CHAPTER 3 ERRORS, RANDOM ERRORS, AND STATISTICAL DATA IN CHEMICAL ANALYSES Introduction to Analytical Chemistry

CHAPTER 3ERRORS, RANDOM ERRORS,

AND STATISTICAL DATA IN CHEMICAL ANALYSES

Introduction toIntroduction toAnalytical ChemistryAnalytical Chemistry

Copyright © 2011 Cengage Learning3-2

Errors, Random Errors, and Statistical Data in Chemical

Analyses

Impossible: The analytical results are free of errors or uncertainties.

Possible: Minimize these errors and estimate their size with acceptable accuracy.

Statistical calculations for use to judge the quality of experimental measurements are many.



Analyses

Only two will be considered: The confidence interval and the least-squares method for constructing calibration curves.


Figure 3-1

Figure 3-1 Results from six replicate determinations for iron in aqueous samples of a standard solution containing 20.00 ppm of iron(III).



Analyses

Measurements are always accompanied by uncertainty. The true value always falls within a range due to uncertainty.

The probable magnitude of the error defines the limit within which the true value lies.

Data of unknown quality are worthless.The true value of a measurement is never known

exactly.


Reliability Can Be Assessed In Several Ways

Standards of known composition can be analyzed and the results compared with the known composition.

Calibrating equipment enhances the quality of data.


Reliability Can Be Assessed In Several Ways

Questions to answer before beginning an analysis: What is the maximum error that I can tolerate in the result? No one can afford to waste time generating data that are more

reliable than is needed.


Mean, arithmetic mean, and average ( ) are synonyms.

The symbol means to add all the valuesxi for the replicates.

3A-1 The Mean and Median

ix

where xi represents the individual values of x making up a set of N replicate measurements.

(3-1)

x

ix



The median is the middle result when replicate data are arranged in order of size.

Equal numbers of results are larger and smaller than the median.

For an odd number of data points, the median can be evaluated directly. For an even number, the mean of the middle pair is used.



The mean of two or more measurements is their average value.

The median is used advantageously when a set of data contains an outlier, a result that differs significantly from others in the set.


Example 3-1

Calculate the mean and the median for the data shown in Figure 3-1.

Because the set contains an even number of measurements, the median is the average of the central pair:


3A-2 What Is Precision?

Precision describes the reproducibility of measurements; the closeness of results to each other.

Precision is determined by repeating the measurement on replicate samples.

Three terms to describe the precision of a set of replicate data: standard deviation, variance, and coefficient of variation.


3A-2 What Is Precision?

Precision is a function of the deviation from the mean di, or just the deviation, which is defined as

Precision is the closeness of results to others that have been obtained in exactly the same way.

(3-2)


3A-3 How About Accuracy?

Accuracy indicates the closeness of the measurement to its true or accepted value and is expressed by the error. Accuracy measures agreement between a result and its true

value. Precision describes the agreement among several results that

have been obtained in the same way.


Figure 3-2

Figure 3-2 Illustration of accuracy and precision using the pattern of darts on a dartboard



Absolute Error The absolute error E in the measurement of a quantity xi is

given by the equation

where xt is the true, or accepted, value of the quantity. Note that we retain the sign in stating the error.

(3-3)



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

(3-4)



Relative error Relative error is also expressed in parts per thousand (ppt).

e.g.


3A-4 Classifying Experimental Errors

To determine accuracy, we have to know the true value, and this value is exactly what we are seeking in the analysis.

Do we know the answer precisely, also know it accurately?

The danger is illustrated in Figure 3-3, which summarizes the results for the determination of nitrogen in two pure compounds: benzyl isothiourea hydrochloride and nicotinic acid.


Figure 3-3 Absolute error in the micro-Kjeldahl determination of nitrogen. Each dot represents the error associated with a single determination. Each vertical line labeled (xi - xt ) is the absolute average deviation of the set from the true value. (Data from C. O. Willits and C. L. Ogg, J. Assoc. Anal. Chem., 1949, 32, 561. With permission.)

Figure 3-3



Figures 3-1 and 3-3 suggest that chemical analyses are affected by at least two types of errors.

