ASTM E876-89(1994)e1

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e1
Designation: E 876 – 89 (Reapproved 1994)
Standard Practice for
Use of Statistics in the Evaluation of Spectrometric Data
This standard is issued under the fixed designation E 876; 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.
e NOTE—Section 7 was added editorially in January 1995.
1. Scope 3.2.2 bias—a systematic displacement of all or most deter-
minations from the assumed true value. An acceptable bias
1.1 This practice provides for the statistical evaluation of
should be agreed upon prior to testing a method. Accuracy,
data obtained from spectrometrical methods of analysis. In-
often used to qualify a method, is a measurement which
cluded are definitions used in statistics, methods to determine
includes both imprecision and bias.
variance and standard deviation of data, and calculations for
(1) estimate of variance and pooling estimates of variance, (2)
NOTE 1—Precision and bias are discussed in detail in Practice E 177. In
standard deviation and relative standard deviation, (3) testing analytical methods, precision refers to the distribution of repeat determi-
nations about the average. All analyses are presumed to have been made
for outliers, (4) testing for bias, (5) establishing limits of
under the same set of conditions. Standard deviation provides a measure
detection, and (6) testing for drift.
of this distribution.
NOTE 2—An evaluation of a method will be sample-dependent. Mul-
2. Referenced Documents
tiple samples should be tested for homogeneity since even certified
2.1 ASTM Standards:
reference materials may exhibit significantly different degrees of inhomo-
E 135 Terminology Relating to Analytical Chemistry for
geneity. A measure of both sample and method precision may be made by
Metals, Ores, and Related Materials replicating determinations on specific portions of the sample specimens.
E 177 Practice for Use of the Terms Precision and Bias in
3.2.3 confidence to be placed on the estimate of mu (μ)—the
ASTM Test Methods
average, x¯, is expected to be close toμ and should be very close
E 178 Practice for Dealing with Outlying Observations
if the number of determinations is large, no significant bias
E 305 Practice for Establishing and Controlling Spectro-
exists and the standard deviation, s, is small. The degree of
chemical Analytical Curves
closeness is expressed as a probability (confidence level) that μ
E 456 Terminology Relating to Quality and Statistics
is in a specified interval (confidence interval) centered at x¯.
With a certain probability, limits are placed on the quantity x¯
3. Terminology
which may include the unknown quantity μ. A probability
3.1 Definitions:
level, p %, can be selected so that μ will be within the limits
3.1.1 For definitions of terms used in this practice, refer to
placed about x¯. See 3.2.1
Terminologies E 135 and E 456.
3.2.4 degrees of freedom (df)—the number of contributors
3.1.2 All quantities computed from limited data are defined
to the deviations of a measurement. Since a deviation can be
as estimates of the parameters that are properties of the system
implied only when there are at least two members of a group,
(population) from which the data were obtained.
the degrees of freedom of a set of measurements is generally
3.2 Definitions of Terms Specific to This Standard:
one less than the number of measurements. It is the sample size
3.2.1 average measurement ( x¯)—the arithmetic mean ob-
less the number of parameters estimated. If the group is a
tained by dividing the sum of the measurements by the number
listing of a series of differences of measurements or a series of
of measurements. It is an estimate of μ, the value of the
determinations of variance, the degrees of freedom is the
population that the average would become if the number of
number of these differences or the total of the degrees of
measurements were infinite. Either x¯ or μ may include a
freedom of each series of determinations.
systematic error if there is a bias in the measurement.
3.2.5 detection limit—paraphrasing the definition in Termi-
nology E 135, it is the lowest estimated concentration that
1 permits a confident decision that an element is present. The
This practice is under the jurisdiction of ASTM Committee E-1 on Analytical
actual concentration being measured falls within a confidence
Chemistry for Metals, Ores and Related Materials and is the direct responsibility of
Subcommittee E01.22 on Statistics and Quality Control.
interval that encompasses the estimated concentration. The
Current edition approved Nov. 20, 1989. Published January 1990. Originally
lowest estimate has a confidence interval that reaches to zero
published as E 876 – 82. Last previous edition E 876 – 89.
2 concentration, but not below. It cannot be assumed that the
Annual Book of ASTM Standards, Vol 03.05.
