m |
|
0.07 gm |
|
= |
|
= 0.002, or, if you wish, 0.2% |
M |
|
34.6 gm |
It is a matter of taste whether one chooses to express relative
errors "as is" (as fractions), or as percents. I prefer to work
with them as fractions in calculations, avoiding the necessity for
continually multiplying by 100. Why do unnecessary work?
But when expressing final results, it is often meaningful to
express the relative uncertainty as a percent. That's easily done,
just multiply the relative uncertainty by 100. This one is 0.2%.
3. Absolute or relative form; which to use.
Common sense and good judgment must be used in choosing which form
to use to represent the error when stating a result.
Consider a
temperature measurement with a thermometer known to be reliable to
± 0.5 degree Celsius. Would it make sense to say that this
causes
a 0.5% error in measuring the boiling point of water (100 degrees)
but a whopping 10% error in the measurement of cold water at a
temperature of 5 degrees? Of course not! [And what if the
temperatures were expressed in degrees Kelvin? That would seem
to reduce the percent errors to insignificance!] Errors and
discrepancies expressed as percents are meaningless for some types
of measurements. Sometimes this is due to the nature of the
measuring instrument, sometimes to the nature of the measured
quantity itself, or the way it is defined.
There are also cases where absolute errors are inappropriate and
therefore the errors should be expressed in relative form.
Sometimes both absolute and relative error measures are
necessary to completely characterize a measuring instrument's error. For
example, if a plastic meter stick uniformly shrank with age, the effect
could be expressed as a percent determinate error. If a one half
millimeter were worn off the zero end of a stick, and this were not
noticed or compensated for, this would best be expressed as an
absolute determinate error. Clearly both errors might be present
in a particular meter stick. The manufacturer of a voltmeter (or
other electrical meter) usually gives its guaranteed limits of
error as a constant determinate error plus a `percent'
error.
Both relative and fractional forms of error may appear in the
intermediate algebraic steps when deriving error equations. [This
is discussed in section H below.] This is merely a computational
artifact, and has no bearing on the question of which form is
meaningful for communicating the size and nature of the error in
data and results.
G. IMPORTANCE OF REPEATED MEASUREMENTS
A single measurement of a quantity is not sufficient to convey any
information about the quality of the measurement. You may need to
take repeated measurements to find out how consistent the
measurements are.
If you have previously made this type of measurement, with the same
instrument, and have determined the uncertainty of that particular
measuring instrument and process, you may appeal to your experience
to estimate the uncertainty. In some cases you may know, from past
experience, that the measurement is scale limited, that
is, that
its uncertainty is smaller than the smallest increment you can read
on the instrument scale. Such a measurement will give the same
value exactly for repeated measurements of the same quantity. If
you know (from direct experience) that the measurement is scale
limited, then quote its uncertainty as the smallest increment you
can read on the scale.
Students in this course needn't become experts in the fine
details of statistical theory. But they should be constantly aware
of the experimental errors and do whatever is necessary to find out
how much they affect results. Care should be taken to minimize
errors. The sizes of experimental errors in both data and results
should be determined, whenever possible, and quantified by
expressing them as average deviations. [In some cases common-sense
experimental investigation can provide information about errors
without the use of involved mathematics.]
The student should realize that the full story about experimental
errors has not been given here, but will be revealed in later
courses and more advanced laboratory work.
H. PROPAGATION OF DETERMINATE ERRORS
The importance of estimating data errors is due to the fact that
data errors propagate through the calculations to produce errors
in results. It is the size of a data errors' effect on the
results which is most important. Every effort should be made
to determine reasonable error estimates for every important
experimental result.
We illustrate how errors propagate by first discussing how to
find the amount of error in results by
considering how data errors propagate through simple mathematical
operations. We first consider the case of determinate errors: those
that have known sign. In this way we will discover certain
useful rules for error propagation, then we'll then be able to modify
the rules to apply to other error measures and also to indeterminate
errors.
We are here developing the mathematical rules for
"finite differences," the algebra of numbers which
have relatively small variations imposed upon them. The finite
differences are those variations from "true values" caused by
experimental errors.
