1010 Introduction

1010 INTRODUCTION*
1010 A. Scope and Application of Methods
The procedures described in Standard Methods for the Examination of Water and Wastewater are intended for use in analyzing a wide range of waters, including surface water, ground
water, saline water, domestic and industrial water supplies, cooling or circulating water, boiler water, boiler feed water, and
treated and untreated municipal and industrial wastewaters. In
recognition of the unity of the water, wastewater, and watershed
management fields, the analytical methods are categorized based
on constituent, not type of water.
An effort has been made to present methods that apply generally. When alternative methods are necessary for samples of
different composition, the basis for selecting the most appropriate method is presented as clearly as possible. In specific instances (e.g., samples with extreme concentrations or otherwise
unusual compositions or characteristics), analysts may have to
modify a method for it to be suitable. If so, they should plainly
state the nature of the modification when reporting the results.
Certain procedures are intended for use with sludges and
sediments. Here again, the effort has been made to present
methods with the widest possible application. However, these
methods may require modification or be inappropriate for chemical sludges or slurries, or other samples with highly unusual
composition.
Most of the methods included here have been endorsed by
regulators. Regulators may not accept procedures that were
modified without formal approval.
Methods for analyzing bulk water-treatment chemicals are not
included. American Water Works Association committees prepare and issue standards for water treatment chemicals.
Laboratories that desire to produce analytical results of known
quality (i.e., results are demonstrated to be accurate within a
specified degree of uncertainty) should use established quality
control (QC) procedures consistently. Part 1000 provides a detailed overview of QC procedures used in the individual standard
methods as prescribed throughout Standard Methods. Other sections of Part 1000 address laboratory safety, sampling procedures, and method development and validation. Material presented in Part 1000 is not necessarily intended to be prescriptive
nor to replace or supersede specific QC requirements given in
individual sections of this book. Parts 2000 through 9000 contain
sections describing QC practices specific to the methods in the
respective Parts; these practices are considered to be integral to
the methods. Most individual methods will contain explicit instructions to be followed for that method (either in general or for
certain regulatory applications).
Similarly, the overview of topics covered in Part 1000 is not
intended to replace or be the sole basis for technical education
and training of analysts. Rather, the discussions are intended as
aids to augment and facilitate reliable use of the test procedures
herein. Each Section in Part 1000 contains references that can be
reviewed to gain more depth or details for topics of interest.
* Reviewed by Standard Methods Committee, 2011.
1010 B. Statistics
1. Normal Distribution
summation procedure but with n equal to a finite number of
repeated measurements (10, 20, 30, etc.):
If a measurement is repeated many times under essentially
identical conditions, the results of each measurement (x) will be
distributed randomly about a mean value (arithmetic average)
because of uncontrollable or experimental uncertainty. If an
infinite number of such measurements were accumulated, then
the individual values would be distributed in a curve similar to
those shown in Figure 1010:1. Figure 1010:1A illustrates the
Gaussian (normal) distribution, which is described precisely by
the mean (␮) and the standard deviation (␴). The mean (average)
is simply the sum of all values (xi) divided by the number of
values (n).
x៮ ⫽ 共兺x i兲/n for estimated mean
The standard deviation of the entire population measured is as
follows:
␴ ⫽ 关兺共x i ⫺ ␮ 兲 2/n兴 1/2
The empirical estimate of the sample standard deviation (s) is
as follows:
␮ ⫽ 共兺x i兲/n for entire population
s ⫽ 关兺共x i ⫺ x៮ 兲 2/共n ⫺ 1兲兴 1/2
Because no measurements are repeated infinitely, it is only
possible to make an estimate of the mean (x៮ ) using the same
The standard deviation fixes the width (spread) of the
normal distribution and consists of a fixed fraction of the
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INTRODUCTION (1010)/Statistics
Figure 1010:1. Three types of frequency distribution curves—normal Gaussian (A), positively skewed (B), and negatively skewed
(C)—and their measures of central tendency: mean, median, and mode. Courtesy: L. Malcolm Baker.
measurements that produce the curve. For example, 68.27%
of the measurements lie within ␮ ⫾ 1␴, 95.45% between
within ␮ ⫾ 2␴, and 99.73% within ␮ ⫾ 3␴. (It is sufficiently
accurate to state that 95% of the values are within ␮ ⫾ 2␴ and
99% within ␮ ⫾ 3␴.) When values are assigned to the ⫾ ␴
multiples, they are called confidence limits, and the range
between them is called the confidence interval. For example,
10 ⫾ 4 indicates that the confidence limits are 6 and 14, and
the confidence interval ranges from 6 to 14.
