<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section{Errors and Uncertainties} \label{error_intro}

  Reliability estimates of measurements greatly enhance their value.  Thus,
saying that the average diameter of a cylinder is $10.00\pm 0.02\ mm$ tells much
more than the statement that the cylinder is a centimeter in diameter.

  To physicists the term ``{\bf error}" is interchangeable with ``{\bf
uncertainty}" and does {\bf not} have the same meaning as ``{\bf mistake}".
Mistakes, such as ``errors" in calculations, should be corrected before
estimating the experimental error.  In estimating the reliability of a single
quantity (such as the diameter of a cylinder) we recognize several different
kinds and sources of error:

  FIRST, are {\bf actual variations} of the quantity being measured, e.g.  the
diameter of a cylinder may actually be different in different places.  You must
then specify where the measurement was made; or if one wants the diameter in
order to calculate the volume, first find the average diameter by means of a
number of measurements at carefully selected places.  Then the scatter of the
measurements will give a first estimate of the reliability of the average
diameter.

  SECOND, the micrometer {\em caliper used may itself}
be in error.  The errors thus introduced will of course not lie
equally on both sides of the true value so that averaging
a large number of readings is no help.  To eliminate (or at least
reduce) such errors, we calibrate the measuring instrument:  in the case of
the micrometer caliper by taking the zero error (the reading when the
jaws are closed) and the readings on selected precision gauges of 
dimensions approximately equal to those of the cylinder to be measured.
We call such errors {\bf systematic}, and these cause errors on accuracy.

  THIRD, Another type of {\bf systematic} error can occur in the 
measurement of a cylinder:  The micrometer will always measure the 
largest diameter between its jaws; hence if there are small bumps
or depressions on the cylinder, the average of a large number of
measurements will not give the true average diameter but a quantity
somewhat larger.  (This error can of course be reduced by making the
jaws of the caliper smaller in cross section.) 

  FINALLY, if one measures something of definite size with a 
calibrated instrument, errors of measurement still exist which
(one hopes) are as often {\bf positive} as {\bf negative} and hence will 
{\bf average out} in a large number of trials.  Such errors are called random,
and result in less precision.  For example, the
reading of the micrometer caliper may vary because one can't close it
with the same force every time.  Also the observer's estimate
of the fraction of the smallest division varies from trial to trial.
Hence the average of a number of these measurements should
be closer to the true value than any one measurement.  Also the 
{\bf deviations} of the individual measurements from the {\bf average} 
give an indication of the {\bf reliability} of that average value.  The
typical value of this deviation is a measure of the precision.  This average
deviation has to be calculated from the absolute values of the deviations,
since otherwise the fact that there are both positive and negative deviations
means that they will cancel.  If one finds the average of the absolute
values of the deviations, this {\bf ``average deviation from the mean"}
may serve as a measure of reliability.  For example, let column 1
represent 10 readings of the diameter of a cylinder taken at one place
so that variations in the cylinder do not come into consideration, then
column 2 gives the magnitude (absolute) of each reading's deviation from
the mean.

