These notes are designed to supplement the treatments in any freshman physics laboratory manual. The level of presentation does not use calculus, and is suitable for freshman. Because of the dearth of technical symbols available in the HTML language, and the inconsistent way browsers and readers display them, I have avoided use of the standard (delta)x notation for error in x. Instead, I use upper case letters for measured quantities, and lower case for their errors: A ± a. The notation <x> represents the mean (arithmetic average) value of X. I use s to represent standard deviation (usually it is the Greek lower case sigma). ASCII notation is used for equations to shorten the download time, compared to bitmapped graphics.A new section of examples (Section J) has been added, October 6, 1996. The section letter labels are now in alphabetical order.

Consistent with current practice, the term "error" is used here as a synonym for "experimental uncertainty."No measurement is perfectly accurate or exact. Many instrumental, physical and human limitations cause measurements to deviate from the "true" values of the quantities being measured. These deviations are called "experimental uncertainties," but more commonly the shorter word "error" is used.

What is the "true value" of a measured quantity? We can think of it as the value we'd measure if we somehow eliminated all error from instruments and procedure. This is a natural enough concept, and a useful one, even though at this point in the discussion it may sound like circular logic.

We can improve the measurement process, of course, but since we can
never eliminate measurement errors entirely, *we can never hope
to measure true values*. We have only introduced the concept
of *true value* for purposes of discussion.
When we specify the "error" in a quantity or
result, we are giving an estimate of how much that measurement is
likely to deviate from the true value of the quantity. This
estimate is far more than a guess, for it is founded on a physical
analysis of the measurement process and a mathematical analysis of
the equations which apply to the instruments and to the physical
process being studied.

A measurement or experimental result is of little use if nothing is known about the probable size of its error. We know nothing about the reliability of a result unless we can estimate the probable sizes of the errors and uncertainties in the data which were used to obtain that result.

That is why it is important for students to learn how to determine quantitative estimates of the nature and size of experimental errors and to predict how these errors affect the reliability of the final result. Entire books have been written on this subject.[1] The following discussion is designed to make the student aware of some common types of errors and some simple ways to quantify them and analyze how they affect results.

A warning:Some introductory laboratory manuals still use old-fashioned terminology, defining experimental error as a comparison of the experimental result with a standard or textbook value, treating the textbook value as if it were atrue value. This is misleading, and isnotconsistent with current practice in the scientific literature. This sort of comparison with standard values should be called anexperimental discrepancyto avoid confusion with measures of error (uncertainty). The only case I can think of where this measure is marginally appropriate as a measure of error is the case where the standard value isvery muchmore accurate than the experimental value.Consider the case of an experimenter who measures an important quantity which no one has ever measured before. Obviously no comparison can be made with a standard value. But this experimenter is still obligated to provide a reasonable estimate of the experimental error (uncertainty).

Consider the more usual case where the experimenter measures something to far greater accuracy than anyone previously achieved. The comparison with the previous (less accurate) results is certainly not a measure of the error.

And often you are measuring something completely unknown, like the density of an unknown metal alloy. You have no standard value with which to compare.

So, if you are using one of these lab manuals with the older, inadequate, definition of error, simply substitute "experimental discrepancy" wherever you see "experimental error" in the book. Then, don't forget, that you arealsoobligated to provide an experimental error estimate, and support it. If you determine both the error and the discrepancy, the experimental discrepancy should fall within the error limits of both your value and the standard value. If it doesn't, you have some explaining, and perhaps further investigation, to do.

*1. Indeterminate Errors.*[2]

Indeterminate errors are present in all experimental measurements. The name "indeterminate" indicates that there's no way to determine the size or sign of the error in any individual measurement. Indeterminate errors cause a measuring process to give different values when that measurement is repeated many times (assuming all other conditions are held constant to the best of the experimenter's ability). Indeterminate errors can have many causes, including operator errors or biases, fluctuating experimental conditions, varying environmental conditions and inherent variability of measuring instruments.

The effect that indeterminate errors have on results can be somewhat reduced by taking repeated measurements then calculating their average. The average is generally considered to be a "better" representation of the "true value" than any single measurement, because errors of positive and negative sign tend to compensate each other in the averaging process.

*2. Determinate (or Systematic) Errors.*

The terms *determinate error* and *systematic error*
are synonyms. "Systematic" means that when the measurement of a
quantity is repeated several times, the error has the *same size
and algebraic sign* for every measurement. "Determinate" means
that the size *and sign* of the errors are determinable (if
the determinate error is recognized and identified).

A common cause of determinate error is instrumental or procedural
*bias*. For example: a miscalibrated scale or instrument,
a color-blind observer matching colors.

Another cause is an outright experimental blunder.
**Examples:** using
an incorrect value of a constant in the equations, using the wrong
units, reading a scale incorrectly.