This method is based on a fundamental principle. In any calculation we want to know how much
an error in one input variable will affect the output result. In complex calculations, such as
in meterology weather forecasting, computers allow us to do this directly on each of many input variables, something
one would never attempt "by hand". It is a "brute-force" method, but necessary. In this laboratory
the equations will be much simpler, and usually yield to algebra and a few simple rules.
Suppose that an experimental result is calculated from the sum of
two data quantities A and B. For this discussion we'll use a and
b to represent the errors in A and B respectively. The data
quantities are written to explicitly show the errors:
(A + a) and (B + b)
We allow that a and b may be either positive or
negative, the signs being "in" the symbols "a" and "b." But we must emphasize
that we are here considering the case where the signs of a and b are determinable,
and we know what those signs are (positive, or negative).
The result of adding A and B to get R is expressed by
the equation: R = A + B. With the errors explicitly included,
this is written:
(A + a) + (B + b) = (A + B) + (a + b)
The result with its error, r, explicitly shown, is: (R + r):
(R + r) = (A + B) + (a + b)
The error in R is therefore: r = a + b.
We conclude that the determinate error in the sum of two quantities is just the
sum of the errors in those quantities. You can easily work out for
yourself the case where the result is calculated from the
difference of two quantities. In that case the determinate error in
the result will be the difference in the errors. Summarizing:
- Sum rule for determinate errors. When two quantities are added,
their determinate errors add.
- Difference rule for determinate errors. When two quantities are subtracted,
their determinate errors subtract.
Now let's consider a result obtained by multiplication, R = AB. With
errors explicitly included:
(R + r) = (A + a)(B + b) = AB + aB + Ab + ab
or: r = aB + Ab + ab
This doesn't look promising for recasting as a simple rule. However,
when we express the errors in relative form, things
look better. If the error a is small relative to A, and b is small
relative to B, then (ab) is certainly small relative to AB,
as well as small compared to (aB) and (Ab). Therefore we
neglect the term (ab) (throw it out), since we are interested only
in error estimates to one or two significant figures. Now we
express the relative error in R as
r | | aB + bA | | a | | b |
|
= |
| = |
| + |
|
R | |
AB | | A | | B |
This gives us a very simple rule:
- Product rule for determinate errors. When two quantities are multiplied, their
relative determinate errors add.
A similar procedure may be carried out for the quotient of two
quantities, R = A/B.
|
|
A + a |
|
A |
|
(A + a) B |
|
A (B + b) |
|
|
|
– |
|
|
|
– |
|
r | | B + b |
|
B |
| (B + b) B |
| B (B + b) |
|
= |
|
= |
|
R | |
A/B |
|
A/B |
| (A + a) B – A (B + b) |
|
(a)B – A(b) | | a | | b |
= |
|
≈ |
|
≈ |
|
– |
|
| A(B + B) |
|
AB |
|
A | | B |
The approximation made in the next to last step was to neglect b
in the denominator, which is valid if the relative errors are
small. So the result is:
Quotient rule for determinate errors. When two quantities are divided, the
relative determinate error of the quotient is the relative
determinate error of the numerator minus the relative determinate
error of the denominator.
A consequence of the product rule is this:
Power rule for determinate errors. When a quantity Q is raised to a power, P,
the relative determinate error in the result is P times the
relative determinate error in Q. This also holds for negative
powers, i.e. the relative determinate error in the square root of
Q is one half the relative determinate error in Q.
One illustrative practical use of determinate errors is the case of correcting a result when you discover, after completing lengthy measurements and calculations, that there was a determinate error in one or more of the measurements. Perhaps a scale or meter had been miscalibrated. You discover this, and fine the size and sign of the error in that measuring tool. Rather than repeat all the measurements, you may construct the determinate-error equation and use your knowledge of the miscalibration error to correct the result. As you will see in the following sections, you will usually have to construct the error equation anyway, so why not use it to correct for the discovered error, rather than repeating all the calculations?
I. PROPAGATION OF INDETERMINATE ERRORS
Indeterminate errors have unknown sign. If their
distribution is symmetric about the mean, then they are unbiased
with respect to sign. Also, if indeterminate errors in different
quantities are independent of each other, their signs have a
tendency offset each other in computations.[11]
When we are only concerned with limits of error (or
maximum error) we must assume a "worst-case" combination of signs.