One type, called random (or indeterminate) error, causes data to be scattered more or less symmetrically around a mean value.

Refer to Figure 3-3: Notice that the scatter in the data, and thus the random error, for analysts 1 and 3 is significantly less than that for analysts 2 and 4.



A second type of error, called systematic (or determinate) error, causes the mean of a set of data to differ from the accepted value.

The results of analysts 1 and 2 in Figure 3-3 have little systematic error, but the data of analysts 3 and 4 show systematic errors of about –0.7 and –1.2% nitrogen.



Random, or indeterminate, errors are errors that affect the precision of measurement.

Systematic, or determinate, errors affect the accuracy of results.



A third type of error is 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.

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 done to determine if a data point is an outlier.


3B Systematic Errors

Systematic errors have a definite value, an assignable cause, and are of about the same magnitude for replicate measurements made in the same way.

Systematic errors lead to bias in measurement technique. Note that bias affects all the data in a set in approximately the same way and that it bears a sign.


3B-1 How do Systematic Errors Arise?

Three types of systematic errors: (1) Instrument errors are caused by imperfections in

measuring devices and instabilities in their components. (2) Method errors arise from non-ideal chemical or physical

behavior of analytical systems. (3) Personal errors result from the carelessness, inattention, or

personal limitations of the experimenter.



Instrument Errors All measuring devices are sources of systematic errors: pipets,

burets, and volumetric flasks. These differences arise from using glassware at a temperature

that differs significantly from the calibration temperature, from distortions in container walls due to heating while drying, from errors in the original calibration, or from contaminants or scratches on the inner surfaces of the containers.



Instrument errors Electronic instruments are subject to instrumental systematic

errors. These uncertainties have many sources. For example, errors emerge as the voltage of a battery-operated power supply decreases with use.

Instrument Errors are detectable and correctable.



Method Errors The nonideal chemical or physical behavior of the reagents

and reactions upon which an analysis is based often introduce systematic method errors.

Such sources of nonideality include the slowness and incompleteness of reactions, the instability of species, non-specificity of most reagents, and possible interference.



Method Errors In Figure 3-3, the results by analysts 3 and 4 show a negative

bias that can be traced to the chemical nature of the sample, nicotinic acid.

The compounds containing a pyridine ring are incompletely decomposed by the sulfuric acid ; hence, the negative errors in Figure 3-3 are likely systematic errors from incomplete decomposition of the samples.



Method Errors Errors inherent in a method are often difficult to detect and

are thus the most serious of the three types of systematic error.



Personal Errors Measurements requiring personal judgments. Judgments of this type are often subject to systematic,

unidirectional errors. An analyst who is insensitive to color changes tends to use

excess reagent in a volumetric analysis. Physical disabilities are often sources of personal determinate errors.

A universal source of personal error is prejudice.



Personal Errors Number bias is another source of personal error that varies

considerably from person to person. The most common number bias encountered in estimating the

position of a needle on a scale involves a preference for the digits 0 and 5. Also prevalent is a prejudice favoring small digits over large and even numbers over odd.

Color blindness amplifies personal errors in a volumetric analysis.


3B-2 What Effects Do Systematic

Errors Have on Analytical Results?

Systematic errors may be either constant or proportional.

The magnitude of a constant error does not depend on the size of the quantity measured.

Proportional errors increase or decrease in proportion to the size of the sample taken for analysis.


3B-3 Detecting Systematic Instrument and Personal Errors

Systematic instrument errors are usually corrected by periodic calibration of equipment. The response of most instruments changes with time.

Most personal errors can be minimized by care and self-discipline.


3B-4 Detecting Systematic Method Errors

Bias in an analytical method is particularly difficult to detect. One or more of the following steps can recognize and adjust for a systematic error in an analytical method.



Analyzing Standard Samples Analyzing standard reference materials, SRM, is the best way

to estimate the bias of an analytical method. The SRM Materials contain one or more analytes at well-

known or certified concentration levels. Standard materials can sometimes be prepared by synthesis. Standard reference material can be purchased from a number

of governmental and industrial sources.



