ASTM E122-99
(Practice)Standard Practice for Calculating Sample Size to Estimate, With a Specified Tolerable Error, the Average for Characteristic of a Lot or Process
Standard Practice for Calculating Sample Size to Estimate, With a Specified Tolerable Error, the Average for Characteristic of a Lot or Process
SCOPE
1.1 This practice covers simple methods for calculating how many units to include in a random sample in order to estimate with a prescribed precision, a measure of quality for all the units of a lot of material, or produced by a process. This practice will clearly indicate the sample size required to estimate the average value of some property or the fraction of nonconforming items produced by a production process during the time interval covered by the random sample. If the process is not in a state of statistical control, the result will not have predictive value for immediate (future) production. The practice treats the common situation where the sampling units can be considered to exhibit a single (overall) source of variability; it does not treat multi-level sources of variability.
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Designation: E 122 – 99
Standard Practice for
Choice of Sample Size to Estimate a Measure of Quality for
a Lot or Process
This standard is issued under the fixed designation E 122; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (e) indicates an editorial change since the last revision or reapproval.
1. Scope
n 5 size of sample j.
j
n 5 size of the sample from a finite lot (7.4).
1.1 This practice covers simple methods for calculating how L
p8 5 fraction of a lot or process whose units have the
many units to include in a random sample in order to estimate
nonconforming characteristic under investigation.
with a prescribed precision, a measure of quality for all the
p 5 an advance estimate of p8.
units of a lot of material, or produced by a process. This
p 5 fraction nonconforming in the sample.
practice will clearly indicate the sample size required to
R 5 range of a set of sampling values. The largest minus
estimate the average value of some property or the fraction of
the smallest observation.
nonconforming items produced by a production process during
R 5 range of sample j.
j
the time interval covered by the random sample. If the process
k
¯
R 5
is not in a state of statistical control, the result will not have
R /k, average of the range of k samples, all of the
(
j
j 5 1
predictive value for immediate (future) production. The prac-
same size (8.2.2).
tice treats the common situation where the sampling units can
s5 lot or process standard deviation of X, the result of
be considered to exhibit a single (overall) source of variability;
measuring all of the units of a finite lot or process.
it does not treat multi-level sources of variability.
s 5 an advance estimate of s.
n
s 5
2 1/2
2. Referenced Documents
[ (X − X¯) /(n−1)] , an estimate of the standard
(
i
i 5 1
2.1 ASTM Standards:
deviation s from n observation, X , i 51ton.
i
k
E 456 Definitions of Terms Relating to Statistical Methods
s¯ 5
S /k, average s from k samples all of the same size
(
j
j 5 1
3. Terminology
(8.2.1).
3.1 Definitions—Unless otherwise noted, all statistical
s 5 pooled (weighted average) s from k samples, not all of
p
terms are defined in Definitions E 456.
the same size (8.2).
3.2 Symbols:Symbols:
s 5 standard derivation of sample j.
j
t 5 a factor (the 99.865th percentile of the Student’s
distribution) corresponding to the degrees of freedom
E 5 maximum allowable sampling error, the difference
f of an advance estimate s (5.1).
o o
between the estimate to be made from the sample and
V 5s/μ, the coefficient of variation of the lot or process.
the result of measuring, by the same methods, all the
V 5 an advance estimate of V (8.3.1).
o
units in the lot or process.
¯
v 5 s/ X, the coefficient of variation estimated from the
e 5 E/μ, maximum allowable sampling error expressed as
sample.
a fraction of μ.
v 5 coefficient of variation from sample j.
j
k 5 the total number of samples available from the same
X 5 numerical value of the characteristic of an individual
or similar lots.
unit being measured.
μ 5 lot or process mean or expected value of X, the result
n
¯
X 5
of measuring all the units in the lot or process. X /n average of n observations, X,i 51ton.
(
i i i
i 5 1
μ 5 an advance estimate of μ.
4. Significance and Use
N 5 size of the lot.
n 5 size of the sample taken from a lot or process.
4.1 This practice is intended for use in determining the
sample size required to estimate, with prescribed precision, a
measure of quality of a lot or process. The practice applies
when quality is expressed as either the lot average for a given
This practice is under the jurisdiction of ASTM Committee E-11 on Statistical
Methods and is the direct responsibility of Subcommittee E11.10 on Sampling and
property, or as the lot fraction not conforming to prescribed
Data Analysis.
standards. The level of a characteristic may often be taken as an
Current edition approved June 10, 1999. Published August 1999. Origianally
indication of the quality of a material. If so, an estimate of the
published as E122–89. Last previous edition E122–89.
Annual Book of ASTM Standards, Vol 14.02. average value of that characteristic or of the fraction of the
Copyright © ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959, United States.
