
\subsection{\bf Central Limit Estimate of Error}

\begin{slide}
\begin{itemize}
\item We can use this example to illustrate an interesting theoretical
point about the error, and now compare two methods of finding an
error.

\item (a) According to the central limit theorem, the variable
$y=\frac{1}{n} {\displaystyle \sum^n_{i=1}} t_i$ is a sum over
independent random variables and has mean $\langle y\rangle = \tau_0$
and standard deviation $\sigma$.
\begin{eqnarray}
\sigma^2 & = & \frac{1}{n} \left\lbrace \left\langle t^2_i
\right\rangle - \left\langle t_i \right\rangle^2 \right\rbrace
\nonumber \\
& = & \frac{1}{n} \left\lbrace \frac{1}{\tau_0} {\displaystyle \int} t^2
e^{-\frac{t}{\tau_0}}\, dt - \tau^2_0 \right\rbrace \hspace{1em}
\mbox{or} \nonumber \\
\sigma & = & \frac{\tau_0}{\sqrt{n}}
\end{eqnarray}

\item The mean and standard deviation of $y$ refer to distribution of
values of $y$ in {\bf many\/} experiments---{\bf each\/} of $n$
observations.
\end{itemize}
\end{slide}