Annual Book of ASTM Standards, Vol 14.02. estimated concentration is an actual concentration. Neither can
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
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E 876 – 89 (1994)
it be assumed that an actual concentration that equals the number of parameters estimated). If correction is made for
detection limit will always give a positive detection. The spectral interference, the number of constants used in the
definition in Terminology E 135 properly characterizes this correction should be counted as being calibration constants.
detection as being a limiting value. See also determination This would be true whether correction was made to readings or
limit which follows. to final concentrations. When the measurement is made with
3.2.6 determination limit—the estimated low concentration new data and applied to a previously determined calibration,
where the range of the encompassing confidence interval bears the degrees of freedom is the number of references used to
some specified maximum ratio to that concentration. The ratio make the test. In either case, the significance of the measure-
would depend upon what is acceptable in a specific application. ment is limited to the range of concentration in the reference
3.2.7 drift—a gradual, systematic change in measurements materials and implies that concentrations will be fairly well
spread within that range.
(either increasing or decreasing) from start to completion of a
set of replicate determinations of the same material.
3.2.8 estimate of standard deviation (s)—the square root of 4. Significance and Use
the estimate of variance. It is a measure of the variability of a
4.1 The data obtained in spectrometrical analyses may be
set of values representing the whole population. It is an
evaluated as statistical measurements. Use of the various
estimate of s, the actual standard deviation of an infinite
determinations of precision which follow permits a consistent
number of measurements. With normal distribution, 68 % of
basis for comparing methods of analysis or for monitoring their
the values in a population will fall within 6s of the true value,
performance.
μ; 95 % within 62 s of μ; and 99.7 % within 63 s of μ.
4.2 Some explanations are included to clarify the function
3.2.9 estimate of variance (s )—a measure of precision of a
of the statistical calculations being made.
measurement based on summing the squares of the deviations
4.3 Examples of all calculations are given in the appen-
of individual determinations from the average and dividing by
dixes.
the degrees of freedom.
3.2.10 outlier—a measurement that, for a specific degree of
5. Calculation
confidence, is not part of the population.
5.1 Average ( x¯):
3.2.11 pooled estimate of variance (s )—the combined
p
x¯ 5 (x/n (1)
estimate of variance calculated from two or more estimates
under the same or similar conditions. Pooling estimates in-
where:
creases the degrees of freedom and improves the quality of the
(x = the sum of all measurements, and
estimate if the variance is approximately the same for each
n = the number of measurements.
measurement.
NOTE 4—Where all items of a category are included in a summation,
NOTE 3—If the concentration level varies considerably within the the simple summation symbol ( will be used. However, the strict
pooled data, the pooled variance may be inaccurate. It may be possible in
mathematical statement of Eq 1 is:
such cases, however, to determine a valid estimate by pooling relative
n
standard deviations.
x¯ 5 ~x !/n
(
i
i 5 1
3.2.12 precision—the agreement among repeat measure-
ments, usually expressed as either repeatability or reproduc- where x ,x , . x is the population of all n determinations which were
1 2 n
made. Since no other constrictions are being made on the summation, the
ibility as defined in Terminology E 456 (see Note 1 and Note
simpler statement of Eq 1 clearly shows the required operation.
2).
3.2.13 range (w)—the difference between the highest and 5.2 Variance (s ):
lowest measurements for a series of values obtained under
5.2.1 Following directly from the definition of 3.2.9 (see
identical conditions. Range is useful for estimating standard Note 4 and Note 5):
deviation and for determining if certain values are outliers.
2 2
s 5 (~x 2 x¯! /~n 2 1! (2)
i
3.2.14 relative standard deviation (RSD)—the standard de-
viation as a percentage of the average analysis or reading. By
providing a means of expressing the precision in relative rather where:
x = an individual determination, and
than absolute terms, it may serve to show a more consistent
i
n = the number of determinations.
measure of precision for widely different values of x¯.
