How do you find the mean of a grouped frequency table?

The first thing to be aware of with grouped frequency is that we are not given the exact data, but instead we are told that we have some data and we have been able to divide it into set groups. Imagine we have a bookshelf on which the lowest shelf is for books with 0-100 pages, the second is 101-200 pages the third 201-300 pages and the top shelf has books with 301 pages or more. We don't know how many pages every one of the books has but we know how many books are on each shelf. Notice that the groupings do not overlap - this is important so it is very clear which shelf to put a book with 100 pages on, for example. To turn grouped data into something we can use, we need to assign one value to best represent the number of pages in the books on each shelf. We could say that we estimate all the books on the lowest shelf have 5 pages, but this seems unlikely because the group can include books up to 100 pages long. So, it makes sense to pick the mid-point (ie 50 pages) as this is the best guess we have. We always use the midpoint of each group so, when facing a grouped frequency table in a question, we should first add another column containing all the midpoints of the groups.
You will be familiar with the mean of a list of numbers. Normally, we add up all the numbers and divide by the number of numbers. It might be tempting to add up all the frequency values and divide by the number of rows in the table (ie the number of groups), but this isn't actually the right thing to do. Instead, for every row we need to multiply the midpoint by the frequency, to give us a new column, which we can call f x m.p. The sum of the f x m.p. column is just like the sum of all the numbers in a list when we usually calculate the mean. The sum of the frequency column is just like the number of numbers, so now we can divide the first sum by the second to give us the mean of a grouped frequency table.