\begin{tabular}{ccl}
      Measurements   &amp;       Deviation~from~Ave. \\
   ~~~~~~9.943 mm   &amp;               0.000       \\
         9.942      &amp;               0.001       \\
         9.944      &amp;               0.001       \\
         9.941      &amp;               0.002       \\
         9.943      &amp;               0.000       \\
         9.943      &amp;               0.000       \\
         9.945      &amp;               0.002     &amp; Diameter =   \\
         9.943      &amp;               0.000      \\
         9.941      &amp;               0.002     &amp; 9.943$\pm$0.001 mm \\  
 $\underline{~9.942~}$      &amp;      $\underline{~0.001~}$ \\
   Ave = 9.943 mm~~~~ &amp; ~~~~~~~~~~~~~~~
   Ave = 0.0009 mm$\approx $0.001 mm \\
\end{tabular}

Expressed algebraically, the average deviation from the mean 
is $ = (\sum |x_i-\bar{x}|)/n )$, where $x_i$ is the i$^{{th}}$ 
measurement of $n$ taken,   
and $\bar x$ is the mean or arithmetic average of the readings.

Standard Deviation:

\label{normaldist} 
  The average deviation shown above is a measure of the spread in a set of
measurements. A more easily calculated version of this is the
standard deviation $\sigma$ (or {\bf root mean square deviation}).  You calculate
$\sigma$ by evaluating  
$$\sigma=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}$$
where $\overline{x}$ is the  mean or arithmetical average of
the set of n measurements and $x_{i}$ is the $i^{th}$ measurement.

  Because of the square, the standard deviation $\sigma$ weights large deviations more heavily
than the average deviation and thus gives a less optimistic estimate of
the reliability.  In fact, for subtle reasons involving degrees of freedom,
$\sigma$ is really
$$ \sigma  = ~\sqrt{\frac{1}{(n-1)}\sum_{i=1}^n (x_i-\bar{x})^2}$$
$\sigma$ tells you the typical deviation from the mean you will find for an
individual measurement.  {\em The mean $\bar{x}$ itself should be more
reliable.}
That is, if you did several sets of $n$ measurements, the typical 
means from different sets will be closer to each other than the
individual measurements within a set.  In other words, the uncertainty in the
mean should be less than $\sigma$.  It turns out to reduce like $1/\sqrt{n}$,
and is called {\bf the error in the mean $\bf \sigma_\mu$}:
$$\sigma_\mu = \rm{error}\,\rm{in}\,\rm{mean} = \frac{\sigma}{\sqrt{n}} =
\frac{1}{\sqrt{n}}\,\sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (x_i-\bar{x})^2}$$
For an explanation of the $(n-1)$
factor and a clear discussion of erros, see P.R. Bevington and D.K
Robinson, {\em Data Reduction and Error Analysis for the Physical
Sciences}, McGraw Hill 1992, p. 11.

 If the error distribution is 
``normal" (i.e.~the errors, $\epsilon$ have a Gaussian distribution, 
$e^{-\epsilon^{2}}$, 
about zero), then on average 68\% of a large number of measurements will lie
closer than $\sigma$ to the true value.  While few measurement sets have 
precisely a ``normal" distribution, the main differences tend to be in the 
tails of the distributions.  If the set of trial measurements are generally 
bell shaped in the central regions, the ``normal" approximation generally 
suffices.

{\bf Relative error and percentage error:} \\
  Let $\epsilon$ be the error in a measurement whose value is $a$.  
Then $(\frac{\epsilon}{a})$ is 
the relative error of the measurement, and $100\ (\frac{\epsilon}{a})\%$ is the 
percentage error.  These terms are useful in laboratory work.

\newpage

\begin{center}
SYSTEMATIC ERRORS IN THE LABORATORY STANDARDS OF \\ LENGTH, TIME AND MASS
\end{center}
\label{error_top}
  For the experiments in this manual these systematic errors are 
usually negligible compared to other uncertainties.  An exception sometimes
occurs for the larger masses especially the 100 gram, the 500 gram,
and 1 kg masses.  Some contain drilled holes into which lead shot and a
plug have been added to adjust the mass to within tolerance (typically
$1.000\pm 0.003$ kg).  Occasionally a plug works loose and the calibration
lead shot is lost.  You can check the assigned mass values by weighing them
on the triple beam balances.  Report any deviations greater than 0.4\%
to the instructor.

\begin{center}
UNCERTAINTY ESTIMATE FOR A RESULT INVOLVING MEASUREMENTS
OF \\ SEVERAL INDEPENDENT QUANTITIES
\end{center}