Another useful statistic is the standard error of the mean
(␴␮)—the standard deviation divided by the square root of the
number of values 共␴/ 冑n). This is an estimate of sampling accuracy; it implies that the mean of another sample from the same
population would have a mean within some multiple of ␴␮. As
with ␴, 68.27% of the measurements lie within ␮ ⫾ 1␴␮,
95.45% within ␮ ⫾ 2␴␮, and 99.73% within ␮ ⫾ 3␴␮. In
practice, a relatively small number of average values is available,
so the confidence intervals about the mean are expressed as:
where t has the following values for 95% confidence intervals:
n
t
n
t
2
3
4
12.71
4.30
3.18
5
10
⬁
2.78
2.26
1.96
Using t compensates for the tendency of a small number of
values to underestimate uncertainty. For n ⬎ 15, it is common to
use t ⫽ 2 to estimate the 95% confidence interval.
Still another statistic is the relative standard deviation (␴/␮)
with its estimate (s/x̄), also known as the coefficient of variation
(CV), which commonly is expressed as a percentage. This statistic normalizes ␴ and sometimes facilitates direct comparisons
among analyses involving a wide range of concentrations. For
example, if analyses at low concentrations yield a result of 10 ⫾
1.5 mg/L and at high concentrations yield a result of 100 ⫾ 8
mg/L, the standard deviations do not appear comparable. However, the percent relative standard deviations are 100 (1.5/10) ⫽
x៮ ⫾ ts/ 冑n
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INTRODUCTION (1010)/Statistics
15% and 100 (8/100) ⫽ 8%, indicating that the variability is not
as great as it first appears.
The mean, median, and mode for each curve in Figure 1010:1
were calculated as follows:
1) Mean is the value at the 50th percentile level, or arithmetic
average,
2) Mode is the value that appears most frequently, and
3) Median1 is estimated as follows:
TABLE 1010:I. CRITICAL VALUES FOR 5% AND 1% TESTS OF
DISCORDANCY FOR A SINGLE OUTLIER IN A NORMAL SAMPLE
Median ⫽ 1⁄3 共2 ⫻ Mean ⫹ Mode)
2. Log-Normal Distribution
In many cases, the results obtained from analyzing environmental samples will not be normally distributed [i.e., a graph of
the data distribution will be obviously skewed (see Figure
1010:1B and C)] so the mode, median, and mean will be distinctly different. To obtain a nearly normal distribution, convert
the measured variable results to logarithms and then calculate x៮
and s. The antilogarithms of x៮ and s are the estimates of geometric mean (x៮ g) and geometric standard deviation (sg). The
geometric mean is defined as:
x៮ g ⫽ 关兿共x i兲兴 1/n ⫽ antilog 兵1/n关兺 log 共xi 兲兴其
Calibration curve data can be fitted to a straight line or
quadratic curve by the least squares method, which is used to
determine the constants of the curve that the data points best fit.
To do this, choose the equation that best fits the data points and
assume that x is the independent variable and y is the dependent
variable (i.e., use x to predict the value of y). The sum of the
squares of the differences between each actual data point and its
predicted value are minimized.
For a linear least squares fit of
1.15
1.49
1.75
1.94
2.10
2.22
2.32
2.41
2.55
2.66
2.71
2.75
2.82
2.88
3.10
3.24
3.34
3.41
3.60
3.66
冧
关兺y2 ⫺ a0 兺y ⫺ a1 兺xy ⫺ a2 兺x2 y兴
1
兺y2 ⫺ 共兺y兲2
n
冋
册
0.5
4. Rejecting Data
In a series of measurements, one or more results may differ
greatly from the others. Theoretically, no result should be arbitrarily rejected because it may indicate either a faulty technique
(casting doubt on all results) or a true variant in the distribution.