In the case of
subtraction, A - B, the worst-case deviation of the answer occurs
when the errors are either +a and -b or -a and +b. In either case,
the maximum error will be (a + b).
In the case of the quotient, A/B, the worst-case deviation of the
answer occurs when the errors have opposite sign, either +a and -b
or -a and +b. In either case, the maximum size of the relative
error will be (a/A + b/B).
The results for the operations of addition and multiplication are
the same as before. In summary, maximum indeterminate
errors propagate according to the following rules:
Addition and subtraction rule for indeterminate errors. The absolute indeterminate
errors add.
Product and quotient rule for indeterminate errors. The relative indeterminate errors
add.
A consequence of the product rule is this:
Power rule for indeterminate errors. When a quantity Q is raised to a power, P,
the relative error in the result is P times the relative error in
Q. This also holds for negative powers, i.e. the relative error in
the square root of Q is one half the relative error in Q.
These rules apply only when combining independent
errors, that is, individual errors which are not dependent on each
other in size or sign.
It can be shown (but not here) that these rules also apply
sufficiently well to errors expressed as average deviations. The
one drawback to this is that the error estimates made this way are
still overconservative in that they do not fully account for the
tendency of error terms associated with independent errors to
offset each other. This, however, would be a minor correction of
little importance in our work in this course.
Error propagation rules may be derived for other mathematical
operations as needed. For example, the rules for errors in trig
functions may be derived by use of trig identities, using the
approximations: sin ß = ß and cos ß = 1,
valid when ß is small. Rules for exponentials may be derived
also.
When mathematical operations are combined, the rules may be
successively applied to each operation, and an equation may be
algebraically derived[12] which expresses the error in the result
in terms of errors in the data. Such an equation can always be cast
into standard form in which each error source appears in
only one
term. Let x represent the error in x, y the error in y, etc. Then
the error r in any result R, calculated by any combination of
mathematical operations from data values X, Y, Z, etc. is given
by:
r = (cx)x + (cy)y + (cz)z ... etc.
This may always be algebraically rearranged to:
r/R = {Cx}(x/X + {Cy}(y/Y) + {Cz}(z/Z) ... etc.
The coefficients (cx) and {Cx} etc.
in each term are extremely
important because they, along with the sizes of the errors,
determine how much each error affects the result. The relative
size of the terms of this equation shows us the relative importance
of the error sources. It's not the relative size of the errors
(x, y, etc), but the relative size of the error terms which tells
us their relative importance.
If this error equation was derived from the
determinate-error
rules, the relative errors in the above might have + or
- signs. The coefficients may also have + or - signs, so the terms
themselves may have + or - signs. It is therefore possible for
terms to offset each other.
If this error equation was derived from the indeterminate
error
rules, the error measures appearing in it are inherently positive.
The coefficients will turn out to be positive also, so terms cannot
offset each other.
It is convenient to know that the indeterminate error equation may
be obtained directly from the determinate-error equation by simply
choosing the worst-case, i.e., by taking the absolute value of
every term. This forces all terms to be positive. This step is
only done after the determinate-error equation has been
fully derived in standard form.
The error equation in standard form is one of the most useful tools
for experimental design and analysis. It should be derived (in
algebraic form) even before the experiment is begun, as a guide to
experimental strategy. It can show which error sources dominate,
and which are negligible, thereby saving time one might spend
fussing with unimportant considerations. It can suggest how the
effects of error sources might be minimized by appropriate choice
of the sizes of variables. It can tell you how good a measuring
instrument you need to achieve a desired accuracy in the
results.
The student who neglects to derive and use this equation may spend
an entire lab period using instruments, strategy, or values
insufficient to the requirements of the experiment. And he may end
up without the slightest idea why the results were not as
good as they ought to have been.
A final comment for those who wish to use standard deviations as
indeterminate error measures: Since the standard deviation is
obtained from the average of squared deviations, equation
(7) must be modifiedeach term of the equation (both sides) must
be squared:
(r/R) = (Cx)2(x/X) + (Cy)2(y/Y) + (Cz)2(z/Z)
This rule is given here without proof.
J. EXAMPLES
Example 1: A student finds the constant acceleration of a slowly
moving object with a stopwatch. The equation used is s = (1/2)at2.
