1The Nature of Random Errors [600240]

11/30/2015
1The 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. What Are the Sources of
Random 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 probabi lity
of occurring and that each can cause the final resu lt to
be high or low by a fixed amount ±U.

Table 3-1 shows all the possible ways the four erro rs
can combine to give the indicated deviations from t he
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.

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2Figure 3-4

For a very large number of individual errors, a bel l-
shaped curve like that shown in Figure 3-4c results .
Such a plot is called a Gaussian curve or a normal error
curve.
Table 3-2Describing the Distribution
of Experimental Data
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 gro ups,
as in Table 3-3.

26% of the data reside in the cell containing the m ean
and median value of 9.982 mL and that more than hal f
the results are within ±0.004 mL of this mean. Describing the Distribution
of Experimental Data

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

As the number of measurements increases, the
histogram approaches the shape of the continuous
curve shown as plot Bin 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 t he
mean.

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

.

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3Sources of random uncertainties

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

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

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

(4) vibrations and drafts that cause small variatio ns in
the balance readings. 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.
Treating Random Errors
with Statistics

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

We base this discussion entirely on normally
distributed random errors. 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.
Samples and Populations

Statistical laws must be modified substantially whe n
applied to a small sample because a few data points
may not be representative of the population. Characterizing Gaussian Curves

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

The equation for a Gaussian curve has the form

11/30/2015
4Characterizing Gaussian Curves

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= 2 σA. (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 μand the Sample Mean

Sample mean = , where
when N is small

Population mean = μ, where
when N → ∞

The difference between x and μdecreases rapidly as N
reaches over 20 to 30. Characterizing Gaussian Curves
x
x
Characterizing Gaussian Curves

The Population Standard Deviation (σ)

σis a measure of the precision or scatter of a population of
data, which is given by the equation
where Nis the number of data points making up the
population. (3-5)
Characterizing Gaussian Curves

The Population Standard Deviation (σ)

The two curves in Figure 3-6a are for two populations of dat a
that differ only in their standard deviations.

The standard deviation for the data set yielding the broader
but lower curve Bis twice that for the measurements yielding
curve A.

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

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5Characterizing Gaussian Curves

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= 2 σ, z is equal to two. (3-6)
Characterizing Gaussian Curves

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.
Characterizing Gaussian Curves

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. Characterizing Gaussian Curves

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 ±2σof 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= ±σFeature 3-2
For μ= 0, x= ±2σ

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6Feature 3-2
For μ= 0, x= ±3σ Finding the Sample Standard
Deviation

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

The quantity N– 1 is called the number of degrees of
freedom . (3-7)
(3-8)
Finding the Sample
Standard Deviation

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/ N x
Substituting into Equation 3-8 leads to Example 3-3

11/30/2015
7Finding the Sample
Standard Deviation

Note in Example 3-3 that the difference between Σ x2
i
and (Σ xi)2/Nis 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. Finding the Sample
Standard Deviation

What Is the Standard Error of the Mean?

For replicate samples, each containing N measurements, ar e
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 s m.
(3-9)
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 the
Reliability 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 the
Reliability of s

To obtain a pooled estimate of the standard deviati on,
spooled, deviations from the mean for each subset a re
squared; the squares of all subsets are then summed
and divided by an appropriate number of degrees of
freedom. Equation for Calculating Pooled
Standard Deviations

The equation for computing a pooled standard
deviation from several sets of data takes the form
where N1is the number of results in set 1, N2is the
number in set 2, and so forth. The term Ntis the
number of data sets that are pooled.

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8Example 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 deviati on for
the method, based on the first three columns of dat a:
Example 3-4

The values in the last two columns for specimen 1 w ere
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.
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)

11/30/2015
9Alternative Terms for
Expressing the Precision of Samples of Data

Spread or Range (w)

Another term to describe the precision of a set of replicat e
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) t he
variance, (b) the relative standard deviation in pa rts 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) s 2 = (0.0038) 2= 1.4 10 -5

(b) RSD = 1000ppt = 5.0ppt

(c) CV = 100% = 0.50%

(d) w =0.760 – 0.751 = 0.009 x
0.0038
0.754
0.0038
0.754 The Standard Deviation
of Sums and Differences

Consider the summation
Absolute standard deviations
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, S2
y is given by

11/30/2015
10 Table 3-4
The Standard Deviation of
Products and Quotients
The Standard Deviation of
Products and Quotients