Analyzing Standard Samples The concentration of one or more of the components in these

materials has been determined in one of three ways: (1) by analysis with a previously validated reference method; (2) by analysis by two or more independent, reliable measurement

methods; or (3) by analysis by a network of cooperating laboratories that are

technically competent and thoroughly knowledgeable with the material being tested.



Standard reference materials from NIST. (Photo courtesy of the National Institute of Standards and Technology.)



Using an Independent Analytical Method If standard samples are not available, a second independent

and reliable analytical method can be used in parallel with the method being evaluated.

A statistical test must be used to determine whether any difference is a result of random errors in the two methods or due to bias in the method under study.



Performing Blank Determinations Blank determinations are useful for detecting certain types of

constant errors. In a blank determination, or blank, all steps of the analysis are

performed in the absence of a sample. The results from the blank are then applied as a correction to

the sample measurements. Blank determinations reveal errors and correct data.



Varying the Sample Size Constant errors can often be detected by varying the sample

size.


3C The Nature of Random Errors

All measurements contain random errors. Random, or indeterminate, errors occur whenever a

measurement is made. Caused by many small but uncontrollable variables. The errors are accumulative.


3C-1 What Are the Sources ofRandom Errors?

Imagine a situation in which just four small random errors combine to give an overall error.

We will assume that each error has an equal probability of occurring and that each can cause the final result to be high or low by a fixed amount ±U.

Table 3-1 shows all the possible ways the four errors can combine to give the indicated deviations from the mean value.


Table 3-1


Figure 3-4

Figure 3-4 Frequency distribution for measurements containing (a) four random uncertainties, (b) ten random uncertainties, and (c) a very large number of random uncertainties.


Figure 3-4

For a sufficiently large number of measurements, we can expect a frequency distribution like that shown in Figure 3-4a. The ordinate is the relative frequency of occurrence of the five possible combinations.


Figure 3-4

Figure 3-4b shows the theoretical distribution for ten equal-sized uncertainties.


Figure 3-4

For a very large number of individual errors, a bell-shaped curve like that shown in Figure 3-4c results. Such a plot is called a Gaussian curve or a normal error curve.


Table 3-2


3C-2 Describing the Distribution

of Experimental Data

This 0.025 mL spread of data, from a low of 9.969 mL to a high of 9.994 mL, results directly from an accumulation of all the random uncertainties in the experiment.

Rearrange Table 3-2 into frequency distribution groups, as in Table 3-3.

26% of the data reside in the cell containing the mean and median value of 9.982 mL and that more than half the results are within ± 0.004 mL of this mean.


3C-2 Describing the Distribution

of Experimental Data

The frequency distribution data in Table 3-3 are plotted as a bar graph, or histogram (labeled A in Figure 3-5).

As the number of measurements increases, the histogram approaches the shape of the continuous curve shown as plot B in Figure 3-5 (a Gaussian curve, or normal error curve).


Figure 3-5

Figure 3-5 A histogram (A) showing distribution of the 50 results in Table 3-3 and a Gaussian curve (B) for data having the same mean and same standard deviation as the data in the histogram.


Sources of random uncertainties

Many small and uncontrollable variables affect even the simple process of calibrating a pipet.

The cumulative effect of random uncertainties is responsible for the scatter of data points around the mean.

Statistics only reveal information that is already present in a data set.

Do not confuse the statistical sample with the analytical sample.


Sources of random uncertainties

(1) visual judgments, such as the level of the water with respect to the marking on the pipet and the mercury level in the thermometer

(2) variations in the drainage time and in the angle of the pipet as it drains

(3) temperature fluctuations, which affect the volume of the pipet, the viscosity of the liquid, and the performance of the balance

(4) vibrations and drafts that cause small variations in the balance readings.


3D Treating Random Errors with Statistics

The random, or indeterminate, errors in the results of an analysis can be evaluated by the methods of statistics.

Ordinarily, statistical analysis of analytical data is based on the assumption that random errors follow a Gaussian, or normal, distribution.


3D Treating Random Errors with Statistics

Sometimes analytical data depart seriously from Gaussian behavior, but the normal distribution is the most common.

We base this discussion entirely on normally distributed random errors.


3D-1 Samples and Populations

In statistics, a finite number of experimental observations is called a sample of data; this is different from the term used in chemical analysis.