The time is measured with a stopwatch, the distance, s, with a meter stick.
s = 2 ± 0.005 meter. This is 0.25%.
t = 4.2 ± 0.2 second. This is 4.8%.
What is the acceleration and its estimated error?
We'll use capital letters for measured quantities, lower case for their errors.
Solve the equation for the result, a. A = 2S/T2.
Its indeterminate-error equation is:
a t s
- = 2 - + -
A T S
The factor of 2 in the time term causes that term to dominate, for
application of the
rule for errors in quantities raised to a power causes the 4.8% error
in the time to be doubled, giving over 9.5% error in T2.
The 1/4 percent error due to the distance measurement is clearly
negligible compared to the 9.5% error due to the time measurement,
so the result (the acceleration) is written:
A = 0.23 ± 0.02 m/s2.
Example 2: A result is calculated from the equation
R = (G+H)/Z, the data values being:
G = 20 ± 0.5
H = 16 ± 0.5
Z = 106 ± 1.0
The ± symbol tells us that these errors are indeterminate.
The calculation of R requires both addition and division,
and gives the value R = 3.40. The error calculation requires
both the addition and multiplication rule, applied in succession,
in the same order as the operations performed in calculating R itself.
The addition rule says that the absolute errors in G and H add, so the
error in the numerator is 1.0/36 = 0.28.
The division rule requires that we use relative
(fractional errors). The relative error in the numerator is
1.0/36 = 0.028. The relative error in the denominator is
1.0/106 = 0.0094. The relative error in the denominator is added
to that of the numerator to give 0.0374, which is the relative error in R.
If the absolute error in R is required, it is
(0.0374)R = 0.0136.
The result, with its error, may be expressed as:
Example 3: Write a determinate-error equation
for example 1.
We follow the same steps, but represent the errors symbolically.
Let N represent the numerator, N=G+H. The determinate error in N
is then g+h. The relative error in the numerator is (g+h)/N.
The relative error in the denominator is z/Z.
The relative error in R is then:
r g + h z g h z
= = +
R G + H Z G+H G+H Z
r G g H h z
= +
R G+H G G+H H Z
This equation is in standard form;
each error, g, h, and z appears in only one term,
that term representing that error's contribution to the error in R.
Example 4: Derive the indeterminate error equation for this
same formula, R = (G+H)/Z.
Here's where our previous work pays off.
Look at the determinate error equation of example 3 and rewrite it
for the worst case of signs of the terms.
That's equivalent to making all of the terms of the standard
form equation positive:
r G g H h z
= + +
R G+H G G+H H Z
Example 5: Rework example 2, this time using the indeterminate error
equation obtained in example 4.
Putting in the values:
r 20 0.5 16 0.5 1
= + +
R 20+16 20 20+16 16 106
r 20 0.5 16 0.5 1
= + +
R 36 20 36 16 106
r
= 0.555(0.025) + 0.5(0.031) + 0.0094
R
r
= 0.014 + 0.014 + 0.0094 = 0.0374
R
This is less than 4%.
Example 6: A result, R, is calculated from the equation
R = (G+H)/Z, with the same data values as the previous example.
After the experiment is finished, it is discovered that the value
of Z was 0.05 too small because of a systematic error in the
measuring instrument.
The result was obtained from averaging large amounts of data,
and the task of recalculating a correction to each value is
daunting. But that's not necessary
Use this information to correct the result.
Look at the determinate error equation:
r G g H h z
= +
R G+H G G+H H Z
The -0.05 error in Z represents a relative error of -0.05/106
in Z. Assuming zero determinate error in G and H, we have:
r/R = -(z/Z) = -(-0.05/106)
So: r = (0.05/106)(0.338) = 0.0001594
Example 7: The density of a long copper rod is to be obtained.
Its length is measured with a meter stick, its diameter with micrometer
calipers, and its mass with an electronic balance.
L = 60.0 ± 0.1 cm (0.17%)
D = 0.632 ± 0.002 cm (0.32%) [The error in D2 is therefore 0.64%]
m = 16.2 ± 0.1 g (0.006%)
The cross sectional area is πr2 =
πD2/4.
So the density is = m/v = 4m/LπD2.
