2 ) is correct, if the following inequality holds.
PX ({x | t3,1 ≤ x ≤ t3,2 }) ≥ p
As the variance is assumed to be known, the procedure is completely based on X:
= X − kσ
T3,1 (X)
= X + kσ
T3,2 (X)
(53)
(54)
(55)
The procedure given by T3 shall yield a correct result with probability 1 − α, i.e., there is
a γ ∈ [p, 1] that the following inequality holds.
(u)
(u)
(t3,1 , t3,2 ) t3,1 ≤ qX (γ − p), t3,2 ≥ qX (γ)
≥1−α
(56)
PT3
The only random part which enters T3 is X and the shortest event with respect to X
which occurs with probability 1 − α is given by:
σ
σ
(57)
E = µ − λ1+ α2 √ , µ + λ1+ α2 √
n
n
Thus, k is determined so that for any x̄ ∈ E the inequality (53) is met. The largest value
of k for x̄ ∈ E becomes necessary, if x adopts one of the boundary values of E. Therefore,
if (53) is met, for instance, for x̄ = µ + λ1+ α2 √σn , then (53) is met for any x̄ ∈ E. It follows
that the desired value of k may be obtained as solution of equation (58).
σ
σ
PX
µ + λ1+ α2 √ − kσ, µ + λ1+ α2 √ + kσ
n
n
λ1+ α2
λ1+ α2
=Φ √ +k −Φ √ −k =p
(58)
n
n
In the standard DIN 55303 (Part 5) formula (58) is used to calculate the tolerance interval
for a “central” prediction interval. However, one should be aware, that the resulting
tolerance interval does not necessarily be central and that generally it does not cover the
central prediction interval defined by (21).
The probability of covering the central prediction interval decreases rapidly with increasing
sample size n. In Table 1 the coverage probability for some values of p, α and n are
displayed.
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
95
Table 1: Probability that the tolerance intervals given in the standard cover the central
prediction interval.
n
1
10
20
50
100
500
6.4
1 − α = 0.95
p = 0.90 p = 0.95 p = 0.99
0.892
0.901
0.914
0.633
0.688
0.763
0.512
0.553
0.662
0.355
0.380
0.495
0.275
0.311
0.341
0.092
0.178
0.249
1 − α = 0.95
p = 0.90 p = 0.95 p = 0.99
0.973
0.977
0.980
0.859
0.893
0.926
0.746
0.805
0.865
0.543
0.604
0.724
0.419
0.452
0.600
0.265
0.178
0.249
Unknown Expectation and Unknown Variance (DIN 55303)
Similar as in the previous case, the tolerance intervals for the tail prediction intervals are
identical with (51) and the tolerance interval for the central prediction interval is different
from (52).
The tolerance interval for the center is computed using a approximation method developed
by Wald and Wolfowitz [11], which uses the same approach as in the case of known
variance, i.e. a tolerance interval is defined as being correct if it covers a prediction interval
(u)
(u)
[qX (γ − p), qX (γ)]
(59)
for an arbitrary number γ ∈ [p, 1].
The tolerance intervals given by in the standard do not cover the central prediction interval
with the given confidence level. Consequently, they are slightly shorter then those derived
here. However, one should be aware that the tolerance interval does not necessarily reflect
the center of the distribution and, therefore, its interpretation is somewhat difficult.
7
Remarks on the Normal Distribution
As already mentioned, the methods proposed in the standard are used for making decisions
on given lots of manufactured items. Let N be the lot size and xi , i = 1, . . . , N the values
of the quality characteristic of the items. The mean value in the lot is given by
N
1 x=
xi
N i=1
(60)
and the mean quadratic deviation from x is given by
N
1 s =
(xi − x)2
N i=1
2
(61)
Consider the experiment of randomly drawing one item from the set and determining the
value of its quality characteristic. Let X denote the corresponding random variable, with
96
Elart von Collani and Karl Baur
expectation x, variance s2 and distribution function equal to the frequency distribution of
the given set of items. An approximation by means of the continuous normal distribution
may only work, if the following conditions are met:
• The number of different values xi is sufficiently large.
• The distance between two subsequent values is small.
• The frequency distribution has a maximum at x = x.
• The frequency distribution is more or less symmetric to x = x.
• The frequency distributions has very small tails.
If there is doubt that any one of these conditions is met sufficiently well, then one should
not approximate the frequency distribution by a normal distribution. Already S.S. Wilks
[13] mentions in his pioneering paper that in case of “even slight non-normality, particular
when skewness is present” the tolerance limits calculated based on the normal distribution
“are apt to give very erroneous results with reference to the proportion of the universe
included in the tails”.
There are three cases which may occur in industrial practice:
1. The process produces more or less on target, process variability is small and the
desired prediction interval is rather small compared with the specification interval.
2. The process produces off target, process variability is large and the prediction interval is large compared with the specification interval.
