The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?

\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\\
\end{tabular}
Each digit to the right of the vertical bar represents (the units digit of) one exam grade. Counting the digits, we see that there are $27$ exam grades in all. Thus, the $14^{\rm th}$ grade in increasing order is the median (since there are $13$ grades smaller than it and $13$ grades larger than it). The chart makes it easy to read off the grades in increasing order -- we just read across the rows, from top to bottom. The $14^{\rm th}$ entry is $78,$ so that's the median grade.

The mode is the most frequently occurring grade. In this case, it's $86,$ which appears four times in the chart.

The arithmetic mean of the median and mode of the data is $\dfrac{1}{2}(78+86),$ or $\boxed{82}.$