The relative error in the result (the density) should be no more than
(0.17% + 0.64% + 0.006% = 0.816%) or about 0.8%. This is written:
density = 8.606 ± 0.07 g/cm3
A reference book gives 8.87 g/cm3 as the density of copper.
The experimental discrepancy is 0.26, indicating that something is wrong.
The student who took this data may have blundered in a measurement.
Maybe the material wasn't pure copper, but a copper alloy.
If it is a measurement blunder, the diameter measurement is the most
likely suspect.
K. THE OBJECTIVES OF LABORATORY WORK
A good way to conclude this chapter is to consider what the
students' objectives in laboratory ought to be. The freshman
laboratory is not the same as a research lab, but we hope
that the student will become aware of some of the concerns,
methods, instruments, and goals of physics researchers.
Experiments in freshman lab fall into several categories. In each
case below, we indicate what the student's responsibility should
be.
1. To measure a fundamental physical quantity.
The student designs an experimental strategy to obtain the most
accurate result with the available equipment. The student must
understand the operation of the equipment and investigate the
inherent uncertainties in the experiment fully enough to state the
limits of error of the data and result(s) with confidence that the
"true" values (if they were known) would not lie outside of the
stated error limits.
2. To confirm or verify a well-known law or principle.
In this case it is not enough to say "The law was (or was not)
verified." The experimenter must state to what error limits the
verification holds, and for what limits on range of data,
experimental conditions, etc. It is too easy to over-generalize.
A student in freshman lab does not verify a law, say F = ma,
for all possible cases where that law might apply. The
student
probably investigated the law in the more limited case of the
gravitational force, near the earth's surface, acting on a small
mass falling over distances of one or two meters. The student
should state these limitations. One should not broadly claim to
have "verified Newton's law." Even worse would be to claim
to have "proved Newton's law."
3. To investigate a phenomena in order to formulate a law or
relation which best describes it.
Here it is not enough to find a law that "works," but to show that
the law you find is a better representation of the data than other
laws you might test. For example, you might have a graph of
experimental data which "looks like" some power of x. You find a
power which seems to fit. Another student says it "looks like" an
exponential function of x. The exponential curve is tried and seems
to fit. So which is the "right" or "best" relation? You may be able
to show that one of them is better at fitting the data. One
may be more physically meaningful, in the context of the larger
picture of established physics laws and theory. But it may be that
neither one is a clearly superior representation of the data. In
that case you should redesign the experiment in such a way that it
can conclusively decide between the two competing hypotheses.
The reader of your report will look very carefully at the "results
and conclusions" section, which represents your claims about the
outcome of the experiment. The reader will also look to see whether
you have justified your claims by specific reference to the data
you took in the experiment. Your claims must be supported by the
data, and should be reasonable (within the limitations of the
experiment). This is a test of your understanding of the
experiment, of your judgment in assessing the results, and your
ability to communicate.
L. CONCLUSION
Error analysis is not an "after-the-fact" activity; it pervades the
entire experimental process from experiment design through
data-taking to the final analysis of the results. Nor is it a
"cut-and-dried" procedure or set of recipes for "calculating
errors." While there are statistical mathematical criteria which
underlie the entire process, considerable insight and judgment
and common sense must be brought to bear on the experiment to
properly assess the dynamical interaction of the error sources. The
experimenter must understand the physics which bears on
the experiment to do a proper job of this. The experimenter must
exercise
judgment and common sense in choosing experimental strategies to
improve results, and in choosing methods for determine the effect
of experimental uncertainties. When error analysis is treated as
a "mindless" calculation process, the gravest blunders of analysis
and interpretation can occur.
APPENDIX I. MEASURES OF UNCERTAINTY
The size of the experimental uncertainty in a set of measurements
may be expressed in several ways, depending on how "conservative"
you want to be.
1. Limits of error.
An attempt to specify the entire range in which all
measurements will lie. In practice one specifies the range within
which the measured values lie.
2. Average deviation.
The average deviation of a set of measurements
from its mean is found by summing the deviations of the
n measurements, then dividing the sum by (n-1). This measure
describes the "spread" of the set of measurements.