E 122
observed values that do not conform to a specification for that
n 5 3s /E! (1)
~
o
characteristic becomes a measure of quality with respect to that
where:
characteristic. This practice is intended for use in determining
s 5 the advance estimate of the standard deviation of the
o
the sample size required to estimate, with prescribed precision,
lot or process,
such a measure of the quality of a lot or process either as an
E 5 the maximum allowable error between the estimate to
average value or as a fraction not conforming to a specified
be made from the sample and the result of measuring
value.
(by the same methods) all the units in the lot or
process, and
5. Empirical Knowledge Needed
3 5 a factor corresponding to a low probability that the
5.1 Some empirical knowledge of the problem is desirable
difference between the sample estimate and the result
in advance.
of measuring (by the same methods) all the units in
5.1.1 We may have some idea about the standard deviation
the lot or process is greater than E. The choice of the
of the characteristic.
factor 3 is recommended for general use. With the
5.1.2 If we have not had enough experience to give a precise
factor 3, and with a lot or process standard deviation
estimate for the standard deviation, we may be able to state our
equal to the advance estimate, it is practically certain
belief about the range or spread of the characteristic from its
that the sampling error will not exceed E. Where a
lowest to its highest value and possibly about the shape of the
lesser degree of certainty is desired a smaller factor
distribution of the characteristic; for instance, we might be able
may be used (Note 1).
to say whether most of the values lie at one end of the range,
or are mostly in the middle, or run rather uniformly from one
NOTE 1—For example, the factor 2 in place of 3 gives a probability of
end to the other (Section 9). about 45 parts in 1000 that the sampling error will exceed E. Although
distributions met in practice may not be normal (Note 2), the following
5.2 If the aim is to estimate the fraction nonconforming,
text table (based on the normal distribution) indicates approximate
then each unit can be assigned a value of 0 or 1 (conforming or
probabilities:
nonconforming), and the standard deviation as well as the
Factor Approximate Probability of Exceeding E
shape of the distribution depends only on p8, the fraction
3 0.003 or 3 in 1000 (practical certainty)
nonconforming to the lot or process. Some rough idea con-
2.56 0.010 or 10 in 1000
2 0.045 or 45 in 1000
cerning the size of p8 is therefore needed, which may be
1.96 0.050 or 50 in 1000 (1 in 20)
derived from preliminary sampling or from previous experi-
1.64 0.100 or 100 in 1000 (1 in 10)
ence.
NOTE 2—If a lot of material has a highly asymmetric distribution in the
5.3 Sketchy knowledge is sufficient to start with, although
characteristic measured, the factor 3 will give a different probability,
more knowledge permits a smaller sample size. Seldom will
possibly much greater than 3 parts in 1000 if the sample size is small.
there be difficulty in acquiring enough information to compute
There are two things to do when asymmetry is suspected.
the required size of sample. A sample that is larger than the
7.1.1 Probe the material with a view to discovering, for
equations indicate is used in actual practice when the empirical
example, extra-high values, or possibly spotty runs of abnor-
knowledge is sketchy to start with and when the desired
mal character, in order to approximate roughly the amount of
precision is critical.
the asymmetry for use with statistical theory and adjustment of
5.4 In any case, even with sketchy knowledge, the precision
the sample size if necessary.
of the estimate made from a random sample may itself be
7.1.2 Search the lot for abnormal material and segregate it
estimated from the sample. This estimation of the precision
for separate treatment.
from one sample makes it possible to fix more economically
7.2 There are some materials for which s varies approxi-
the sample size for the next sample of a similar material. In
mately withμ , in which case V (5s/μ) remains approximately
other words, information concerning the process, and the
constant from large to small values of μ.
material produced thereby, accumulates and should be used.
7.2.1 For the situation of 7.2, the equation for the sample
6. Precision Desired
size, n, is as follows:
6.1 The approximate precision desired for the estimate must 2
n 5 ~3 V /e! (2)
o
be prescribed. That is, it must be decided what maximum
deviation, E, can be tolerated between the estimate to be made where:
from the sample and the result that would be obtained by V 5 (coefficient of variation)5s /μ the advance estimate
o o o
of the coefficient of variation, expressed as a fraction
measuring every unit in the lot or process.
6.2 In some cases, the maximum allowable sampling error is (or as a percentage),
e 5 E/μ, the allowable sampling error expressed as a
expressed as a proportion, e, or a percentage, 100 e. For
fraction (or as a percentage) of μ, and
example, one may wish to make an estimate of the sulfur
μ 5 the expected value of the characteristic being mea-
content of coal with maximum allowable error of 1 %, or e
sured.
5 0.01.
If the relative error, e, is to be the same for all values of μ,
7. Equations for Calculating Sample Size
then everything on the right-hand side of Eq 2 is a constant;
7.1 Based on a normal distribution for the characteristic, the hence n is also a constant, which means that the same sample
equation for the size, n, of the sample is as follows: size n would be required for all values of μ.