Statisticians call the theoretical infinite number of data a population, more specifically a parent population, or a universe, of data.


3D-1 Samples and Populations

Statistical laws must be modified substantially when applied to a small sample because a few data points may not be representative of the population.


3D-2 Characterizing Gaussian Curves

Figure 3-6a shows two Gaussian curves in which the relative frequency y of occurrence of various deviations from the mean is plotted as a function of the deviation from the mean.

The equation for a Gaussian curve has the form



The equation contains just two parameters, the population mean μ and the population standard deviation σ.


Figure 3-6 (a)

Figure 3-6 Normal error curves. The standard deviation for curve B is twice that for curve A; that is, σB = 2σA. (a) The abscissa is the deviation from the mean in the units of measurement. (b) The abscissa is the deviation from the mean in units of σ. Thus, the two curves A and B are identical here.


Figure 3-6 (b)


• The Population Mean μ and the Sample Mean

Sample mean = , where when N is small

Population mean = μ, where when N → ∞

The difference between x and μ decreases rapidly as N reaches over 20 to 30.


x

x



The Population Standard Deviation (σ) σ is a measure of the precision or scatter of a population of

data, which is given by the equation

where N is the number of data points making up the population.

(3-5)



The Population Standard Deviation (σ) The two curves in Figure 3-6a are for two populations of data

that differ only in their standard deviations. The standard deviation for the data set yielding the broader

but lower curve B is twice that for the measurements yielding curve A.

The precision of the data leading to curve A is twice as good as that of the data that are represented by curve B.



The Population Standard Deviation (σ) Figure 3-6b shows another type of normal error curve in which

the abscissa is now a new variable z, which is defined as

z is the deviation of a data point from the mean relative to one standard deviation. That is, when x – μ = σ, z is equal to one; when x – μ = 2σ, z is equal to two.

(3-6)



The Population Standard Deviation (σ) A plot of relative frequency versus this parameter yields a

single Gaussian curve that describes all populations of data regardless of standard deviation.



The Population Standard Deviation (σ) Variance: The square of the standard deviation σ2. A normal error curve has several general properties:

(1) The mean occurs at the central point of maximum frequency, (2) there is a symmetrical distribution of positive and negative

deviations about the maximum, (3) there is an exponential decrease in frequency as the magnitude of

the deviations increases. Small random uncertainties are more common.



Areas under a Gaussian Curve Regardless of its width, 68.3% of the data making up the

population will lie within the bounds bracketed by ±1σ. Approximately 95.4% of all data points are within ±2σ of the

mean and 99.7% within ±3σ. These are shown in Figure 3-6. Because of such area relationships, the standard deviation of a

population of data is a useful predictive tool.


Feature 3-2

For μ = 0, x = ±σ


Feature 3-2

For μ = 0, x = ±2σ


Feature 3-2

For μ = 0, x = ±3σ


3D-3 Finding the Sample Standard Deviation

Equation 3-5 must be modified for a small sample of data. Thus, the sample standard deviation s is given by the equation

The quantity N – 1 is called the number of degrees of freedom.

(3-7)


(3-8)


An Alternative Expression for Sample Standard Deviation


The following results were obtained in the replicate determination of the lead content of a blood sample: 0.752, 0.756, 0.752, 0.751, and 0.760 ppm Pb. Calculate the mean and the standard deviation of this set of data.

Example 3-3


To apply Equation 3-8, we calculate and

Example 3-3

2

ix 2( )

i/ Nx


Substituting into Equation 3-8 leads to

Example 3-3



Note in Example 3-3 that the difference between Σx2i

and (Σxi)2/N is very small. If we had rounded these numbers before subtracting them, a serious error would have appeared in the computed value of s. To avoid this source of error, never round a standard deviation calculation until the very end.



What Is the Standard Error of the Mean? For replicate samples, each containing N measurements, are

taken randomly from a population of data, the mean of each set will show less and less scatter as N increases.

The standard deviation of each mean is known as the standard error of the mean and is given the symbol sm.

(3-9)


3D-4 The Reliability of s as a Measure of Precision

The rapid improvement in the reliability of s with increases in N makes it feasible to obtain a good approximation of σ when the method of measurement is not excessively time consuming and when an adequate supply of sample is available.


