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A.
Introduction
1. Accuracy
When a quantitative
value is determined experimentally, our main concern
is that we obtain the "right" answer. The comparison
between our experimentally determined value and
the true value is a measure of accuracy.
Definition:
Accuracy
is the degree of agreement between the experimental
value and the true value.
The accuracy
has a sign associated with it and that sign indicates
whether the experimental value is high (+) or
low (-) with respect to the true value.
Example:
A solution
known to have a pH of 8.00 pH units is measured
with a pH meter and the average of several trials
is found to be 7.90 pH units.
The accuracy
is 7.90 pH units - 8.00 pH units = -0.10 pH units.
This
gives an absolute error because the units
of the error are the same as the units of the
measurement. A relative error can also
be calculated with respect to the true value:
Note that
the relative error is unitless.
The accuracy
can only be measured if the true value is known
which is not always the case; someone has to have
found it in the first place by some method. Different
methods might inherently give slightly different
answers. Which method will ultimately be used
to provide the "true" answer? Even standards change
occasionally as the ability to make certain types
of measurements change. Up until 1948 the coulomb
was defined as the quantity of electricity which
must pass through a circuit to deposit 0.0011180
grams of silver from a solution of silver nitrate.
Now it is defined as the quantity of electricity
on the positive plate of a condenser of one farad
capacity when the electromotive force is one volt.
We will
not deal here with the problem of what is the
true value. You should be aware, however, that
it is not as straightforward as you might have
thought. For that reason, perhaps accepted
value is a better term.
2. Determinate Errors
and Accuracy
What is it
that keeps every measurement or analysis from
being completely accurate? That is, what causes
the discrepancy between the accepted value and
the measured value? The discrepancy is caused
by many small deviations which can be divided
into two groups: determinate and indeterminate
errors.
Definition:
Determinate
errors are errors which have a definite value
that can, in principle, be measured and accounted
for.
Determinate
errors, also called systematic errors, can be
divided into arbitrary categories. The most common
divisions are instrumental, operator, and method
errors. Determinate errors are often unidirectional,
that is they are all positive or all negative
with respect to the accepted value. Be aware that
determinate errors can be corrected for but only
after the cause is determined. This might take
some detective work. For example, an incorrectly
calibrated instrument might give results which
are too high; this determinate error would be
the fault of the operator. On the other hand,
an instrument signal drifting downward might give
a low value and illustrates instrumental error.
3.
Indeterminate Error
and Precision
Even when
all determinate errors are corrected and compensated
for, the same measurement taken several times
will not necessarily give the same answer. This
is because of indeterminate errors also called
random or statistical errors.
Definition:
Indeterminate
errors are errors which fluctuate randomly
and do not have a definite value; They cannot
be positively identified.
To further
understand indeterminate errors, consider the
weight of an object obtained by doing five different
weighings on a four place analytical balance.
trial
1: 0.7952 g
trial
2: 0.7950 g
trial
3: 0.7951 g
trial
4: 0.7953 g
trial
5: 0.7951 g
The first
three figures are the same in all cases. The last
figure has an uncertainty associated with it.
This uncertainty is a function of the type of
sample, the conditions under which it is being
weighed, the balance, and the person doing the
weighing. Even when all factors are optimized,
there will still be some variation in the weight.
This variation or uncertainty is the result of
pushing the balance to its limit.
We could
cut the last figure off; then all the weights
would be the same, but the weight would be known
only to the nearest milligram. We obtain more
information if we keep that last figure but remain
aware of its uncertainty. That uncertainty arises
because of indeterminate error; and is indicative
of the precision of the measurement.
Definition:
Precision is the degree
of agreement between replicate measurements of
the same quantity.
Note
that even if the precision of a measurement is
excellent, the value obtained can have poor accuracy
if there has been a determinate error. For example,
an incorrectly calibrated instrument cannot give
an accurate reading although the precision very
likely will not be affected. On the other hand,
poor precision rarely results in an accurate value
being obtained.
B.