When one wishes to make inferences about how far an estimated mean
is likely to deviate from the "true" mean value of the parent
distribution, use the average deviation of the mean. To
calculate it, sum the deviations of the n measurements, then divide
this sum by n(n-1)1/2. This measure expresses the
quality of your estimate of the mean. This is the measure we
call the uncertainty (or error) in the mean.
This last definition automatically includes two mathematical
corrections, one required to make inferences about the parent
distribution from a finite sample of data, and one to correct for
the fact that you have used only a small sample.
3. Standard deviation.
The standard deviation has become a "standard" method for
expressing uncertainties because it is supported by a
well-developed mathematical model. Unfortunately it is only
appropriate when the experimenter (a) has large data samples, and
(b) knows that the distribution of the data is really Gaussian, or
near-Gaussian. Therefore its use in the freshman lab is seldom
justified—something like using a sledgehammer to crack a walnut.
APPENDIX II. CALCULATIONS USING STANDARD DEVIATIONS
The rules for error propagation for the elementary algebraic
operations may be restated to apply when standard deviations are
used as the error measure for random (indeterminate) errors:
- When independently measured quantities are added or
subtracted,
the standard deviation of the result is the square root of the sum
of the squares of the standard deviations of the quantities.
- When independently measured quantities are multiplied or
divided, the relative (fractional or percent) standard deviation
of the result is the square root of the sum of the squares of the
relative standard deviations of the quantities.
These are cumbersome to write. The simple underlying idea is
this:
When using standard deviations, the rules for combining average
deviations are modified in this way: Instead of simply summing the
error measures, you square them, sum the squares and then take the
square root of the sum. This is called "summing in quadrature."
Are Standard Deviations Better? Too many elementary
laboratory
manuals stress the standard deviation as the one standard way to
express error measures. However, one can find, from standard
statistical theory that when very few measurements are made, the
error estimates themselves will have low precision. The uncertainty
of an error estimate made from n pieces of data is
So we'd have to average 51 independent values to obtain a 10% error
in the determination of the error. We would need 5000 measurements
to get an error estimate good to 1%. If only 10 measurements were
made, the uncertainty in the standard deviation is about 24%. This is why
we have continually stressed that error estimates of 1 or 2
significant figures are sufficient when data samples are small.
This is just one reason why the use of the standard deviation in
elementary laboratory is seldom justified. How often does one take
more than a few measurements of each quantity? Does one even take
enough measurements to determine the nature of the error
distribution? Is it Gaussian, or something else? One usually
doesn't know. If it isn't close to Gaussian, the whole apparatus
of the usual statistical error rules for standard deviation must
be modified. But the rules for maximum error, limits of error, and
average error are sufficiently conservative and robust that they
can still be reliably used even for small samples.
However, when three or more different quantities contribute to a
result, a more realistic measure of error is obtained by using the
`adding in quadrature' method described at the beginning of this
section.
Just as it's bad form to display more significant figures than are
justified, or to claim more significance for results than is
warranted by the experiment, so, too, it is bad form to use
statistical techniques and measures of error to express results
when the data does not justify those error measures nor the
mathematical rules used to obtain them. This implies more quality
significance to the results than may be the case, and borders on
scientific fraud.
APPENDIX III. IMPORTANCE OF INDEPENDENCE WHEN USING
ALGEBRAIC ERROR PROPAGATION EQUATIONS
The algebraic rules given for propagation of indeterminate errors
are one way to derive correct error equations, but must be used
with care. Here's an example which illustrates a pitfall you must
avoid.
A student wishes to calculate the error equation for the effective resistance, R,
of two resistors, X, and Y,
in parallel. The equation for parallel resistors is:
1 1 1
- = - + -
R X Y
The student solves this for R, obtaining:
XY
R =
X + Y
The error in the denominator is, by the sum rule, x+y. To proceed,
we must use the quotient rule, which requires relative
error
measures. So the student converts the error in the denominator to
relative form, (x+y)/(X+Y). The rest involves products and
quotients, so the relative determinate error in R is found to
be:
r x y x + y
= +
R X Y X + Y
The next step requires some algebra to cast this in standard
form, but let's not waste the effort, for this equation is already
wrong!
Why? Eq. 11 has X and Y in both numerator and
denominator. Therefore the numerator and denominator are not
independent. The quotient rule is not valid when the
numerator and denominator aren't independent.