5.2.2 An alternative determination that can be readily
3.2.15 standard error—a term sometimes used synony-
handled with a calculator without first determining x¯ is (see
mously with standard deviation but which will be used here to
Note 4):
measure how consistently the accepted true concentrations of a
2 2 2
series of reference materials compare to the apparent concen-
s 5 @(~x ! 2 ~(x! /n#/~n 2 1! (3)
trations determined from a calibration. It is an estimate that is
similar to standard deviation except for the degrees of freedom
NOTE 5—To prevent significant errors in calculating s do not round the
used. When the measurement is used to define the effectiveness
sum of the squares of differences, ((x −x¯) in Eq 2, nor the sum of the
i
of the calibration established by these reference materials, the 2 2
squares of the measurement, ((x ), and the square of the sum, ((x) ,ofEq
degrees of freedom is the number of data points minus the
3. Although these equations are algebraically identical, they may give
number of constants in the calibration (the sample size less the slightly different results with large numbers or large summations on a
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E 876 – 89 (1994)
computer because of greater round off errors from using Eq 3 instead of
1 . 1 ~(x ! /n # /~N 2 k!
%
k k
Eq 2.
where:
NOTE 6—The numerators of Eq 2 and Eq 3 are often referred to as the
“sum of squares,” meaning the sum of the squares of deviation. Using Eq
x represents all readings, and
i
3, its effectiveness in being a measure of the degree of deviation of a series
N = the total number of readings.
of values from each other can be seen by considering some simple lists.
For example, 2, 3, 4 totals up to 9 for an average of 3. Summing the
NOTE 8—Pooling two or more estimates of variance is valid only if
squares of the three values yields 4+9+16=29. The square of the sum
each set of analyses was obtained under similar or identical conditions
divided by the number of values is 9 /3 = 81/3 = 27. The sum of squares
with samples of similar composition and history. The pooling is valid only
becomes 29 − 27 = 2. If the list were 1, 3, 5, the sum would still be 9 and
if variabilities are statistically the same.
the term ((x) /n would remain as 27. The summing of the squares of the
5.3 Standard Deviation (s)—The estimate of standard de-
three values, however, would yield 1+9+25=35, and the sum of
squares would now become 35 − 27 = 8, giving a quantitative statement as viation follows directly from variance as:
to how much more deviation there is in the second set than the first. The
s 5 =s (8)
sum of squares is insensitive to the level of the readings. Thus, if the list
were 3, 4, 5, the summing would yield 9 + 16 + 25 − 12 /3 = 50−48=2,
5.3.1 A close estimate of s can be determined from the value
resulting in the same difference as the first set as it should since the
of range, w, as defined in 3.2.13:
deviations are the same. If the list were 3, 3, 3, the ((x ) term would
become 27 and the sum of squares would properly be zero since the
s 5 w/ n8 5 ~x 2 x !/ n8 (9)
= =
2 r h l
((x) /n would also be 27 and the list shows no deviation.
where:
5.2.3 Estimate from Duplicate Determinations—The differ-
x = the highest measurement in a set,
h
ence between duplicate determinations can be used to estimate
x = the lowest measurement in a set, and
2 2 l
variance from s = D /2 where D is the difference. It is a special
n8 = number of measurements, limited from 4 through 12.
case of Eq 3, for then, if the two measurements are x and x :
1 2
2 2 2 2
s 5 @x 1 x 2 ~x 1 x ! /2#/~2 2 1! (4)
1 2 1 2 Reliable estimates from range by Eq 8 can be made only for
2 2 2 2
sets of measurements from four through twelve. If extended
5 @2~x 1 x ! 2 ~x 1 2x x 1 x !#/2
1 2 1 1 2 2
beyond twelve measurements, the estimate will be low.
1 1
2 2 2
5 ~x 2 2x x 1 x ! 5 ~x 2 x !
1 1 2 2 1 2
2 2 5.3.2 Repeat measurements, even when made on different
days, might be biased because the second and subsequent
When there are K duplicates, the pooling of the individual
values are expected to agree with the initial value. In coopera-
estimates, as in 5.2.4, becomes (see Note 4):
tive analyses, a laboratory might make extra determinations
2 2
s 5 (D /2K (5)
and report only those that show good agreement. To overcome
If many duplicates are used, the degrees of freedom in- the possibility of such prejudiced results, an estimate of
creases and the quality of the estimate of s improves. standard deviation may be calculated from single determina-
tions made on pairs of samples having similar composition in
NOTE 7—The estimate from duplicates is particularly useful in produc-
a number of different laboratories or by a number of different
tion laboratories where routinely analyzed samples can be analyzed a
analysts. For a single pair, determine the differences in mea-
second time to obtain a measure of precision under practical conditions.
su
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