3. The process produces more or less on target, process variability is reasonable and
the prediction interval is not considerably smaller than the specifications.
Generally, the first two cases make no difficulty. An meaningful procedure decides correctly by accepting the lot in the first case and rejecting the lot in the second case.
Problems with the procedure arise only in the last situation, where it is difficult to make
a correct decision.
Already slight deviations from the above formulated necessary conditions for applying the
normal approximation will lead to erroneous results and, therefore, the actual procedure’s
reliability is not at all controlled by the desired confidence level 1 − α.
8
Conclusions
There are several remarks to be made with respect to tolerance intervals and their application in quality control. The first one refers to their appropriateness in principle, the
second one to the normal approximation and the third one to the industrial standard.
Prediction Intervals, Tolerance Intervals and Standards in Quality Control - Part 1
97
If the quality of a given lot or a future lot shall be evaluated for making a decision, a
suitable lot quality characteristic has to be defined. As soon as the quality characteristic
is available, an appropriate method for controlling it, can be selected. Tolerance intervals
represent a method without a quality characteristic and, therefore, the question arises,
which quality characteristic can be effectively controlled by tolerance intervals. If lot
quality is measured by the proportion of nonconforming items a simple acceptance sampling plan would be more efficient. Thus, the underlying objective for applying tolerance
intervals in industrial quality control remains unclear.
Clearly, the frequency distribution of a given lot is almost never symmetric around x̄, even
if the process distribution may be approximated by a normal distribution. There is no
possibility to quantify the deviations from symmetry implying that the actual reliability
of the procedure cannot be assessed. Therefore, application of tolerance intervals based
on the normal distribution appears to be hazardous for both consumer and producer.
The industrial standard on tolerance intervals exhibits additional weaknesses. The cases
treated in the standard do not include the case of known mean but unknown variance,
although it is of significance for practical application. Moreover, the “central” tolerance
intervals given in the standard are not at all central. This fact is not mentioned in the
standard and, therefore, misinterpretations of the obtained results will necessarily occur.
To sum up, efficiency and even appropriateness of the tolerance intervals in the industrial
standard based on the normal approximation are doubtful and, therefore, should not be
recommended for application as a general tool.
Besides the tolerance intervals based on the normal approximation, the standard DIN
55303 also offers methods which are not based on a special distribution family. These
methods will be evaluated in a second part of this paper.
References
[1] Anghel, C. (2001): Statistical Process Control Methods from the Viewpoint of Industrial Application. Economic Quality Control 16, 49-63.
[2] v. Collani, E. (1999): Control of Production Processes Subject to Random Shocks.
Annals of Operations Research 91, 289-304.
[3] v. Collani, E. und Dräger, K. (2001): Binomial Distribution Handbook for Scientists
and Engineers. Birkhäuser, Boston.
[4] DIN 55303 (Teil 5) (1987): Statistische Auswertung von Daten - Bestimmung eines
statistischen Anteilsbereichs. Beuth Verlag, Berlin.
[5] Hall, I.J. and Sheldon, D.D. (1979): Improved Bivariate Normal Tolerance Regions
With Some Applications. J. Quality Technology 11, 13-19. QK-01.
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Elart von Collani and Karl Baur
[6] Kabe, D.G. (1976): On Confidence Bands for Quantiles of a Normal Distribution.
JASA 71, 417-419.
[7] Krewski, D. (1976): Distribution-Free Confidence Intervals for Quantile Intervals.
JASA 71, 420-422.
[8] Owen, D.B. (1964): Control of Percentages in Both Tails of the Normal Distribution.
Technometrics 6, 377-387.
[9] W.A. Shewhart (1931): Economic Control of Quality of Manufactured Product. Van
Nostrand, Princeton.
[10] Wald, A. (1943): An Extension of Wilk’s method of Setting Tolerance Limits. Annals
of Mathematical Statistics 14, 45-55.
[11] Wald, A. and Wolfowitz, J. (1946): Tolerance Limits for a Normal Distribution.
Annals of Mathematical Statistics 17, 208-215.
[12] Weissberg, A. and Beatty, G.H. (1960): Tables of Tolerance Limit Factors for Normal Distributions. Technometrics 2, 483-500.
[13] Wilks, S.S. (1941): Determination of Sample Sizes for Setting Tolerance Limits.
Annals of Mathematical Statistics 12, 91-96.
[14] Wilks, S.S. (1942): Statistical Prediction With Special Reference to the Problem of
Tolerance Limits. Annals of Mathematical Statistics 13, 400-409.
[15] Wilks, S.S. (1962): Mathematical Statistics. Wiley & Sons, new York.
Elart von Collani
University of Würzburg
Sanderring 2
D-97070 Würzburg
collani@mathematik.uni-wuerzburg.de
Karl Baur
E.ON Kernkraft GmbH
Treskowstr. 5
D-30457 Hannover
karl.baur@eon-energie.com