6) Errors in Chemical Analyses
Topic Objectives
By the end of this topic the learner should be able to:
1) State and explain the various types of errors encountered in chemical analyses.
2) State and explain the methods we can use to detect errors in chemical analyses.
Introduction
Analytical results are often used in the diagnosis of disease, in the assessment of hazardous wastes and
pollution, in the solving of major crimes, and in the quality control of industrial products. Errors in these
results can have serious personal and societal effects.
Measurements invariably involve errors and uncertainties. Only a few of these are due to mistakes on the
part of the experimenter. More commonly errors are caused by faulty calibrations or standardizations or
random variations and uncertainties in results.
Frequent calibrations, standardizations, and analyses of known samples can sometimes be used to lessen all
but the random errors and uncertainties. In the limit, however, measurement errors are an inherent part of
the quantized world in which we live.
Because of this, it is impossible to perform a chemical analysis that is totally free of errors or uncertainties.
We can only hope to minimize errors and estimate their size with acceptable accuracy.
This chapter explores the nature of experimental errors and their effects on the results of chemical analyses.
Every measurement is influenced by many uncertainties, which combine to produce a scatter of results.
Because measurement uncertainties can never be completely eliminated, measurement data can give us only
an estimate of the "true" value. However, the probable magnitude of error in a measurement can often be
evaluated.
It is then possible to define limits within which the true value of a measured quantity lies with a given level
of probability. Although it is not always easy to estimate the reliability of experimental data, it is important
to do so whenever we collect laboratory results, because data of unknown quality are worthless. On the other
hand, results that do not seem especially accurate may be of considerable value if the limits of uncertainty
are known.
Unfortunately, there is no simple and widely applicable method for determining the reliability of data with
absolute certainty. Often, estimating the quality of experimental results requires as much effort as collecting
the data.
Reliability can be assessed in several ways. Experiments designed to reveal the presence of errors can be
performed. Standards of known composition can be analyzed and the results compared with the known
composition. A few minutes in the library to consult the chemical literature can be profitable. Calibrating
equipment usually enhances the quality of data.
Finally, statistical tests can be applied to the data. Because none of these options is perfect, we must
ultimately make judgments as to the probable accuracy of our results.
The quality assurance of analytical methods and the ways to validate and report results
One of the first questions to answer before beginning an analysis is, "What maximum error can I tolerate
in the result?" The answer to this question often determines the method chosen and the time required to
complete the analysis.
1
,Some Important Terms
1) Replicates
Because one analysis gives no information about the variability of results, chemists usually carry two to
five portions (replicates) of a sample through an entire analytical procedure.
Individual results from a set of measurements are seldom the same, so we usually consider the "best"
estimate to be the central value for the set.
First, the central value of a set should be more reliable than any of the individual results. Usually, the mean
or the median is used as the central value for a set of replicate measurements.
Second, an analysis of the variation in data allows us to estimate the uncertainty associated with the central
result.
2) The Mean and Median
The most widely used measure of central value is the mean, x. The mean, also called the arithmetic mean,
or the average, is obtained by dividing the sum of replicate measurements by the number of measurements
in the set:
(Equation 1)
Where Xi represents the individual values of X making up the set of N replicate measurements.
The median is the middle result when replicate data are arranged according to increasing or decreasing
value. There are equal numbers of results that are larger and smaller than the median. For an odd number
of results, the median can be evaluated directly. For an even number, the mean of the Middle pair is used.
In ideal cases, the mean and Median are identical, but when the number of measurements in the set is small,
the values often differ.
3) Precision
Precision describes the reproducibility of measurements in other words, the closeness of results that have
been obtained in exactly the same way. Generally, the precision of a measurement is readily determined by
simply repeating the measurement on replicate samples.
Three terms are widely used to describe the precision of a set of replicate data:
Standard deviation,
variance
Coefficient of variation
These three are functions of how much an individual result Xi differs from the mean, which is called the
deviation from the mean di.
(Equation 2)
4) Accuracy
Accuracy indicates the closeness of the measurement to the true or accepted value and is expressed by the
error:
Accuracy measures agreement between a result and the accepted value.