Every effort should be made to minimize the possibility of these errors, by careful calibration of the apparatus and by use of the best possible measurement techniques.

Determinate errors can be more serious than indeterminate errors
for three reasons. (1) There is no sure method for discovering and
identifying them just by looking at the experimental data. (2)
Their
effects can *not* be reduced by averaging repeated
measurements. (3) A determinate error has the same size and sign
for each measurement in a set of repeated measurements, so there
is no opportunity for positive and negative errors to offset each
other.

When a set of measurements is made of a physical quantity, it is
useful to express the difference between each measurement and the
average (mean) of the entire set. This is called the
*deviation* of the measurement from the mean. Use the word
*deviation* when an individual measurement of a set is being
compared with a quantity which is representative of the entire set.
Deviations can be expressed as absolute amounts, or as percents.

*2. Difference.*

There are situations where we need to compare measurements or results which are assumed to be about equally reliable, that is, to express the absolute or percent difference between the two. For example, you might want to compare two independent determinations of a quantity, or to compare an experimental result with one obtained independently by someone else, or by another procedure. To state the difference between two things implies no judgment about which is more reliable.

*3. Experimental discrepancy.*

When a measurement or result is compared with another which is
assumed or known to be more reliable, we call the difference
between the two the *experimental* discrepancy.
Discrepancies may be
expressed as absolute discrepancies or as percent discrepancies.
It is customary to calculate the percent by dividing the
discrepancy by the more reliable quantity (then, of course,
multiplying by 100). However, if the discrepancy is only a few
percent, it makes no practical difference which of the two
is in the denominator.

*1. Limits of error*

Error limits may be expressed in the form Q ± q where Q is the
measured
quantity and q is the magnitude of its limit of error.[3] This
expresses the experimenter's judgment that the "true" value of Q
lies between Q - q and Q + q. This entire interval within which the
measurement lies is called the *range of error*.
Manufacturer's
performance guarantees for laboratory instruments are often
expressed this way.

*2. Average deviation*[4]

This measure of error is calculated in this
manner: First calculate the mean (average) of a set of successive
measurements of a quantity, Q. Then find the magnitude of the
deviations of each measurement from the mean. Average these
magnitudes
of deviations to obtain a number called the *average
deviation* of
the data set. It is a measure of the dispersion (spread) of the
measurements with respect to the mean value of Q, that is, of how
far a typical measurement is likely to deviate from the mean.[5]
But this is not quite what is needed to express the quality of the
mean itself. We want an estimate of how far the mean
value of Q is likely to deviate from the "true" value of Q. The
appropriate statistical estimate of this is called the *average
deviation of the mean*. To find this rigorously would involve
us in
the theory of probability and statistics. We will state the result
without proof.[6]

For a set of n measurements Q_{i} whose mean value is
<Q>,[7] the average deviation of the mean (A.D.M.) is:

n | | SUM |Q - <Q>| i=1 | i | Ave. Dev. of the Mean = -------------- (1) 1/2 (n-1) (n)The vertical bars enclosing an expression mean "take the absolute value of" that expression. That means that if the expression is negative, make it positive.

If the A.D.M. is quoted as the error measure of a mean,
<Q>_{exp}, this is equivalent to saying that the
probability of <Q>_{exp} lying within one A.D.M. of
the "true" value of Q, Q_{true}, is 58%, and the odds
against it lying outside of one A.D.M. are 1.4 to 1.

As a rough rule of thumb, the probability of
<Q>_{exp}
being within three A.D.M. (on either side) of the true value is
nearly 100% (actually 98%). This is a useful relation for
converting (or comparing) A.D.M. to limits of error.[8]

*3. Standard Deviation.*

[This section is included for completeness, and may be skipped or skimmed unless your instructor specifically assigns it.]

The *standard deviation* is a well known, widely used, and
statistically well-founded measure of error. For a set of n
measurements Q_{i} whose mean value is <Q>, the
standard deviation of the mean is found from:

----------------- | | |2 | SUM |Q - <Q>| Std. Dev. of the Mean = | | i | | -------------- (2) \| n(n-1)The sum is from i=1 to n.

This form of the equation is not very convenient for calculations. By expanding the summand it may be recast into a form which lends itself to efficient computation with an electronic calculator:

--------------------- | 2 2 | SUM (Q ) - n(<Q>) Std. Dev. of the mean = | i (3) | ------------------ \| n(n-1)[Note that the n(<Q>)

The calculation of the standard deviation requires two summations,
one a sum of the data values (to obtain <Q>), and one a sum
of the
squares of the data values. Many electronic calculators allow these
two sums to be obtained with only one entry of each data value.
This is a good feature to have in a scientific calculator. When n
is large, the quantity n(n-1) becomes approximately n^{2},
further simplifying the work.

The use of the standard deviation is hardly justified unless the
experimenter has taken a large number of repeated measurements of
*each* experimentally determined quantity. This is seldom
the case in the freshman laboratory.

