Frequency Tables and Tallies

When you collect data you end up with a long, messy list, and counting it straight off the page is slow and easy to get wrong. Tally marks fix that. You draw one upright stroke for each answer, and when you reach the fifth you draw it across the previous four to close off a bundle. Now you count in fives instead of ones, so seven is one bundle plus two extra strokes. The other half of the method matters just as much: work through your list once, from the start to the end. Read an answer, put a stroke on the right row, move on. Going back through the list a second time is exactly how values get missed or counted twice.

Here is the method on real data: the number of brothers and sisters reported by fifteen students. First write every different answer down the left in order, from nought up to four. Then work through the list once, putting a tally stroke beside the value you read. Finally write the count next to each row, and that count is called the frequency. Three students have no siblings, six have one, four have two, one has three and one has four. Now do the check that saves you every time: the frequencies must add to the number of answers you collected. Three plus six plus four plus one plus one is fifteen, which matches, so the table is right.

Once the table exists it answers questions in seconds. How many students altogether? Add the frequency column and you get fifteen. What is the most common number of siblings? Look for the largest frequency, which is six, sitting beside the value one. How many are an only child? Read straight across from nought, giving three. How many have two or more? Add the rows for two, three and four, so four plus one plus one is six. The last one works differently. To find how many siblings there are altogether you multiply each value by its frequency and then add, giving nought plus six plus eight plus three plus four, which is twenty-one siblings among fifteen students.

Frequency tables are not only for numbers. Ask twenty students their favourite fruit and you can tally the words in exactly the same way: seven chose apple, six banana, four orange and three grapes. The rows are categories rather than values, but everything else is identical, including the check — seven plus six plus four plus three is twenty. There is one real difference, though. With categorical data you cannot build a value times frequency column, because apple times seven is not a number. So with categories you can count how many are in each group and name the biggest group, and that is all the arithmetic there is. Remember that from the last lesson: categories can be counted but never averaged.

Before you hand a frequency table in, run three checks. First, add the frequency column and compare it with how many answers you actually collected — those two numbers must be equal. Second, recount the tally strokes on each row against the frequency you wrote, which is quick because the bundles of five do most of the work. Third, make sure every value has a row, including any value nobody chose, which gets a frequency of nought. Try it on this one: fifteen students were surveyed, the frequencies read three, six, four, one and one, but the total row says sixteen. Adding them gives fifteen, so the total is simply wrong and needs fixing.