Uncertainty
The element
of uncertainty in experimental data can be quantified
and is often reported along with the actual experimental
value itself. The value of the uncertainty gives
one an idea of the precision inherent in a measurement
of an experimental quantity. So important is the
concept of uncertainty it actually has a principle
named after it. With the uncertainty being reported
along with the experimental quantity one has an
idea of how "good" the reported experimental value
is. Consequently, comparisons of the numbers obtained
in a series of measurements with the same or different
techniques are made more meaningful by the inclusion
of uncertainties.
There
are many ways to quantify uncertainty, ranging
from very simple techniques to highly sophisticated
methods. The method used will depend upon how
many measurements of a single quantity are made
and on how crucial the reporting of the value
of uncertainty is with regard to the interpretation
of the experimental data. The following is a list
of many of the ways in which uncertainty is reported.
1.
Standard Deviation
a.
Deviation of the Expression for Standard Deviation
The
most widely used method for calculating uncertainty
is representing it by the standard deviation of
a series of measurements. This can be easily accomplished
if it is assumed that a gaussian distribution
function best describes the spread in the error
values for a series of measurements made on a
single observable. The distribution function is
then termed a "normal" error probability function
and written
where
represents the error in a measurement (negative
or positive) and 
is a parameter known as the standard deviation.
One can see from this equation that 
is related to the width of the distribution of
errors. If 
has a small value, the probability function decreases
rapidly from its maximum value (where 
=0) indicating that the probability of large errors
in the measurement is small. If
has a large value, the probability function is
broad indicating a high degree of probable error
in the measurement. Mathematically 
is defined as the root-mean-square error associated
with this probability function.
where
 2
is the mean of the squared errors. If each individual
error, 
is known exactly (through knowledge of the true
value of the observable) an approximate formula
for
can be derived; it is given by
where 
is the number of independent data or degrees of
freedom on which the calculation of  is
based, and
is the difference between the observed value and
the true value of a quantity (i.e.,= ).
In a
typical experiment one would not know the true
value of an observable (or why do the experiment?).
A good approximation to the true value of an observable
is the arithmetic mean, ,
of a series of n measurements (neglecting systematic
error),
where
the 
are the values for the individual measurements.
Because it is known that the sum of an entire
set of deviations from the mean must necessarily
add up to zero, i.e.,
then a knowledge
of only n-1 values of
is necessary to define the nth or last value.
That is any value
can be determined from the remaining
values provided the mean 
has been calculated. Thus, it is found that calculating
the mean value, ,
reduces the number of independent variables or
degrees of freedom with which one can determine
the precision inherent in a series of measurements.
The problem
of determining the uncertainty (which is a measure
of precision) in a series of measurements of a
single observable by the standard deviation method
is solved by using the formula
where
s2
is termed the variance, and the n-1 term
in the denominator follows from the above discussion.
This represents the best estimate of the standard
deviation for a finite set of data. Note that
is being approximated by  ,
hence n-1 appears in this equation.
b.
Calculation of the Standard Deviation
The standard
deviation of a set of five weights can be calculated
as shown:
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0.7962
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0.0010
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1.0
 10-6
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0.7950
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-0.0002
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4.0
10-8
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0.7941
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-0.0011
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1.2
 10-6
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0.7953
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0.0001
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1.0
 10-8
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0.7955
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0.0003
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9.0
 10-8
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Note
that the standard deviation has the same units
as the measurement itself, i.e., g, and is thus
a measure of absolute precision. The number
of figures used to express x and s
is explained in Section IV, Significant Figures.
The
standard deviation can be expressed relative to
the mean giving a measure of relative precision.
Relative precision can be expressed as a fraction,
a percentage, parts per thousand, or any other
desired relative measure. When it is expressed
as a fraction, it is usually referred to as the
coefficient of variation.
Notice
that the relative error is a unitless number.
2.
Average Deviation
Sometimes
to get a quick estimate of the uncertainty or
precision, it is easier to use the average deviation.
The average absolute deviation is simply the mean
of the sum of the deviation of individual trials
from the mean without regard to sign (if the sign
were taken into account, the sum of deviations
would of course be zero).
