Tally marks and frequency: organising data so it can be read

0:00A teacher wants to buy sweets for the class. Not a random assortment — the right number of each kind. So she asks everybody which sweet they would like. Five kinds are on offer. And the answers come back one at a time. Doughnut. Ice cream. Ice cream. Cookie. Brownie. Chocolate cake. Ice cream. Now she has to record those as they arrive, because thirty-eight children are not going to wait while she thinks.

0:31Which turns out to be a genuinely different problem from counting things that sit still and let you. And it has a genuinely different solution. Suppose you try to keep running totals in numerals. Doughnut, one. Ice cream, one. Ice cream again — so cross out the one, write two. Cookie, one. Brownie, one. Chocolate cake, one. Ice cream — cross out the two, write three. You can already feel where this is going.

1:03Every single answer means finding the right row, reading a number, crossing it out, and writing a new one. Four operations per child, on a page that is getting messier, while somebody is talking to you. And if you lose your place, you cannot go back. That child has already answered and walked off. So here is the notation, and it is beautifully simple. One stroke for one answer. That is the whole of it.

1:35Doughnut? Draw a stroke in the doughnut row. Ice cream? Draw a stroke in the ice cream row. No reading, no crossing out, no arithmetic at all. One mark, and you are ready for the next child. But there is one more piece, and it is the clever bit. Every fifth stroke is not drawn upright. It is drawn across the other four. So the marks come in bundles of five, tied off as you go along.

2:05And now ask why that matters, because it is certainly not decoration. Here is a row with one bundle and one loose stroke. You do not count that. You see five, and you see one, and it is six. Here is a bundle and four loose ones. Five and four. Nine. Now compare that with nine upright strokes in a row. Try to read nine strokes at a glance, and you will count them — and you will sometimes get eight.

2:35The bundling means the record can be read without being recounted. Which is the entire reason for that crossing stroke. It buys you accuracy later, for free, at the moment of writing. Right. Your book prints the finished tallies, fills in two of the numbers, and leaves three of them for you. Doughnuts: one bundle and one loose. Six — that one is printed. Chocolate cake: one bundle and four loose. Nine — also printed.

3:08Now the three blanks. Ice cream: two bundles and three loose. Ten and three. Thirteen. Brownies: three loose strokes and no bundle at all. Three. Cookies: one bundle and two loose. Seven. And now the table can be read in a single look. Ice cream leads, with thirteen. Before we go on, do the check that almost nobody remembers to do. Add the five numbers up. Six, plus nine, plus thirteen, plus three, plus seven.

3:43Thirty-eight. Now — is thirty-eight the size of the class? If it is, then every child has been recorded exactly once, and you can trust the table. If it came to thirty-seven, somebody was missed. If it came to thirty-nine, somebody got two strokes. Your book does not print this sum. It is worth doing every single time, because it is the only check the table can give you. Now, the number in that last column has a name, and your book puts it in a box of its own.

4:20Frequency. The frequency of a value is simply how many times it turned up. The frequency of chocolate cake is nine. The frequency of brownies is three. And it is worth noticing what this word is for. It answers how many of these — not how much. Nine chocolate cakes has a frequency of nine. Nine kilograms of chocolate cake does not have a frequency at all. Counting things, rather than measuring stuff. That distinction runs through the whole of data handling.

4:57And now the question your book asks next, which is sharper than it looks. Is this table enough to hand each sweet to the child who asked for it? And no. Obviously not, the moment you say it out loud. The table says thirteen children want ice cream. It does not say which thirteen. That information was in the original list of answers — and the moment you turned it into tallies, you threw it away.

5:26Which was the right thing to do. The teacher's job was to place an order, and for that job the table is perfect. But it is worth being precise about it. Organising data is not tidying up. Every way of organising keeps some questions answerable and closes others off. Let me show you that with a second data set, organised a different way. Twenty-seven shoe sizes, written on a board in the order they were measured.

5:59Four, five, three, four, three, four, five, five, four, and so on, for twenty-seven numbers. Now, these are not categories. They are measurements, and measurements have an order — a five really is bigger than a four. So there is something available here that was not available with the sweets. You can sort them. Put them in ascending order, smallest first, and look what falls out straight away. Three, three, three, four, four, four, and on up to a single seven right at the end.

6:36The smallest is three. The largest is seven. You read both off the ends without counting anything at all. How many wear size five? They are all together now, so you count one block. Ten. And how many wear larger than four? Everything to the right of the fours. Ten fives, four sixes, one seven. Fifteen. That last question would have been horrible on the unsorted list. You would have had to scan all twenty-seven and keep a running count.

7:09On the sorted list it is one cut and one addition. So compare the two things we have just done. With the sweets we made a frequency table. Five rows, five numbers. We kept the counts, and we lost who chose what. With the shoe sizes we sorted. We still have all twenty-seven measurements — not one of them was thrown away. What we lost is the order they arrived in.

7:38Two different organisings. Two different things given up. And that is what to take from this section. There is no neutral way to organise data. Every arrangement makes some questions easy and some questions impossible — and you want to know which, before you choose. So go and build one. Your book gives you three, and they get better as they go. One. Count the trees on your walk to school. Kinds of tree in the first column, tallies in the next.

8:12Two. That letter count in a piece of newspaper. Five letters — and you will need the bundles, because e will run into the hundreds. Three, and this is the best of them. Roll a die thirty times and tally the six faces. Then read off which face came up most often, which least, and which ones tied. And notice that you cannot predict that table. Nobody can. Which makes it the first genuinely honest data collection in the chapter. Nobody knows the answer before you start — including your teacher.

8:50Next time, we stop writing the counts down, and start drawing them.