jmc

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 $14^{\mathrm{th}}$ 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 $14^{\mathrm{th}}$ 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 $\frac{1}{2}(78+86),$ or $\boxed{82}.$