In
the case of the weights which we have been looking
at, the average absolute deviation is:
The average
absolute deviation is a measure of absolute uncertainty
because it is in the same units as the measurement,
because it is calculated from the absolute values
of the deviation.
The average
relative deviation usually measured in parts per
thousand (ppt) is:
The
average absolute and relative deviations are slightly
smaller than the corresponding standard deviations,
but they are still reasonable estimates of the
magnitude of uncertainty.
3. Range
When
a series of measurements is made on a single observable,
the uncertainty can be crudely approximated by
the range, which is given as the difference between
the maximum and minimum values of the measured
quantity. In the case of the set of five weights
given in Section I. C., the range is 0.7953
g - 0.7950 g = 0.0003 g.
4.
The Graduation
Method
When a measurement
is done only once or when it is not possible to
repeat an experiment enough times for a statistical
treatment to be used, the uncertainty must be
approximated. Very often the uncertainty in a
single measurement is given by a value one-half
of the smallest level of graduation in the measuring
instrument. For example, if a single measurement
of the length of an object is to be made using
a meter stick marked with millimeter graduations,
the length should be reported ± .5 mm (or ± .0005
m).
In some
cases, it may be possible to divide the space
between scale divisions into five equal parts.
For example, a pH meter usually is calibrated
in tenths of a pH unit but is read to 0.02 pH
units.
You must
use your own judgment in choosing an increment
to calculate the precision using the Graduation
Method. If in doubt, always be conservative; i.e.
report the largest of possible uncertainties.
5.
Uncertainties Inherent
in Graphing
When graphical
techniques are used to determine a quantity of
interest, special methods are needed for determining
the uncertainty in the measurement of the quantity.
The method to be used depends on the number of
observations (data points) used to determine the
quantity of interest. In all cases the ranges
of data points should span the graph sufficiently
to illustrate the experimental behavior with clarity.
A distinction here is (arbitrarily) made between
a small number of data points (less than 10) and
a large number of data points (greater than 25).
For situations in-between subjectivity is required
to make a choice of the following two methods:
a.
Method of Limiting Slopes
When few
observations are made, the "method of limiting
slopes" is the best technique for determining
the uncertainty in the measurement of the slope
and intercept of a linear plot. The technique
consists of drawing a rectangle around every data
point where the dimensions of the rectangle represent
the uncertainties of the measured quantities.
A best straight line is then drawn throughout
the individual points. Two other lines which represent
the maximum or minimum slopes are then drawn so
that both lines pass through every rectangle.
The differences in the slopes or intercepts of
the limiting lines can be taken as twice the uncertainty
in the slope or intercept of the best line. The
following figure produced by C. David illustrates
the procedure.
b.
Limiting Slopes Method Consistent with Scatter
in the Data
When a large
number of observations is made, an estimate of
the uncertainty in the intercept can be made from
the scatter in the data. The two lines corresponding
to the limiting slopes can be drawn so as to remain
within the scatter of points. The uncertainties
in the slope or intercept of the best line is
thus straightforwardly obtained.
c.
Propagation of Error
Because of
the impossibility of measuring an experimental
quantity to infinitely small precision, there
will always exist random error. When the quantity
of interest is directly measurable, any one of
the techniques set forth in Section II can be
used to estimate the precision of the experimental
measurement. When the quantity of interest is
not directly measurable but results from a calculation
involving two or more experimentally determined
quantities, it is necessary to determine how the
error in each individual quantity propagates through
to the final result. This concept of the propagation
of errors is important in determining ways of
improving experimental design. For example, it
would be a waste of money to buy an expensive
analytical balance to replace an ordinary trip
scale if the uncertainty in the weight of a sample
contributes insignificantly to the precision or
accuracy of the final result.
1. Expressions derived
by differentiation
We designate
the quantity of interest as x, where x
is understood to be a function of more than one
experimental quantity (independent variable),
i.e.,
where
a, b, c... are the measured quantities.
