%\newpage

\subsection{MC-1a~~Measurement and Error}

\noindent
OBJECTIVES: 
\begin{enumerate}
\item[] \latex{\vspace*{-.10in}}The major objectives of this first, short (est.~1 hr) computer lab  are to begin to develop
an basic understanding of what it means to make an experimental measurement and provide a methodology for assessing random and systematic errors in this
measurement process. In addition this lab will also give you a minimal framework in which to introduce you to the 
PASCO{\small \copyright} (page~\pageref{PASCO}) interface hardware and software.
\end{enumerate}\latex{\vspace*{-.05in}}

\noindent
THEORY:  
\begin{enumerate}
\item[] \latex{\vspace*{-.10in}}By now you should have had numerous opportunities to become familiar with time and the concept of a time interval. The increment of one second will be used  as an intuitive reference point.  In this lab you will test your ability to internalize this one second time interval by making and recording a repetitive flicking motion with your finger.  By flicking your finger back and forth you will move it though an infrared beam sensor (i.e.~the PASCO photogate) and each full cycle (back and forth, approximately 2 seconds) 
will be simultaneously recorded, plotted and tabulated by the PASCO interface software.

Your goal in this experiment is to assess the size of systematic and random
errors in your data set and learn a simple methodology for 
distinguishing between the two.\\
SYSTEMATIC ERRORS:  These are errors which affect the {\em accuracy} of a
measurement.  Typically they are reproducible so that they always affect the
data in the same way.  For instance if a clock runs slowly you will make
a time measurement which is less than the actual reading.\\
RANDOM ERRORS:  These are errors which affect the {\em precision} of a
measurement.  A process itself may have a random component (as in radioactive
decay) or the measurement technique may introduce noise that causes the
readings to fluctuate.  If many measurements are made, a statistical
analysis will reduce the uncertainty from random errors by averaging. \\
\end{enumerate}


\noindent APPARATUS:
\begin{enumerate} 
\item[$\Rightarrow$] \latex{\vspace*{-0.09in}}Computer  with monitor, keyboard and
mouse.
\item[$\Rightarrow$]\latex{\vspace*{-0.09in}}A PASCO photogate and stand:
This device emits a narrow infrared beam in the gap and occluding the beam 
prevents it from reaching a photodetector.  When the beam is interrupted the red LED
should become lit. (Plugged into DIGITAL CHANNEL \#1.)
\item[$\Rightarrow$] \latex{\vspace*{-0.09in}}A PASCO Signal Interface (CI-700 or CI-750) monitors the photodetector output {\it vs} time and can be configured to tabulate, plot and analyze this data.
\end{enumerate}

\noindent{PROCEDURE:}
\begin{enumerate} 
\item[] \latex{\vspace*{-0.09in}}To configure the experiment you should refer to Fig.~1 below.  Adjust the photogate so that one member can easily and repetitively flick his/her finger through the gap.  The phone-jack cable from the photogate should be plugged into the  DIGITAL CHANNEL \#1 socket.  Ignore the other sensors which may already be plugged into other sockets.  It is {\em important} that the PASCO interface be turned on before the computer.  If not the computer will not recognize it and, therefore, it must be rebooted to properly communicate with the PASCO module.
\end{enumerate}

\begin{figure}[h] \centering 
\includegraphics[height=1.9in]{figs/m01-1rev.eps}
\caption {A schematic of the MC1a components and layout.}
\end{figure}

\begin{enumerate} 
\item[]\latex{\vspace*{-0.15in}}To initiate the PASCO (page~\pageref{PASCO}) interface software you will need to click the computer mouse on the telescope icon in the ``toolkit'' area below (web version).  Fig.~2 below gives a good idea of how the display should appear.  Note that, while you are able to reconfigure the display parameters, the default values that are specified on start-up will allow you to do most of this experiment without necessitating any major changes.

You will note that a ``dummy'' first data set already exists on start-up showing a typical data run.  In the table you can view all 47 data points and the statistical analysis, including mean and standard deviation. In addition there should be a plot of this data and a histogram.   
\latex{\vspace*{-0.15in}}\end{enumerate}

\input{toolbar/m00c_tb1} 

\begin{figure}[!h] \centering 
\includegraphics[height=3.85in]{figs/mc0-02revA.eps}
\caption {The PASCO Data Studio display window}
\end{figure}

\noindent{SUGGESTED PROCEDURE:} 

\begin{enumerate} 

\item \vspace*{-.05in} Start the preliminaries by CLICKing on the \includegraphics[height=0.18in]{figs/start_button.eps} icon and practice ``flicking'' a finger back and forth so that a two second interval appears in the window. CLICK on the {\bf Stop} icon when done.   The same person need not perform both operations.  This will produce a second data set.  (There is also a ``monitor'' function which can permit adjustments and trials without storing the results in memory.  To access this type ALT-M). 

\item \vspace*{-.05in} Each run gets its own data set in the ``Data'' display window.  (If there are any data sets in existence  you will {\em not} be able to reconfigure the interface  parameters or sensor inputs.) A data set can be deleted by moving the mouse cursor to the ``Run \#~1" position, CLICKing the left mouse button and then striking the ``Delete'' key.

\item \vspace*{-.05in} Once you are comfortable with the procedure then click on the \includegraphics[height=0.18in]{figs/start_button.eps} icon and cycle a finger back and forth over fifty times.  The click on the ``Stop" icon. DO NOT watch the time display while you do this, since you want to find out how accurately and precisely you can reproduce a time interval of 2 s using only your mind. 

\item What is the mean time per cycle?  What is the standard deviation?
The mean $ \overline{t}$ and standard deviation $ \sigma$ are given by:
$$ \overline{t}= \sum_{i=1}^N t_i/N \mbox{~~~and~~~~} 
 \sigma = \sqrt{{\sum_{i=1}^N  (t_i-\overline{t})^2}/[N-1]}~~. $$
Questions to consider:\\
\indent ~I. Is your mean suggestive of a systematic error? \\
\indent II. Does your data qualitatively give the appearance of a normal
distribution (i.e.~a Gaussian bell curve.)

\item \vspace*{-.05in} For analyzing and quantifying random errors, you need
to asses how a data set is distributed about the mean.  The standard deviation $\sigma$ is one common calculation that does this.  In the case of a normal distribution approximately 68\% of the data points fall within $\pm 1 \sigma$ of the mean (90\% within $\pm 2 \sigma$).   Is your data consistent with this attribute?   

\item  \vspace*{-.05in} Assessing the possibility of systematic behavior is somewhat more subtle. In general $\sigma$ is a measure of how much a {\em single} measurement fluctuates from the mean.  In this run you have made fifty presumed identical measurements.  A better estimate of how well you have really determined the mean is to calculate the {\em standard deviation of the mean}  $\overline{\sigma}= \sigma/\sqrt{N}$.  After recording $\overline{\sigma}$ in you lab book, can you now observe any evidence that there is a systemic error in your data?  Answer this same question with respect to the first ``sample'' data set. 

\item \vspace*{-.05in} OPTIONAL:  Systematic errors can sometimes drift over time.  In the best-case scenario they drift up and down so that they hopefully average out to zero.  (Clearly it would be better if they could be eliminated entirely.) With respect to $\overline{t}$ and $\overline{\sigma}$ for the first 25 and second 25 cycles do you observe any systematic trends?  Use the ``Zoom Select'' feature on the Graph.  Simply by draging the mouse will highlight a subset of the data.  The mean, $\sigma$ and other statistical attributes will appear at the bottom of the table.

\end{enumerate}




