Hi Christopher,
A frequency table simply lists the number of times a certain event occurs, usually by keeping a tally. I will try my best to represent it here:
a) Age Tally Frequency
12  6
13  7
14  10
15  12
16  5
The mode is simply the value that appears the most often in a data set. Since "15" occurs the greatest amount of times, 12 times to be exact, this is the modal age.
b) The arithmetic mean is the sum of every age divided by the number of students, in this case 40 [ie.
(12+12+12+12+12+12+13+13+13+13+...+16+16+16+16+16)/40]
Carrying out this calculation to completion gives you 563/40, which is equal to 14.075. Since the problem asks the mean to the nearest month, you would multiply 0.075*12(months in the year), which gives you 0.9. Rounding up to the nearest month results in an arithmetic mean of 14 years, 1 month.
As you can see, the mode and the mean represent two different statistical characteristics of a data set, although they are quite close to one another. The mean is more affected by outliers than the mode, so the amount of students aged 12 and 13 reduce the average age.
Finally, a third way to represent the data (not asked for here), is the median. To arrive at the median, you would write out every age from youngest to oldest, and find the value in the middle; if there are two values (in an evennumbered data set), you would average these two values to get the median.
Hope this helps!
Jason
11/13/2013

Jason S.