Pooling Data to Improve theReliability of s

Pooled data from a series of similar samples accumulated over time provide an estimate of s that is superior to the value for any individual subset.

Assume the same sources of random error in all the measurements.


Pooling Data to Improve theReliability of s

To obtain a pooled estimate of the standard deviation, spooled, deviations from the mean for each subset are squared; the squares of all subsets are then summed and divided by an appropriate number of degrees of freedom.


Feature 3-4 Equation for Calculating Pooled Standard

Deviations

The equation for computing a pooled standard deviation from several sets of data takes the form

where N1 is the number of results in set 1, N2 is the number in set 2, and so forth. The term Nt is the number of data sets that are pooled.


Example 3-4

The mercury in samples of seven fish taken from Chesapeake Bay was determined by a method based on the absorption of radiation by gaseous elemental mercury.


Example 3-4

Calculate a pooled estimate of the standard deviation for the method, based on the first three columns of data:


Example 3-4

The values in the last two columns for specimen 1 were computed as follows:


Example 3-4

The other data in columns 4 and 5 were obtained similarly. Then

One degree of freedom is lost for each of the seven samples.


3D-5 Alternative Terms for Expressing the Precision of Samples of

Data

Other than sample standard deviation , three other terms are often employ in reporting the precision.

1. The variance (s2) is

People who do scientific work tend to use standard deviation rather than variance as a measure of precision.

(3-10)


Relative Standard Deviation (RSD) and Coefficient of Variation (CV)

2. Relative standard deviation:

3. Coefficient of variation (CV)

(3-11)


3D-5 Alternative Terms for Expressing the Precision of Samples of

Data

Spread or Range (w) Another term to describe the precision of a set of replicate

results. It is the difference between the largest value in the set and the

smallest. Example: The spread of the data in Figure 3-1 is (20.3 – 19.4) =

0.9 ppm Fe.


Example 3-5

For the set of data in Example 3-3, calculate (a) the variance, (b) the relative standard deviation in parts per thousand, (c) the coefficient of variation, and (d) the spread.


Example 3-5

In Example 3-3, we found

= 0.754 ppm Pb and s = 0.0038 ppm Pb

(a) s2 = (0.0038)2 = 1.410-5

(b) RSD = 1000ppt = 5.0ppt (c) CV = 100% = 0.50%(d) w = 0.760 – 0.751 = 0.009

x

0.0038

0.7540.0038

0.754


3E-1 The Standard Deviation of Sums and Differences

Consider the summation

Absolute standard deviations


3E-1 The Standard Deviation of Sums and Differences

The summation could be as large as +0.02 + 0.03 + 0.05 = + 0.10,

or as small as – 0.02 – 0.03 – 0.05 = – 0.10, or any value lies between these two extremes, or even – 0.02 – 0.03 + 0.05 = 0

or + 0.02 + 0.03 – 0.05 = 0


Variance and Propagation of Errors

The variance of a sum or difference is equal to the sum of the individual variances, which demonstrate how the errors propagate.

If y = a + b – c For the computation

The variance of y, S2y is given by


Table 3-4


3E-2 The Standard Deviation of Products and Quotients



As shown in Table 3-4, the relative standard deviation of a product or quotient is determined by the relative standard deviations of the numbers forming the computed result.


Applying this equation to the numerical example gives

we can write the answer and its uncertainty as 0.0104(±0.0003).



3F Reporting Computed Data

One of the best ways of indicating reliability is to give a confidence interval at the 90% or 95% confidence level as we describe in Section 3G-2.

Another method is to report the absolute standard deviation or the coefficient of variation of the data.

A less satisfactory but more common indicator of the quality of data is the significant figure convention.


3F-1 The Significant Figure Convention

A simple way of indicating the probable uncertainty associated with an experimental measurement is to round the result so that it contains only significant figures.

The significant figures in a number are all the certain digits plus the first uncertain digit.


3F-1 The Significant Figure Convention

A zero may or may not be significant depending on its location in a number.

A zero that is surrounded by other digits is always significant (such as in 30.24 mL).

Zeros that only locate the decimal point for us are not.