As shown in Table 3-4, the relative standard deviat ion
of a product or quotient is determined by the relat ive
standard deviations of the numbers forming the
computed result.
Applying this equation to the numerical example giv es

we can write the answer and its uncertainty as
0.0104( ±0.0003). The Standard Deviation of
Products and Quotients
Reporting Computed Data

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

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

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

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11 The Significant Figure Convention

A zero may or may not be significant depending on i ts
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. 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 pla ce
Significant Figures in
Numerical Computations

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 origi nal
number with the smallest number of significant digits. Significant Figures in
Numerical Computations

Logarithms and Antilogarithms
1. In a logarithm of a number, keep as many digits to the righ t
of the decimal point as there are significant figures in t he
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 significan t
digits are retained: (a) log 4.000 10 5= –4.3979400
and (b) antilog 12.5 = 3.162277 10 12 .
(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
Rounding Data

A good guide to follow when rounding a 5 is always to
round to the nearest even number. For example, 0.6 35
rounds to 0.64 and 0.625 rounds to 0.62.

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

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12 Rounding the Results from
Chemical Computations

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

It is especially important to postpone rounding unt il
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” digi t. 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 associate d
with its subsequent use.
Assumptions of the Least-
Squares Method

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

The mathematical relationship that describes this
assumption is called the regression model, which ma y be
represented as
where b is the y intercept (the value of y when x i s zero) and
mis the slope of the line.
Computing the Regression Coefficients and
Finding the Least-Squares Line

The vertical deviation of each point from the strai ght
line is called a residual.
Computing the Regression Coefficients and
Finding the Least-Squares Line

1. The slope of the line m:

2. The intercept b:

3. The standard deviation about regression sr:
Computing the Regression Coefficients and
Finding the Least-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:

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13 Computing the Regression Coefficients and
Finding the Least-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 r esults
from the least-squares analysis:
Computing the Regression Coefficients and
Finding the Least-Squares Line

The standard deviation about regression is often ca lled
the standard error of the estimate or the standard error
in y .
Instrument Calibration Methods
Calibration Methods are techniques that are use to
find the relationship between analyte concentration
and instrument response or signal. The most
common approaches are;
• External Calibration Method (Calibration curve)
• Standard Addition Method
• Internal Standard Method Calibration curve
•A calibration curve (also known as analytical curve
or working curve ) is a general method for
determining the concentration of a substance in
an unknown sample by comparing the unknown to
a set of standard samples of known concentration.
•The basis of quantitative analysis is that the
magnitude of the measured property is
proportional to concentration of analyte.
Calibration curve Calibration curve
•This is typically obtained by measuring the
analytical signals for a series of standards i.e.
analyte solutions of known concentration.
•Ideally the analyte content and the measured
signal should exhibit a perfectly linear relationsh ip
of formula:
Y = mX + b
or Signal = m (Conc.) + Sblank Where mis the
gradient and bis
the y-intercept

11/30/2015
14 Calibration curve
•In reality such perfectly linear relationships don’ t
exist when real samples and standards are used.
•Therefore the “best” line relationship among the
experimental points is used finding the best-fit li ne
of the data.
•Visual estimation of the “best” line through a set
of points is subject to error, both in plotting the
points and in fitting the line.
•A preferable approach for finding such a line is th e
method of least squares or sometimes referred to
as linear regression. Advantages to using calibration curves
•First, the calibration curve provides a reliable wa y to
calculate the uncertainty of the concentration
calculated from the calibration curve (using the
statistics of the least squares line fit to the data).
•Second, the calibration curve provides data on an
empirical relationship. The mechanism for the
instrument's response to the analyte may be
predicted or understood according to some
theoretical model, but most such models have
limited value for real samples.
•(Usually instrumental response is highly dependent
on the condition of the analyte, solvents used and
impurities it may contain; it could also be affecte d by
external factors such as pressure and temperature.)
Disadvantages to using calibration curves
•The chief disadvantages are that the standards
require a supply of the analyte material,
preferably, of high purity and in known
concentration.
•(Some analytes – e.g., particular proteins – are
extremely difficult to obtain pure in sufficient
quantity.)
Schematic of a calibration curve plot showing limit of detection (LOD),
limit of quantification (LOQ), dynamic range, and l imit of linearity
(LOL).
Regions of calibration curve
Dynamic Range the range over which measurements
can be made. Extends from LOQ to LOL.
Regions of calibration curve
LOQ (limit of quantitation): [lowest] at which
quantitative measurements can reliably be made.
Equal to 10 x Average Signal for blank i.e. 10Sbl.
Regions of calibration curve