In practice, it is permissible to reject the result of any analysis in
which a known error occurred. In environmental studies, extremely high and low concentrations of contaminants may indicate either problematic or uncontaminated areas, so they should
not be rejected arbitrarily.
An objective test for outliers has been described.4 If a set of
data is ordered from low to high (xL, x2 . . . xH) and the mean and
standard deviation are calculated, then suspected high or low
outliers can be tested via the following procedure. First, calculate
the statistic T using the discordancy test for outliers:
共兺x兺y/n ⫺ 兺xy兲
关共兺x兲 2/n ⫺ 兺x 2兴
兺y ⫺ m兺x
n
共兺x兺y/n兲 ⫺ 兺xy
兺y 2 ⫺ 共兺y兲 2/n
1.15
1.46
1.67
1.82
1.94
2.03
2.11
2.18
2.29
2.37
2.41
2.44
2.50
2.56
2.74
2.87
2.96
3.03
3.21
3.27
冦
The correlation coefficient1–3 (degree of fit) is:
冋
3
4
5
6
7
8
9
10
12
14
15
16
18
20
30
40
50
60
100
120
r⫽ 1⫺
the slope (a1) and the y intercept1–3 (a0) are computed as follows:
r⫽m
1%
Critical Value
a more detailed description of the algebraic manipulations, see
the cited references.
In this case, the correlation coefficient1 is:
y ⫽ mx ⫹ b
b⫽
5%
SOURCE: BARNET, V. & T. LEWIS. 1995. Outliers in Statistical Data, 3rd ed. John
Wiley & Sons, New York, N.Y.
3. Least Square Curve Fitting
m⫽
Number of
Measurements
n
册
0.5
The best fit is when r ⫽ 1. There is no fit when r ⫽ 0.
For a quadratic least squares fit of
y ⫽ a 2 x 2 ⫹ a 1 x ⫹ a 0,
T ⫽ (x H – x̄)/s for a high value, or
the constants (a0, a1, and a2)1⫺3 must be calculated. Typically,
these calculations are performed using software provided by
instrument manufacturers or independent software vendors. For
T ⫽ (x៮ ⫺ x L )/s for a low value.
Second, compare T with the value in Table 1010:I for either a
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INTRODUCTION (1010)/Glossary
5% or 1% level of significance for the number of measurements
(n). If T is larger than that value, then xH or xL is an outlier.
Further information on statistical techniques is available elsewhere.5–7
3. TEXAS INSTRUMENTS, INC. 1975. Texas Instruments Programmable
Calculator Program Manual ST1. Statistics Library, Dallas, Texas.
4. BARNETT, V. & T. LEWIS. 1995. Outliers in Statistical Data, 3rd ed.,
John Wiley & Sons, New York, N.Y.
5. NATRELLA, M.G. 1963. Experimental Statistics, Handbook 91. National Bur. Standards, Washington, D.C.
6. SNEDECOR, G.W. & W.G. COCHRAN. 1980. Statistical Methods. Iowa
State University Press, Ames.
7. VERMA, S.P. & A. QUIROZ-RUIZ. 2006. Critical values for 22 discordancy test variants for outliers in normal samples up to sizes 100, and
applications in science and engineering. Revista Mexicana de Ciencias Geologicas 23(3):302.
5. References
1. SPIEGEL, M.R. & L.J. STEPHENS. 1998 Schaum’s Outline—Theory and
Problems of Statistics. McGraw-Hill, New York, N.Y.
2. LAFARA, R.L. 1973. Computer Methods for Science and Engineering.
Hayden Book Co., Rochelle Park, N.J.
1010 C. Glossary
This glossary defines concepts, not regulatory terms. It is not
intended to be all-inclusive.
Accuracy— estimate of how close a measured value is to the true
value; includes expressions for bias and precision.