Precision on the other hand, describes the agreement among several results obtained in the same way. We
can determine precision just by measuring replicate samples.
2
, Accuracy is often more difficult to determine because the true value is usually unknown. An accepted value
must be used instead. Accuracy is expressed in terms of either absolute or relative error.
Absolute Error
The absolute error E in the measurement of a quantity x is given by the equation
(Equation 3)
Where, Xt is the true or accepted value of the quantity.
Relative Error
Often, the relative error Er is a more useful quantity than the absolute error. The percent relative error is
given by the expression
(Equation 4)
Relative error is also expressed in parts per thousand (ppt).
5) Types of Errors in Experimental Data
The precision of a measurement is readily determined by comparing data from carefully replicated
experiments. Unfortunately, an estimate of the accuracy is not as easy to obtain. To determine the accuracy,
we have to know the true value, which is usually what we are seeking in the analysis.
Note: Results can be precise without being accurate and accurate without being precise.
(a) Random (or indeterminate) error
- Causes data to be scattered more or less symmetrically around a mean value. In general, random error in
a measurement is reflected by its precision.
(b) Systematic (or determinate) error
Causes the mean of a data set to differ from the accepted value. In general, a systematic error in a series
of replicate measurements causes all the results to be too high or too low.
An example of a systematic error is the unsuspected loss of a volatile analyte while heating a sample.
(c) Gross error
Gross errors differ from indeterminate and determinate errors. They usually occur only occasionally, are
often large, and may cause a result to be either high or low. They are often the product of human errors.
For example, if part of a precipitate is lost before weighing, analytical results will be low. Touching a
weighing bottle with your fingers after its empty mass is determined will cause a high mass reading for a
solid weighed in the contaminated bottle.
Gross errors lead to outliers, results that appear to differ markedly from all other data in a set of replicate
measurements.
Various statistical tests can be performed to determine if a result is an outlier
A) Systematic errors
Systematic errors have a definite value and an assignable cause, and are of the same magnitude for replicate
measurements made in the same way.
Systematic errors lead to bias in measurement results. Note that bias affects all of the data in a set in the
same way and that it bears a sign.
3
Topic Objectives
By the end of this topic the learner should be able to:
1) State and explain the various types of errors encountered in chemical analyses.
2) State and explain the methods we can use to detect errors in chemical analyses.
Introduction
Analytical results are often used in the diagnosis of disease, in the assessment of hazardous wastes and
pollution, in the solving of major crimes, and in the quality control of industrial products. Errors in these
results can have serious personal and societal effects.
Measurements invariably involve errors and uncertainties. Only a few of these are due to mistakes on the
part of the experimenter. More commonly errors are caused by faulty calibrations or standardizations or
random variations and uncertainties in results.
Frequent calibrations, standardizations, and analyses of known samples can sometimes be used to lessen all
but the random errors and uncertainties. In the limit, however, measurement errors are an inherent part of
the quantized world in which we live.
Because of this, it is impossible to perform a chemical analysis that is totally free of errors or uncertainties.
We can only hope to minimize errors and estimate their size with acceptable accuracy.
This chapter explores the nature of experimental errors and their effects on the results of chemical analyses.
Every measurement is influenced by many uncertainties, which combine to produce a scatter of results.
Because measurement uncertainties can never be completely eliminated, measurement data can give us only
an estimate of the "true" value. However, the probable magnitude of error in a measurement can often be
evaluated.
It is then possible to define limits within which the true value of a measured quantity lies with a given level
of probability. Although it is not always easy to estimate the reliability of experimental data, it is important
to do so whenever we collect laboratory results, because data of unknown quality are worthless. On the other
hand, results that do not seem especially accurate may be of considerable value if the limits of uncertainty
are known.
Unfortunately, there is no simple and widely applicable method for determining the reliability of data with
absolute certainty. Often, estimating the quality of experimental results requires as much effort as collecting
the data.
Reliability can be assessed in several ways. Experiments designed to reveal the presence of errors can be
performed. Standards of known composition can be analyzed and the results compared with the known
composition. A few minutes in the library to consult the chemical literature can be profitable. Calibrating
equipment usually enhances the quality of data.
