In Chapter 16, the text says that with repeated chance processes, the absolute error goes up as the number of draws increases, and the relative error goes down as the number of draws increases. The authors call this statement the "Law of Averages". The language is vague, but the authors presume that the statements are intuitive, and the reader is supposed to get the meaning through examples in the reading. Using the square root law from Chapter 17, we can clarify the meaning of the Law of Averages.
In a box model, with random draws (taken with replacement), we have
\begin{align*}
\text{ expected(sum of draws) }\amp = (\text{average of the box}) \cdot (\text{number of draws})\\
\text{ SE(sum of draws) } \amp = (\text{SD of the box}) \sqrt{\text{number of draws}}.
\end{align*}
The absolute error and the relative error are
\begin{align*}
\text{ absolute error }\amp = (\text{actual sum of draws)} -(\text{expected(sum of draws)})\\
\text{ relative error } \amp = \frac{ \text{absolute error}
}{\text{number of draws}}.
\end{align*}
The absolute error is estimated by
\begin{equation*}
\text{ absolute error }\approx
\text{ SE(sum of draws) }=(\text{SD of the box}) \sqrt{\text{number of draws}}.
\end{equation*}
Dividing both sides of the last equation by \(\text{(number of
draws)}\text{,}\) we have
\begin{equation*}
\text{ relative error }\approx \frac{(\text{SD of the
box})\sqrt{\text{number of draws}} }{\text{number of draws}} = \frac{(\text{SD of the
box}) }{\sqrt{\text{number of draws}}}\text{.}
\end{equation*}
Summarizing, we have
\begin{align*}
\text{ absolute error }\amp \approx (\text{SD of the box}) \sqrt{\text{number of draws}}\\
\text{ relative error }\amp \approx \frac{(\text{SD of the
box}) }{\sqrt{\text{number of draws}}}.
\end{align*}
From the last two equations, we can see clearly that the absolute error goes up and the relative error goes down as the number of draws increases.