3F-2 Significant Figures in Numerical Computations

Sums and Differences For addition and subtraction, the number of significant figures

can be found by visual inspection.

the second and third decimal places in the answer cannot be significant because 3.4 is uncertain in the first decimal place



Products and Quotients A rule of thumb sometimes suggested for multiplication and

division is that the answer should be rounded so that it contains the same number of significant digits as the original number with the smallest number of significant digits.


The first answer would be rounded to 1.1 and the second to 0.96. If we assume a unit uncertainty in the last digit of each number in the first quotient, however, the relative uncertainties associated with each of these numbers are




Because the first relative uncertainty is much larger than the other two, the relative uncertainty in the result is also ； the absolute uncertainty is then

Therefore, the first result should be rounded to three significant figures or 1.08, but the second should be rounded to only two; that is 0.96.

124



Logarithms and Antilogarithms1. In a logarithm of a number, keep as many digits to the right

of the decimal point as there are significant figures in the original number.

2. In an antilogarithm of a number, keep as many digits as there are digits to the right of the decimal point in the original number.


Example 3-7

Round the following answers so that only significant digits are retained: (a) log 4.000 105 = –4.3979400 and (b) antilog 12.5 = 3.162277 1012.(a) Following rule 1, we retain 4 digits to the right of the

decimal point

(b) Following rule 2, we may retain only 1 digit


3F-3 Rounding Data

A good guide to follow when rounding a 5 is always to round to the nearest even number. For example, 0.635 rounds to 0.64 and 0.625 rounds to 0.62.

We should note that it is seldom justifiable to keep more than one significant figure in the standard deviation because the standard deviation contains error as well.


3F-4 Rounding the Results from Chemical Computations

The uncertainty of the result is estimated using the techniques presented in Section 3E. Finally, the result is rounded so that it contains only significant digits.

It is especially important to postpone rounding until the calculation is completed. At least one extra digit beyond the significant digits should be carried through all the computations to avoid a rounding error.

This extra digit is sometimes called a “guard” digit.


3G Confidence Limits

1. Defining a numerical interval around the mean of a set of replicate analytical results within which the population mean can be expected to lie with a certain probability. This interval is called the confidence interval.

2. Determining the number of replicate measurements required to ensure at a given probability that an experimental mean falls within a certain confidence interval.

3. Using the least-squares method for constructing calibration curves.


3G Confidence Limits

Confidence limits define a numerical interval around that contains μ with a certain probability.

A confidence interval is the numerical magnitude of the confidence limit.


3G-1 Finding the Confidence Interval When s Is a Good Estimate

of σ

confidence limits (CL)

(3-15)


3G-1 Finding the Confidence Interval When s Is a Good Estimate

of σ


3G-2 Finding the Confidence Interval When σ Is Unknown

(3-17)


Table 3-6


3H Analyzing Two-dimensional Data: The Least-squares Method

A statistical technique called regression analysis provides the means for objectively obtaining such a line and also for specifying the uncertainties associated with its subsequent use.


3H-1 Assumptions of the Least-Squares Method

The method of least squares is used to generate a calibration curve, two assumptions are required. The first is that there is actually a linear relationship between the measured variable (y) and the analyte concentration (x).

The mathematical relationship that describes this assumption is called the regression model, which may be represented as

where b is the y intercept (the value of y when x is zero) and m is the slope of the line.


3H-2 Computing the Regression Coefficients and Finding the Least-

Squares Line

The vertical deviation of each point from the straight line is called a residual.



Squares Line

1. The slope of the line m:

2. The intercept b:

3. The standard deviation about regression sr :



Squares Line

4. The standard deviation of the slope sm :

5. The standard deviation of the intercept sb :

6. The standard deviation for results obtained from the calibration curve sc :



Squares Line

The standard deviation about regression sr (Equation 3-24) is the standard deviation for y when the deviations are measured not from the mean of y (as is usually the case) but from the straight line that results from the least-squares analysis:



Squares Line

The standard deviation about regression is often called the standard error of the estimate or the standard error in y.


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CHAPTER 3 ERRORS, RANDOM ERRORS, AND STATISTICAL DATA IN CHEMICAL ANALYSES Introduction to Analytical Chemistry