Finally, statistical tests can be applied to the data. Because none of these options is perfect, we must
ultimately make judgments as to the probable accuracy of our results.
The quality assurance of analytical methods and the ways to validate and report results
One of the first questions to answer before beginning an analysis is, "What maximum error can I tolerate
in the result?" The answer to this question often determines the method chosen and the time required to
complete the analysis.
1
,Some Important Terms
1) Replicates
Because one analysis gives no information about the variability of results, chemists usually carry two to
five portions (replicates) of a sample through an entire analytical procedure.
Individual results from a set of measurements are seldom the same, so we usually consider the "best"
estimate to be the central value for the set.
First, the central value of a set should be more reliable than any of the individual results. Usually, the mean
or the median is used as the central value for a set of replicate measurements.
Second, an analysis of the variation in data allows us to estimate the uncertainty associated with the central
result.
2) The Mean and Median
The most widely used measure of central value is the mean, x. The mean, also called the arithmetic mean,
or the average, is obtained by dividing the sum of replicate measurements by the number of measurements
in the set:
(Equation 1)
Where Xi represents the individual values of X making up the set of N replicate measurements.
The median is the middle result when replicate data are arranged according to increasing or decreasing
value. There are equal numbers of results that are larger and smaller than the median. For an odd number
of results, the median can be evaluated directly. For an even number, the mean of the Middle pair is used.
In ideal cases, the mean and Median are identical, but when the number of measurements in the set is small,
the values often differ.
3) Precision
Precision describes the reproducibility of measurements in other words, the closeness of results that have
been obtained in exactly the same way. Generally, the precision of a measurement is readily determined by
simply repeating the measurement on replicate samples.
Three terms are widely used to describe the precision of a set of replicate data:
Standard deviation,
variance
Coefficient of variation
These three are functions of how much an individual result Xi differs from the mean, which is called the
deviation from the mean di.
(Equation 2)
4) Accuracy
Accuracy indicates the closeness of the measurement to the true or accepted value and is expressed by the
error:
Accuracy measures agreement between a result and the accepted value.
Precision on the other hand, describes the agreement among several results obtained in the same way. We
can determine precision just by measuring replicate samples.
2
, Accuracy is often more difficult to determine because the true value is usually unknown. An accepted value
must be used instead. Accuracy is expressed in terms of either absolute or relative error.
Absolute Error
The absolute error E in the measurement of a quantity x is given by the equation
(Equation 3)
Where, Xt is the true or accepted value of the quantity.
Relative Error
Often, the relative error Er is a more useful quantity than the absolute error. The percent relative error is
given by the expression
(Equation 4)
Relative error is also expressed in parts per thousand (ppt).
5) Types of Errors in Experimental Data
The precision of a measurement is readily determined by comparing data from carefully replicated
experiments. Unfortunately, an estimate of the accuracy is not as easy to obtain. To determine the accuracy,
we have to know the true value, which is usually what we are seeking in the analysis.
Note: Results can be precise without being accurate and accurate without being precise.
(a) Random (or indeterminate) error
- Causes data to be scattered more or less symmetrically around a mean value. In general, random error in
a measurement is reflected by its precision.
(b) Systematic (or determinate) error
Causes the mean of a data set to differ from the accepted value. In general, a systematic error in a series
of replicate measurements causes all the results to be too high or too low.
An example of a systematic error is the unsuspected loss of a volatile analyte while heating a sample.
(c) Gross error
Gross errors differ from indeterminate and determinate errors. They usually occur only occasionally, are
often large, and may cause a result to be either high or low. They are often the product of human errors.
For example, if part of a precipitate is lost before weighing, analytical results will be low. Touching a
weighing bottle with your fingers after its empty mass is determined will cause a high mass reading for a
solid weighed in the contaminated bottle.
Gross errors lead to outliers, results that appear to differ markedly from all other data in a set of replicate
measurements.
Various statistical tests can be performed to determine if a result is an outlier
A) Systematic errors
Systematic errors have a definite value and an assignable cause, and are of the same magnitude for replicate
measurements made in the same way.
Systematic errors lead to bias in measurement results. Note that bias affects all of the data in a set in the
same way and that it bears a sign.
3