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15 LOL (limit of linearity) : point where signal is no longer
proportional to concentration.
Regions of calibration curve Example 1
•The table below shows the calibration data for the chromatog raphic
determination of isooctane in a hydrocarbon mixture.
•Calculate (a) the equation for the least-squares line (b) the standard
deviation about regression (c) the standard deviation of the sl ope (d)
standard deviation of the intercept
•A peak area of 2.65 was obtained for the chromatographic
determination of isooctane in a sample of hydrocarbon mixture .
Calculate (a) the mole percent of isooctane in the mixture and the
standard deviation if the area was (i) the result of a singl e measurement
and (ii) the mean of four measurements.
0.352 1.09
0.803 1.78
1.08 2.60
1.38 3.03
1.75 4.01
Solution 1
0.352 1.09 0.123904 1.1881 0.38368
0.803 1.78 0.644809 3.1684 1.42934
1.08 2.60 1.1664 6.76 2.808
1.38 3.03 1.9044 9.1809 4.1814
1.75 4.01 3.0625 16.0801 7.0175
5.365 12.51 0.123904 1.1881 0.38368
Solution 1
Solution 1
1. Gradient or slope of line, m
2. intercept, b
Finish the rest parts of the example FIGURES OF MERIT FOR ANALYTICAL METHODS
Analytical procedures are characterized by a number
of figures of merit such as accuracy, precision,
sensitivity, detection limit, and dynamic range. So me
of these have already been discussed.
1. Sensitivity
•Indicates the response of the instrument to
changes in analyte concentration or a measure of a
method’s ability to distinguish between small
differences in concentration in different samples.

11/30/2015
16 •Effected by slope of calibration curve &
precision.
•Indicated by the slope of the calibration curve or
in other words, a change in analytical signal per
unit change in [analyte].
•For two methods with equal precision, the one
with steeper calibration curve is more sensitive.
This is referred to as Calibration sensitivity
•If two methods have calibration curves with
equal slopes, the one with higher precision is
more sensitive. This is referred to as Analytical
sensitivity Sensitivity cont. •Calibration sensitivity ( S):
The slope of the calibration curve evaluated in the [analyte] range of
interest.
S = mc + Sbl (m = slope; c = conc; Sbl = Signal of blank)
Advantage: With linear calibration curve, sensitivity independent of
[analyte].
Disadvantage : does not account for precision of individual
measurements.
•Analytical Sensitivity (γ)
Defined by Mandel and Stiehler to include precision in sensitivity.
Definition.
γ = m/ss(m = slope; ss=standard deviation of measurement)
Advantage: Insensitive to amplification factors i.e. increasing gain also
increases m but Ss also increases by same factor hence γ stays
constant. Independent of measurement units for s.
Disadvantage : concentration dependent as ssusually varies with
[analyte].
2. Detection Limit
•The smallest [analyte] that can be determined with
statistical confidence.
•Analyte must produce an analytical signal that is
statistically greater than the random noise of
blank. ( i.e. analytical signal = 2 or 3 times std. dev.
of blank measurement (approx. equal to the peak-
peak noise level).
Calculation of detection limit
•The minimum detectable analytical signal ( Sm) is
given by:
Sm= Sbl + k(sbl ); for detection use k =3
Sbl (mean blank signal) and sbl (std. dev. of blank signals)To Determine the detection limit Experimentally
•Perform 20 – 30 blank measurements over an
extended period of time.
•Treat the resulting data statistically to obtain Sbl
(mean blank signal) and sbl (std. dev. of blank
signals). Use these to obtain Sm value.
•Using slope (m) from calibration curve. Detection
limit ( Cm) is calculated by: (Rearranged from S m=
Cm+ Sbl )
Example
The analysis of the calibration data for the determ ination of
lead based upon its flame emission spectrum yielded an
equation: S = 1.12C + 0.312 where C is the concentr ation of
Pb in ppm and S is a measure of relative emission in tensity.
The following replicate data were obtained:
•Calculate (a) The calibration sensitivity (b) The a nalytical
sensitivity at 1 and 10 ppm of Pb (c) The detection limit Concn. (C)
ppm No. of
replicates Mean value
of SStd. dev.
10.0 10 11.62 0.15
1.00 10 1.12 0.025
0.000 24 0.0296 0.0082