(Strictly speaking x is only proportional
to f in that x may result from a
calculation involving nonmeasurable quantities,
e.g. 
or h. For simplicity, we ignore this finer
point.) These independent variables, a, b,
c, have corresponding differential changes
da, db, dc... . From the chain rule we
have
If
it is assumed that the uncertainties 
of the individual experimental determinations
are reasonably small then it is possible to write
There
is a problem, however, in that the uncertainties
come as positive and negative quantities for random
errors. For systematic errors where the exact
values (and signs) of the uncertainties are known,
one uses the above expression for D
for the propagation of the error. For the case
of random error, the ambiguity is removed by considering
the square of the above expression for  .
Thus, it is found
Averaging
this expression over all possible values
yields
because the
average values of 
will always be zero for a normal gaussian distribution
of errors; i.e., the frequency of positive deviations
equals that of negative deviations. In particular,
therefore, the above expression suffices for use
in the determination of the absolute error propagated
through a calculation. The relative error is found
by dividing 
by x.
Examples:
1. 
2. 
These formulas
were obtained by taking partial derivatives as
shown above. Notice that the absolute error of
a value obtained by either addition or subtraction
is the same; the square root of the sum of the
squares of the absolute error of each individual
measurement. This means that we want to minimize
absolute errors when a result will be obtained
by addition or subtraction of measurements.
3. 
This gives
an expression for the absolute error of a product
of two variables.
It
is interesting to manipulate this equation to
obtain the relative error,
Realizing
that b = x/a and a = x/b, we obtain:
But this
is simply:
This gives
us the result that the relative error of a product
is the square root of the sum of the relative
errors of the variables. This means that we want
to minimize relative errors when a result will
be obtained by multiplication of measurements.
4. 
This gives
the expression for the absolute error of a quotient
of two variables.
Let
us once again manipulate this equation to obtain
the relative error of the resulting value,
 .
since 
then 
This
is a very interesting result; although the absolute
error of the result of a quotient is very
different from that of a product, the expression
for the relative error of a result obtained
from the quotient of the two measurements is the
same as that of a result obtained from the product
of two measurements. Thus, once again we want
to minimize relative errors when a result will
be obtained by division of one measurement by
another.
The reader
might want to derive the expression for absolute
and relative errors in the case of an exponential
relationship, ,
and in the case of a logarithmic relationship,
 .
2.
Expressions derived
by the "worst case" method
The formulas
presented above are useful for the propagation
of error from measured quantities through to the
quantity of interest. A less rigorous approach
to error propagation presented by Gordon et al.
can be achieved by examining the result of an
arithmetic operation in the limits of the uncertainties
of the variables upon which the quantity of interest
depend. The idea is to deduce the uncertainty
of a result of an operation by exploring the high
and low limits of the result using the ranges
imposed by the precisions in the independent variables.
The following
figure shows two observables, a and b,
and their respective uncertainties, Da
and Db. The "worst case", highest
value of a is a + Da, and
its lowest value is a - Da. The
"worst case", lowest value of b
is b - Db, and its highest value
is b + Db. The figure shows that,
for example, if the quantities a and b
are added, the uncertainty in the result, x
= a + b, will be bound on the high
side by 
and on the low side by  .
The difference between these two limits gives
two times the uncertainty, D x. This
argument can be extended to all arithmetic operations.
The expressions for addition, subtraction, multiplication
and division are given below.
Examples:
1.
2.
3.
4.
D.
Significant Figures
1. Rounding off
When rounding
off a number, the figure in the place immediately
preceding the dropped figure remains the same
if the dropped figure is less than five and is
rounded up if the dropped figure is greater than
five.
Example:
If the dropped
figure is five, then the general rule is that
the preceding figure is treated in such a way
as to make it an even number. This means that
half the time the preceding figure remains the
same (if it is even) and half the time it is rounded
up (if it is odd).
Example:
2.
Which
Zeros are Significant?
First, let
us make sure we know how to count significant
figures. The biggest confusion arises with zeros.
a. Zeros
to the right of the decimal point but to the left
of the first non-zero digit are not significant.