Analyte—the element, compound, or component being analyzed.
Bias— consistent deviation of measured values from the true
value, caused by systematic errors in a procedure.
Calibration check standard—standard used to determine an
instrument’s accuracy between recalibrations.
Confidence coefficient—the probability (%) that a measurement
will lie within the confidence interval (between the confidence
limits).
Confidence interval—set of possible values within which the
true value will lie with a specified level of probability.
Confidence limit—one of the boundary values defining the
confidence interval.
Detection levels—various levels in use are:
Instrument detection level (IDL)—the constituent concentration
that produces a signal greater than five times the instrument’s
signal:noise ratio. The IDL is similar to the critical level and
criterion of detection, which is 1.645 times the s of blank
analyses (where s is the estimate of standard deviation).
Lower level of detection (LLD) [also called detection level and
level of detection (LOD)]—the constituent concentration in
reagent water that produces a signal 2(1.645)s above the mean
of blank analyses. This establishes both Type I and Type II
errors at 5%.
Method detection level (MDL)—the constituent concentration
that, when processed through the entire method, produces a
signal that has 99% probability of being different from the
blank. For seven replicates of the sample, the mean must be
3.14s above the blank result (where s is the standard deviation
of the seven replicates). Compute MDL from replicate
measurements of samples spiked with analyte at concentrations more than one to five times the estimated MDL.
The MDL will be larger than the LLD because typically
7 or less replicates are used. Additionally, the MDL will
vary with matrix.
Reporting level (RL)—the lowest quantified level within an
analytical method’s operational range deemed reliable
enough, and therefore appropriate, for reporting by the
laboratory. RLs may be established by regulatory mandate
or client specifications, or arbitrarily chosen based on a
preferred level of acceptable reliability. Examples of
RLs typically used (besides the MDL) include:
Level of quantitation (LOQ)/minimum quantifiable level
(MQL)—the analyte concentration that produces a signal
sufficiently stronger than the blank, such that it can be
detected with a specified level of reliability during
routine operations. Typically, it is the concentration
that produces a signal 10s above the reagent water
blank signal, and should have a defined precision and
bias at that level.
Minimum reporting level (MRL)—the minimum concentration that can be reported as a quantified value for a
target analyte in a sample. This defined concentration
is no lower than the concentration of the lowest calibration standard for that analyte and can only be used
if acceptable QC criteria for this standard are met.
Duplicate—1) the smallest number of replicates (two), or 2)
duplicate samples (i.e., two samples taken at the same time
from one location) (field duplicate) or replicate of laboratory
analyzed sample.
Fortification—adding a known quantity of analyte to a sample or
blank to increase the analyte concentration, usually for the
purpose of comparing to test result on the unfortified sample
and estimating percent recovery or matrix effects on the test to
assess accuracy.
Internal standard—a pure compound added to a sample extract
just before instrumental analysis to permit correction for in
efficiencies.
Laboratory control standard—a standard usually certified by an
outside agency that is used to measure the bias in a procedure.
For certain constituents and matrices, use National Institute of
Standards and Technology (NIST) or other national or international traceable sources (Standard Reference Materials),
when available.
Mean—the arithmetic average (the sum of measurements divided
by the number of items being summed) of a data set.
Median—the middle value (odd count) or mean of the two middle
values (even count) of a data set.
Mode—the most frequent value in a data set.
Percentile—a value between 1 and 100 that indicates what percentage of the data set is below the expressed value.
Precision (usually expressed as standard deviation)—a measure of
the degree of agreement among replicate analyses of a sample.
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INTRODUCTION (1010)/Dilution/Concentration Operations
Quality assessment—procedure for determining the quality of
laboratory measurements via data from internal and external
quality control measures.
Quality assurance—a definitive plan for laboratory operations that
specifies the measures used to produce data with known precision
and bias.
Quality control—set of measures used during an analytical
method to ensure that the process is within specified control
parameters.
Random error—the deviation in any step in an analytical
procedure that can be treated by standard statistical techniques. Random error is a major component of measurement error and uncertainty.
Range—the difference of the largest and smallest values in a data
set.