E 122
TABLE 1 Values of the Correction Factor C and d for Selected
7.3 If the problem is to estimate the lot fraction noncon-
4 2
A
2 Sample Sizes n
j
forming, then s is replaced by p so that Eq 1 becomes:
o o
Sample Size ,(n ) C d
j 4 2
n 5 ~3/E! p ~1 2 p ! (3)
o o
2 .798 1.13
4 .921 2.06
where:
5 .940 2.33
p 5 the advance estimate of the lot or process fraction
o 8 .965 2.85
10 .973 3.08
nonconforming p8 and E # p
1 o
A
7.4 When the average for the production process is not As n becomes large, C approaches 1.000.
j 4
needed, but rather the average of a particular lot is needed, then
the required sample size is less than Eq 1, Eq 2, and Eq 3
¯
R
indicate. The sample size for estimating the average of the
s 5 (8)
o
d
finite lot will be: 2
The factor, d , from Table 1 is needed to convert the average
n 5 n/@1 1 ~n/N!# (4) 2
L
range into an unbiased estimate of s .
o
where:
8.2.3 Example 1—Use of s¯.
n 5 the value computed from Eq 1, Eq 2, or Eq 3, and
8.2.3.1 Problem—To compute the sample size needed to
N 5 the lot size.
estimate the average transverse strength of a lot of bricks when
This reduction in sample size is usually of little importance
the desired value of E is 50 psi, and practical certainty is
unless n is 10 % or more of N.
desired.
8.2.3.2 Solution—From the data of three previous lots, the
8. Reduction of Empirical Knowledge to a Numerical
values of the estimated standard deviation were found to be
Value of s (Data for Previous Samples Available)
o
215, 192, and 202 psi based on samples of 100 bricks. The
8.1 This section illustrates the use of the equations in
average of these three standard deviations is 203 psi. The c
Section 7 when there are data for previous samples.
value is essentially unity when Eq 1 gives the following
8.2 For Eq 1—An estimate of s can be obtained from
equation:
o
previous sets of data. The standard deviation, s, from any given
2 2
n @~3 3 203!/50!# 5 5 149 bricks (9)
sample is computed as:
for the required size of sample to give a maximum sampling
n
2 1/2
¯
s 5 X 2 X! / n 2 1! (5) error of 50 psi, and practical certainty is desired.
@ ~ ~ #
( i
i 5 1
8.3 For Eq 2—If s varies approximately proportionately
The s value is a sample estimate ofs . A better, more stable
o with μ for the characteristic of the material to be measured,
value for s may be computed by pooling the s values obtained
¯
o
compute both the average, X, and the standard deviation, s, for
from several samples from similar lots. The pooled s value s
p several samples that have the same size. An average of the
for k samples is obtained by a weighted averaging of the k
¯
several values of v 5 s/ X may be used for V .
o
results from use of Eq 5.
8.3.1 For cases where the sample sizes are not the same, a
k k
weighted average should be used as an approximate estimate
2 1/2
s 5 n 2 1!s / n 2 1! (6)
@ ~ ~ #
p ( j j ( j
for V
j 5 1 j 5 1 o
k k
1/2
where:
V 5 @ ~n 2 1!v / ~n 2 1!# (10)
o ( j j ( j
j 5 1 j 5 1
s 5 the standard deviation for sample j,
j
n 5 the sample size for sample j.
j
where:
8.2.1 If each of the previous data sets contains the same
v 5 the coefficient of variation for sample j, and
j
number of measurements, n , then a simpler, but slightly less
j
n 5 the sample size for sample j.
j
efficient estimate for s may be made by using an average ( s¯)
o 8.3.2 Example 2—Use of V, the estimated coefficient of
of the s values obtained from the several previous samples. The
variation:
calculated s¯ value will in general be a slightly biased estimate
8.3.2.1 Problem—To compute the sample size needed to
of s . An unbiased estimate of s is computed as follows:
o o
estimate the average abrasion resistance of a material when the
s¯ desired value of e is 0.10 or 10 %, and practical certainty is
s 5 (7)
o
c
desired.
8.3.2.2 Solution—There are no data from previous samples
where the value of the correction factor, c , depends on the
of this same material, but data for six samples of similar
size of the individual data sets (n ) (Table 1 ).
j
materials show a wide range of resistance. However, the values
8.2.2 An even simpler, and slightly less efficient estimate
of estimated standard deviation are approximately proportional
¯
fors may be computed by using the average range ( R) taken
o
to the observed averages, as shown in the following text table:
from the several previous data sets that have the same group
Estimate of Coefficient
size.
Sample Avg Observed
Lot No. s 5 of Varia-
o
¯
Sizes Cycles range, R
A
¯
R/3.08 tion, %
1 10 90 40 13.0 14
2 10 190 100 32.5 17
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