To avoid this blunder, do whatever algebra is necessary to
rearrange the original equation so that application of the rules
will never require
combining errors for non-independent quantities. In fact, the form
of the equation 10 is an ideal starting point, for all
of its operations (+ and /) involve independent quantities.
To do this correctly, begin with Eq. 10 (in which each
quantity appears only once and there is no question that every
operation is independent). The relative error in 1/X is, by the
quotient rule, (0 - x/X) which is simply -x/X. The error in 1/X is
therefore (-x/X)(1/X) = -x/X2. Likewise the error in y
is -y/Y2 and in r is -r/R2. Finally, using
the addition rule for errors, the result is:
2 2
r x y r R x R r R x R y
= + , or = + , or r = +
2 2 2 R X X Y Y X X Y Y
R X Y
Or, using Eq. 11, the right side can be expressed in terms of
measured quantities only.
r Y x X y
= +
R X+Y X X+Y Y
EXERCISES
In the following situations, consider common sense
physical principles to determine which is the most
meaningful way to describe the error: as an absolute
error or a fractional error, an indeterminate error or
a determinate error, a precise measure or an accurate
one. Support your answers by stating your reasoning.
(1) A batch of plastic meter sticks is accurately manufactured, but
a year after leaving the factory the plastic shrank fairly
uniformly by an average amount of 2 mm.
(2) The knife edges of a mechanical balance (used for weighing
objects) have become blunted.
(3) The fast/slow setting screw in a precision mechanical stopwatch
is misadjusted.
(4) The supports of the cone bearing in a mechanical electrical
voltmeter have become loose so that the pointer bearing is very
loosely confined.
(5) The effect (small) of air drag on a measurement of the
acceleration due to gravity by a falling body experiment.
(6) (a) The effect of uncontrolled and unmeasured laboratory
temperature on a delicate mechanical instrument which makes
measurements daily over many months. (b) The effect of temperature
on the instrument if the experiment took 60 seconds to complete.
(7) The effect of air drag on the period of a pendulum.
(8) The effect of very impure alcohol used as the liquid in the
determination of density of a solid by Archimedes' principle. [The
solid is weighed when immersed in the liquid and the formula for
the result contains the density of the liquid.]
In the next group of exercises, assume the following data: A =
10, B = 2, C = 5, D = 20. In each case the formula for the
result, R, is given. Calculate the numeric value of R. Find the
determinate error equation in each case, and then use it to
answer the specific question asked.
(9) Equation: R = (C - B)/A. Use the determinate-error equation to
find what the value of R would be if B were actually 2.1 instead
of 2. Check your answer by direct calculation.
r c - b a
=
R C - B A
Hint: Without actually writing the whole determinate-error
equation, we can write the term of that equation which gives
the contribution due to error in B.
r -B b
= ,
R C - B B
due to error in B alone.
(10) Equation: R = (C/A) - C - 5. Use the error equation to find
R if C were changed to 4.7. Check answer by direct calculation.
(11) Equation: R = (D2C2)-3/(D -
A)2. Find how R changes if D changes to 22, A changes
to 12 and C changes to 5.3 (all at once).
(12) Equation: R = D sin [(A - C)/3B]. Find how R changes if C
increases by 2%. Remember that arguments of trig functions are
always in radians.
(13) Equation: R = exp[(C - B)/D] Find how R changes if B decreases
by 2% and D increases by 4 units. This is standard notation: exp(x)
means the same as ex. Here e is, of course, the base of
natural logarithms.
This last group of questions is more general and requires careful
thought and analysis of all possibilities. Be sure to consider
these in the most general context, considering all possible
measures of error: indeterminate, determinate, relative and
absolute. The statements might be true for one kind of error
measure and false for others. If so, specify this in your
answer.
(14) A student says, "When two measurements are mathematically
combined, the error in the result is always greater than the error
of either of the measurements." Discuss this statement
critically.
(15) Another student says, "When two measurements have 2% error,
and they are used in an equation to calculate a result, the result
will have 4% error." Discuss, critically.
(16) Still another student says, "When several measurements are
used to calculate a result, the error in the result can never be
less than the error of the worst measurement". Discuss,
critically.