Replicate—repeated operation during an analytical procedure.
Two or more analyses for the same constituent in an extract of
one sample constitute replicate extract analyses.
Spike—see fortification.
Surrogate standard—a pure compound added to a sample in the
laboratory just before processing so a method’s overall efficiency can be determined.
Type I error (also called alpha error)—the probability of determining that a constituent is present when it actually is absent.
Type II error (also called beta error)—the probability of not
detecting a constituent that actually is present.
1010 D. Dilution/Concentration Operations
total volume will equal a ⫹ b, which is not always the case).
Most aqueous-solution volumes are additive, but alcoholic solutions or concentrated acid may be only partially volumetrically
additive, so be aware of potential problems when combining
nonaqueous solutions with aqueous diluents.
b. Volumetric dilution to a measured volume (a/c). This
method is used to dilute an aliquot to a given volume via a pipet
and volumetric flask. It is the most accurate means of dilution,
but when fortifying sample matrices, some error can be introduced if a regular Class A volumetric flask is used. The error will
be proportional to the volumes of both spiking solution and flask.
For the most accurate work, measure the unfortified sample
aliquot in a 100-mL Cassia Class A volumetric flask to the
100-mL mark (0.0 on the flask neck*), and then pipet the volume
of fortifying solution. Mix the solution and note the graduated
volume on the neck of the flask. The fortified solution’s true
volume is equal to 100 mL ⫹ graduated volume over 100 mL.
The true total volume is necessary when computing the dilution
factor for the percent recovery of fortified analyte (LFM) in
Sections 1020B.12e and 4020B.3a to obtain the most accurate
analytical estimate of recovery.
Dilution factors for multiple volumetric dilutions are calculated as the product of the individual dilutions. Generally, serial
dilution is preferred when making dilutions of more than two or
three orders of magnitude. Avoid trying to pipet quantities of less
than 1.0 mL into large volumes (e.g., ⬍1.0 mL into 100 or 1000
mL) to avoid large relative error propagation.
Some biological test methods (e.g., BOD or toxicity testing) may
include dilution techniques that do not strictly conform to the preceding descriptions. For example, such techniques may use
continuous-flow dilutors and dilutions prepared directly in test
equipment, where volumes are not necessarily prepared via Class A
volumetric equipment. Follow the method-specific dilution directions.
1. Adjusting Solution Volume
Analysts frequently must dilute or concentrate the amount of
analyte in a standard or sample aliquot to within a range suitable
for the analytical method so analysis can be performed with
specified accuracy. The following equations enable analysts to
compute the concentration of a diluted or concentrated aliquot
based on the original aliquot concentration and an appropriate
factor or fractional constant. (A factor in this context is the ratio
of final adjusted volume to original volume.) They also can
compute the concentration of adjusted aliquot volume based on
the original aliquot volume.
Concentration of diluted aliquot ⫽
original aliquot concentration ⫻ dilution fraction
Concentration of original aliquot ⫽
diluted aliquot concentration ⫻ dilution factor
Concentration of concentrated aliquot ⫽
original aliquot concentration ⫻ concentration factor
Concentration of original aliquot ⫽
concentrated aliquot concentration ⫻ concentration fraction
where:
Dilution fraction ⫽ original volume/adjusted volume,
Dilution factor ⫽ adjusted volume/original volume,
Concentration factor ⫽ original volume/adjusted volume, and
Concentration fraction ⫽ adjusted volume/original volume.
2. Types of Dilutions
3. Bibliography
Several types of dilutions are used in Standard Methods procedures. Two of the most common volumetric techniques critical
to analytical chemistry results are:
a. Volumetric addition [a/(a ⫹ b)]. This method typically is
used to dilute microbiological samples and prepare reagents
from concentrated reagents. It assumes that volumes a and b are
additive (i.e., when a is combined with b in one container, the
NIEMELA, S.I. 2003. Uncertainty of Quantitative Determinations Derived
by Cultivation of Microorganisms, Publication J4/2003 MIKES.
Metrologia, Helsinki, Finland.
* Pyrex, or equivalent.
5