\label{error_addition}
  A.)  If the desired result is the {\em sum} or {\em difference} of two 
  measurements, the {\em ABSOLUTE uncertainties ADD}:  \\ 
 Let $\Delta x$ and $\Delta y$ be the errors in x and y
respectively.  For the {\em sum} we have 
$z=x+\Delta x + y + \Delta y = x + y + \Delta x + \Delta y$
and the relative error is $\frac{\Delta x + \Delta y}{x + y}$.  
Since the signs of $\Delta x$ and $\Delta y$
can be opposite, adding the absolute values gives a pessimistic estimate
of the uncertainty.  If errors have a normal or 
Gaussian distribution and are independent, they combine in quadrature, i.e.
the square root of the sum of the squares, i.e.,
$$\Delta z=\sqrt{\Delta x^{2}+\Delta y^{2}}$$

  For the {\em difference} of two measurements we obtain a relative error of
 $\frac{\Delta x + \Delta y}{x - y}$.  
which becomes very large if x is nearly equal to y.
Hence avoid, if possible, designing an experiment where one 
measures two large quantities and takes their difference to obtain the
desired quantity.

  B.)  If the desired result involves {\bf multiplying (or dividing)} measured
quantities, then the {\bf RELATIVE uncertainty} of the result is the 
{\em SUM of the RELATIVE errors} in each of the measured quantities.

Proof:  
$$ \mbox{Let}~~~ z =\frac{x_1~x_2~x_3 .....}{y_1~y_2~y_3 .....}~~~
\mbox{and hence}  \hspace*{2.1in}$$
$$ \ln{z} =\ln{x_1} +\ln{x_2} +\ln{x_3} + .....
          -\ln{y_1} -\ln{y_2} -\ln{y_3} - .....  $$
Then find the differential, $d(\ln{z)}$:
$$d(\ln{z}) =\frac{dz}{z} =\frac{dx_1}{x_1}+\frac{dx_2}{x_2}+\frac{dx_3}{x_3}
    + ..... -\frac{dy_1}{y_1}+\frac{dy_2}{y_2}+\frac{dy_3}{y_3} 
    - ..... $$
Consider finite differentials, $\Delta z$, etc. and note that the most 
pessimistic
case corresponds to adding the absolute value of each term since
$\Delta x_{i}$ and $\Delta y_{i}$ can be of either sign.  Thus
$$\frac{\Delta z}{z}=\sum_{i}(\frac{\Delta x}{x})+\sum_{i}(\frac{\Delta y}{y})$$
 Again, if the measurement errors are independent and have a 
Gaussian distribution, the relative errors will add in
quadrature:
$$\frac{\Delta z}{z}=
  \sqrt{\sum_{i}(\frac{\Delta x}{x})^{2}+\sum_{i}(\frac{\Delta y}{y})^{2}}$$

  C.)  {\bf Corollary:}  If the desired result is a {\em POWER} of the measured 
quantity, the {\em RELATIVE ERROR} in the result is the relative error in the
measured quantity {\em MULTIPLIED by the POWER:}  Thus $z = x^{n}$ and
$$\frac{\Delta z}{z} = n \frac{\Delta x}{x}. $$

  The above results also follow in more general form:  Let 
$R = f(x,y,z)$ be the functional relationship between three measurements
and the desired result.  If one differentiates $R$, then
$$ dR = \frac{\partial{f}}{\partial{x}} dx + 
        \frac{\partial{f}}{\partial{y}} dy + 
        \frac{\partial{f}}{\partial{z}} dz $$
gives the uncertainty in $R$ when the uncertainties $dx$, $dy$ and $dz$ are 
known. 

  For example, consider the density of a solid (Exp.\ M1).  The relation is
$$ \rho~= ~\frac{m}{\pi r^2 L} $$
where $m$ = mass, $r$ = radius, $L$ = length, are the three measured 
quantities and $\rho$ = density.  Hence
$$
\frac{\partial{\rho}}{\partial{m}}=\frac{1}{\pi r^2L} ~~~~~~~~~~~
\frac{\partial{\rho}}{\partial{r}}=\frac{-2m}{\pi r^3L} ~~~~~~~~~~~
\frac{\partial{\rho}}{\partial{L}}=\frac{-m}{\pi r^2L^2} $$
and so
$$
d\rho=\frac{1}{\pi r^2L} ~dm +~\frac{-2m}{\pi r^3L} ~dr 
+~ \frac{-m}{\pi r^2L^2}~ dL. $$
To get the relative error divide by $\rho=m/\pi r^2 L$.  The result, if one
drops the negative signs, is
$$ \frac{d\rho}{\rho} = \frac{dm}{m} + 2 \frac{dr}{r} + \frac{dL}{L} $$
and represents a worst possible combination of errors.  For small
increments:
$$ \frac{\Delta\rho}{\rho} = \frac{\Delta m}{m} + 2 \frac{\Delta r}{r} 
+ \frac{\Delta L}{L} $$
and
$$ \Delta\rho = \rho \left[ \frac{\Delta m}{m} + 2 \frac{\Delta r}{r} 
+ \frac{\Delta L}{L} \right] $$
Again if the errors have normal distribution, then
$$ \frac{\Delta\rho}{\rho} = \sqrt{\left(\frac{\Delta m}{m}\right)^2 
+ \left(2\frac{\Delta r}{r}\right)^2 + \left(\frac{\Delta L}{L}\right)^2} $$

\begin{center} {\bf SIGNIFICANT FIGURES}  \end{center}

 Suppose you have measured the diameter of a circular disc and wish to 
compute its area $A=\pi d^{2}/{4}=\pi r^{2}$. Let the average value of 
the diameter be $24.326\pm 0.003$ mm ; dividing $d$
by 2 to get $r$ we obtain $12.163\pm 0.0015$ mm with a relative error