They are place keepers and can be eliminated by
using exponential notation.
Example:
0.000151 3
significant figures
1.51
x 10-4 3
significant figures
b. Zeros
to the right of a non-zero digit may or may not
be significant; at times it requires judgment
to decide on their status. It is best to use exponential
notation to avoid ambiguity.
Example:
54.20 4
significant figures; the value is understood to
be between 54.19 and 54.21.
17,000 All
three zeros are probably not significant.
1.7 x 104
Two significant figures. Exponential
notation clears up any ambiguity.
3.
Significant Figures from
Uncertainty of Single Measurements
Definition:
Significant figures
are figures needed
to report a value to the precision with which
a measurement is made.
The concept
of significant figures arises only when we begin
to talk about measurements. It has no meaning
with respect to pure numbers. That is why nobody
ever worries about significant figures when doing
long division in third grade arithmetic; the quotient
3.8948/41.65 can be carried out as far as we want
but in finding the quotient 3.8948 g/41.65 mL
we must be concerned about the precision of each
measurement.
The question
we want to answer is to how many figures should
the quotient be expressed? In order to answer
this, we must know the precision of the measurements
and from those propagate the uncertainty of the
resulting quotient.
For the
purpose of this example, we will assume that the
mass had a precision of 
0.0004 g, and the volume had a precision of
0.04 mL. Remember from the propagation of error
derivations, when x = a/b, then:
In this case,
the quotient, a/b, read from the calculator is:
Thus, the
quotient can be expressed as:
0.09351
0.00009
By the
rule of thumb, which you probably learned in freshman
chemistry, we would have guessed that the number
could be expressed to four significant figures
because the number in the calculation containing
the least figures had four. But we could not have
guessed the value of the uncertainty which
is quite high,
0.00009 to be exact, which tells us that the last
figure, 1, is not very precise at all.
Let us
look at the precision of a sum. Suppose ten aliquots
of solvent are delivered into a beaker from a
pipette which delivers 25.00
0.05 mL. What is the total volume delivered?
We are
adding ten pipettes so:
by error
propagation
Thus,
Again,
this obeys the "freshman chemistry" rule of thumb
for significant figures of a sum which says that
the number of decimal places in the sum is that
of the addend with the least number of decimal
places. Again, however, the rule does not give
us the quantity of the uncertainty. In this case,
the two figures after the decimal point are uncertain.
4.
Significant Figures
from Standard Deviation
Often in
the lab multiple determinations of some value
are obtained and a standard deviation is found.
The standard deviation is a measure of precision
and indicates the significance of the figures
obtained in a mean.
Suppose
the concentration of HCl is determined in triplicate
with the following results:
0.052792
moles/liter
0.052817
moles/liter
0.052835
moles/liter
moles/liter
s
= 0.000022 moles/liter
Thus,
the mean is expressed to five significant figures.
There
is a problem as to how much to round off a calculation
before determining the mean and standard
deviation. Looking at the three trials, the first
three figures are identical, 0.0528, which means
the calculation has to be carried out to at least
one more place to even see an uncertainty. However,
even the fourth digits do not span a really wide
range: 79-83 is a range of 4.
As a
rule of thumb, if the range of the uncertain figure
is less than 6, keep another figure after the
first uncertain one for the calculation of the
mean and standard deviation.
If we
had kept only four figures, we would have obtained
the following:
0.05279
moles/liter
0.05282
moles/liter
0.05284
moles/liter
= 0.05283 moles/liter
s
= 0.000025 moles/liter
reported
as 
or in this example 0.05283 0.00002
moles/liter.
The mean
must be expressed to the same precision as the
uncertainty; otherwise we are not getting the
full value of our measurements!
Note
that the rounding off of individual values too
soon also resulted in a different standard deviation.
When in doubt, it is better to keep an extra figure
until the standard deviation is found; it will
then tell you if you have to round off. Whenever
several trials are done, a standard deviation
should be used to determine the number of significant
figures rather than a propagated error as shown
above.
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