(17) Yet another student says, "When several measurements are used
to calculate a result, and the error of one is 10 times as large
as the next worst one, you might as well neglect all but the worst
one in the error propagation equation." Discuss, critically.
ENDNOTES
1. Some of the better treatments of error analysis are:
- Young, Hugh D. Statistical Treatment of Experimental
Data. McGraw-Hill 1962.
- Baird, D. C. Experimentation, an introduction to
measurement theory and experiment design.. Second edition.
Prentice-Hall, 1988.
- Taylor, John R. An Introduction to Error Analysis.
University Science Books, 1962.
- Meiners, Harry F., Eppenstein and Moore. Laboratory
Physics. Wiley, 1969.
- Swartz, Clifford E. Used Math, for the first two years of
college science. Prentice-Hall, 1973. American Institute of
Physics, 1996. Chapter 1 discusses error analysis at the level
suitable for Freshman.
- Swartz, Clifford E. and Thomas Miner. Teaching Introductory
Physics, A Sourcebook. American Institute of Physics, 1977.
Chapter 2 of this valuable book gives an account of error analysis
which is entirely consistent with my own philosophy on the matter.
It discusses three levels of treatment of errors.
- Significant Figures—a first approximation to error analysis.
(But one not adequate for undergraduate laboratory work in physics.)
- Absolute and Percentage Errors—a second approximation to error
analysis. This is the level we have discussed at length above. Swartz and
Miner say "[These] rules are ... often satisfactory. Indeed, for most
introductory laboratory work, they are the only valid rules.
- Data Distribution Curves—a third approximation to error analysis.
This includes the use of standard deviations as a measure of error, and
the rules for combining them. I cannot resist quoting from this book:
The use of this third approximation to error analysis is justified only
when certain experimental conditions and demands are met. If the formalism
is applied blindly, as it often is, sophisticated precision may be claimed
when it does not exist at all. The situation is aggravated by the easy
availability of statistical programs on many hand calculators. Just enter
a few numbers, press the keys, and standard deviations and correlations
will come tumbling out to 10 insignificant figures.
2. Some books call these "random errors." This is a poor name, for
indeterminate errors in measurements are not entirely random
according to the mathematical definition of random. I've also seen
them called "chance errors." Some other synonyms for indeterminate
errors are: accidental, erratic, and statistical errors.
3. The magnitude of a quantity is its size, without regard to its
algebraic sign.
4. The average deviation might more properly be called the "average
absolute deviation," or "mean absolute deviation," since it is a
mean of the absolute values of the deviations, not of the
deviations themselves. [The mean of the deviations of a symmetric
distribution would be zero.]
5. In the statistical study of uncertainties, the words "average"
and "mean" are not used as if they were complete synonyms. When
referring to the average of a set of data measurements,
the word "mean" is always used, rather than "average." When referring to
other averaging processes the word "average" is preferred.
Perhaps this usage distinction is to avoid generating a clumsy name
like "mean deviation of the mean."
6. See Laboratory Physics by Meiners, Eppensein and Moore
for more details about the average deviation, and other measures
of dispersion.
7. This relatively new notation for mean values is, I think, neater
and easier to read than the old notation of putting a bar over the
Q.
8. For a good discussion see Laboratory Physics by
Meiners, Eppenstein and Moore. There (on p. 36) you will find a
side-by-side calculation of average deviation and standard
deviation, and a discussion of how they compare as measures of
error.
9. The Gaussian distribution, sometimes called the "normal curve
of error" has the equation:
2
-[(X - <X>)/2s]
f(X) = C e
where <X> is the mean value of the measurement X, and s is
the standard deviation of the measurements. C is a scaling
constant. f(X) is the number of measurements falling within a range
of values from X to X + x, where x is small. This is the famous
"bell-shaped curve" of statistics.
10. See Meiners et. al., who comment: "This means that for many
purposes, we can use the average deviation...instead of the
standard deviation. This is an advantage because the average
deviation is easier to compute than the standard deviation."
11. Independent errors are those for which the error of one
individual measurement is not dependent on the errors in other
measurements. No error influences the others, or is mathematically
determinable from the others.
12. Calculus may be used instead.
This document is © 1996, 2017 by Dr. Donald E. Simanek,
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