$\frac{\Delta r}{r}$ of $\frac{0.0015}{12}=0.00012$.
 Squaring $r$ (using a calculator) we have $r^2 = 147.938569$, with a relative
error $2\Delta r/r = 0.00024$, or an absolute error in $r^2$ of
$0.00024\times147.93\cdots = 0.036 \approx 0.04$.  Thus we can write $r^2 =
147.94 \pm 0.04$, any additional places in $r^2$ being unreliable.
Hence for this example the first five figures 
are called {\em significant}.

  Now in computing the area $A = \pi r^{2}$ how many digits of $\pi$ must be
used?  A pocket calculator with $\pi=3.141592654$ gives
$$A = \pi r^{2}= \pi\times(147.94 \pm 0.04) = 464.77 \pm 0.11 \mbox{ mm}^{2}$$
Note that $\frac{\Delta A}{A}=2\frac{\Delta r}{r}=0.00024.$
Note also that the
same answer results from $\pi=3.1416$, but that $\pi=3.142$ gives 
$A=464.83\pm 0.11$~mm$^{2}$
which differs from the correct value by $0.06$~mm$^{2}$, an amount comparable
to the estimated uncertainty.

  A good rule is to use one more digit in
constants than is available in your measurements, and to save
one more digit in computations than the number 
of significant figures in the data.
When you use a calculator you usually get many more digits than you need.
Therefore
{\bf at the end}, be sure to {\bf round off} the final answer to display the
correct number of {\bf significant figures.} \\
 
{\bf SAMPLE QUESTIONS}
\begin{enumerate}
\item How many significant figures are there in the following number?
\begin{enumerate}
\item 976.45
\item 4.000
\item 10
\end{enumerate}
\item Round off each of the following numbers to three significant figures.
\begin{enumerate}
\item 4.455
\item 4.6675
\item 2.045
\end{enumerate}


\item A function has the relationship $Z(A,B)=A+B^3$ where A and B are
found to have uncertainties of $\pm \Delta A$ and   $\pm \Delta B$ respectively.
Find $\Delta Z$ in term of $A$, $B$ and the respective uncertainties assuming
the errors are uncorrelated.

\item What happens to $\sigma$, the standard deviation, as you make more and
more measurements? what happens to $\overline{\sigma}$, the standard deviation 
of the mean? \begin{enumerate}
\item They both remain same 
\item They both decrease 
\item $\sigma$ increases and $\overline{\sigma}$ decreases 
\item $\sigma$ approachs a constant and $\overline{\sigma}$ decreases 
\end{enumerate}


        
\end{enumerate}


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%So, what do think : &lt;BR&gt; &lt;INPUT TYPE="text" SIZE=4,4 NAME="other_wp"&gt;

\newpage
\noindent 
Suggestions on Form for Lab Notebooks:\\ \\

\begin{center}
NUMBER AND TITLE (e.g. E1.  ELECTROSTATICS) \end{center}

\latex{Date performed:\underline{\hspace*{1in}}\hfill
              Partner:\underline{\hspace*{1.5in}} \\ }
\html{Date performed:____________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
              Partner:_________________________\\ }
\\ 
Subdivisions:  
If appropriate, name and number each section as in the manual.\\ \\
DATA:  \begin{enumerate}
\item[] \latex{\vspace*{-.08in}}Label numbers and give units.  
In a few words, state what quantities you measured.  If appropriate, record the data in 
tabular form.  Label the tables and give units.
\end{enumerate}
CALCULATIONS: 
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}State the equations used and present a sample 
calculation.  (Inclusion of the arithmetic is not necessary.)
\end{enumerate}
  CONCLUSIONS:
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}If any important conclusions follow from the
experiments, state them and show by a brief statement 
how they follow.  Compare your results with accepted values if the
experiment has involved the measurements of a physical constant.
\end{enumerate}

  Errors:
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}
Some of your experiments will be qualitative while
others will involve quantitative measurements of physical constants.
Where it is appropriate, estimate the uncertainty of
each measurement used in a calculation and compute the uncertainty of
the result.  Does your estimate of uncertainty indicate satisfactory
agreement between your result and the accepted one (or between your several 
values if you have several)?  Intelligent discussion is welcomed, but don't
make this section a burden on you.
\end{enumerate}
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