I have made the following graph:
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\usepgfplotslibrary{statistics}
\begin{document}
\begin{tikzpicture}
\begin{axis}
[title = {Lengths of Amplification and Deletion Regions},ylabel = {$log_{10}(Length)$},
boxplot/draw direction=y,
xtick={1,2,3},
xticklabels={Amplification, Deletion, Both},
x tick label style={font=\footnotesize, text width=2.5cm, align=center}
]
\addplot+[mark = *, mark options = {red},
boxplot prepared={
lower whisker=2.214844,
lower quartile=3.608312,
median=3.895478,
upper quartile=4.447298,
upper whisker=4.666284
}, color = red
] coordinates{(0,4.832228)(0,5.513942)(0,6.29165)(0,5.216712)(0,5.677036)(0,4.981995)(0,5.172095)(0,4.056417)};
\addplot+[mark = *,mark options = {blue},
boxplot prepared={
lower whisker=3.508799,
lower quartile=5.079821,
median=5.481519,
upper quartile=5.971588,
upper whisker=6.250831
}, color = blue
] coordinates{(0,6.508115)(0,6.486354)(0,6.860059)(0,6.620663)(0,7.312391)(0,7.357306)(0,6.421694)
(0,6.479597)(0,6.690945)(0,6.661593)(0,7.271025)(0,6.396931)(0,7.035161)(0,7.371248)(0,7.033689)
(0,7.002645)(0,6.590617)(0,7.171933)(0,6.416259)(0,5.552438)};
\addplot+[mark = *,mark options = {green},
boxplot prepared={
lower whisker=2.437751,
lower quartile=3.334956,
median=4.336029,
upper quartile=5.068459,
upper whisker=5.265037
}, color = green
] coordinates{(0,5.826492)(0,5.819791)(0,6.21436)(0,4.47166)};
\end{axis}
\end{tikzpicture}
\end{document}
which gives:

The circles near the median lines of each graph represents the average (log-transformed) for that respective group. I would like to make these dots into stars and to write the respective averages in parentheses, black colour: 4.06, 5.55, and 4.47.

averagekey, which you can use in the same way as you are using (for example) themediankey. Then you can use the normal controls to customize its appearance. By default it will appear as a diamond. – Thruston May 23 '15 at 22:10\log_10{\hbox{Length}}– Thruston May 23 '15 at 22:11