It can be shown that when the measurements are distributed according the "normal" ("Gaussian")[11] distribution, average deviations and standard deviations are related by a simple formula:[12]

[average deviation] = 0.80 [standard deviation] This is a useful "rule of thumb" when it is necessary to compare the two measures of error or convert from one to the other.

Uncertainties may be expressed as *absolute* measures,
giving the size of the a quantity's uncertainty in the same units
in the quantity itself.

*Example.* A piece of metal is weighed a number of times,
and the
average value obtained is: M = 34.6 gm. By analysis of the scatter
of the measurements, the uncertainty is determined to be m = 0.07
gm. This *absolute* uncertainty may be included with the
measurement in this manner: M = 34.6 ± 0.07 gm.

The value 0.07 after the ± sign in this example is the estimated absolute error in the value 3.86.

*2. Relative (or Fractional) Error.*

Uncertainties may be expressed as *relative* measures,
giving the ratio of the quantity's uncertainty to the quantity
itself. In general:

absolute error in a measurement Relative error = ------------------------------- (5) size of the measurement

m 0.07 gm - = ------- = 0.002, or, if you wish, 0.2% M 34.6 gmIt 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 cases where absolute errors are inappropriate and therefore the errors should be expressed in relative form. There are also cases where the reverse is true.

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 expanded, 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.

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 won't need to 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.

We illustrate how errors propagate by first discussing how to
determine the error limits (or maximum error) of results by
considering how data errors propagate through simple mathematical
operations. We first confine our attention to *determinate
errors*,
which have a known sign. In this way we will discover certain
useful rules for error propagation, then we'll then be able to
extend the rules to other error measures and also to indeterminate
errors.

We are 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.

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."

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 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 error in
the result will be the difference in the errors. Summarizing:

**Sum and difference rule. When two quantities are added (or subtracted), their determinate errors add (or subtract).**

(R + r) = (A + a)(B + b) = AB + aB + Ab + ab or: r = aB + Ab + abThis doesn't look promising for stating as a simple rule. However, when we express the errors in

r aB + bA a b - = ------- = - + - R AB A BThis gives us a very simple rule:

**Product rule. When two quantities are multiplied, their***relative*determinate errors add.

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 BThe 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. 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.**

**Power rule. 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.**

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. The absolute indeterminate errors add.****Product and quotient rule. The relative indeterminate errors add.**

**Power rule. 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.**

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 = (c ) x + (c ) y + (c ) z ... etc. x y ywhich may always be algebraically rearranged to:

r x y z - = {C } - + {C } - + {C } - ... etc. (5) R x X y Y z ZThe coefficients (c

If this error equation was derived from the
*determinate-error*
rules, the relative errors in the above equation 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
(5) must be modified--each term of the equation (both sides) must
be squared:

2 2 2 2 2 2 2 (r/R) = C (x/X) + C (y/Y) + C (z/z) (6) x y zThis rule is given here without proof.

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 shift to capital letters for quantities, lower case for errors. Solve the equation for the
result, a. A = 2S/T^{2}.
Its indeterminate-error equation is:

a t s - = 2 - + - A T SThe 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 T

**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:

R = 0.338 ± 0.014

**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 ZThis equation is in

**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

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 RThis 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 ZThe -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 (PI)r

density = 8.606 ± 0.07 g/cm^{3}

A reference book gives 8.87 g/cm^{3} 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.

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.

*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.

**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.**

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

100 ----------- percent. (7) 1/2 [2(n-1)]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.

A student wishes to calculate the error equation for two resistors in parallel. The equation for parallel resistors is:

1 1 1 - = - + - (8) R X YThe student solves this for R, obtaining:

XY R = ----- (9) X + YThe error in the denominator is, by the sum rule, x+y. To proceed, we must use the quotient rule, which requires

r x y x + y - = - + - - ----- (10) R X Y X + YThe next step requires some algebra to cast this in standard form, but let's not waste the effort, for this equation is already

Why? Eq. 9 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 given in Eq. 8 is an ideal starting point, for all of its operations (+ and /) involve independent quantities.

To do this correctly, begin with Eq. 8 (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/X^{2}. Likewise the error in y
is -y/Y^{2} and in r is -r/R^{2}. 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 (11)Or, using Eq. 9, the right side can be expressed in terms of measured quantities only.

r Y x X y - = --- - + --- - (12) R X+Y X X+Y Y

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

r -B b - = ----- - , R C - B Bdue 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 = (D^{2}C^{2})-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 e^{x}. 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.

- 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.

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 ewhere <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 copyright 1996 by Dr. Donald E. Simanek, Lock Haven University, Lock Haven, PA, 17745. Commercial uses prohibited without permission of author. The document may be freely used by instructors and distributed to students without charge, so long as this copyright notice is included.

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