Mean and median from a frequency table, by hand and in a spreadsheet

0:00A class is asked a simple question. How big is the average family in this room? Everybody says a number, and instead of writing thirty-six numbers on the board, somebody tallies them. Three students said three. Eleven said four. Nine said five, seven said six, three said seven, and one each said eight, nine and ten. Eight rows. Thirty-six students. That table is not a summary of the data. It IS the data, with the repetitions folded up - and every rule for computing with one comes straight out of unfolding it again.

0:42Here is the answer somebody always gives first. Add up the left-hand column - three, four, five, six, seven, eight, nine, ten - and you get fifty-two. Divide by eight rows. Six point five. It is wrong, and it is worth more than being told so. That computation is not meaningless. It is the average of the eight family sizes that were REPORTED - one vote each, whether one student said it or eleven did.

1:15A real quantity. Just not the one anybody asked for. So unfold the table and look at what the data actually is. The row that says three, three does not mean one three. It means three threes. The row that says four, eleven means eleven fours. Write them all out and there are thirty-six numbers on the board, most of them fours and fives. Now the objection dies on its own. A row with a frequency of one and a row with a frequency of eleven are not two equal things. One of them is eleven pieces of data.

1:58You do not have to write them out, though. You only have to count each value as often as it occurs. Three threes are nine. Eleven fours are forty-four. Nine fives are forty-five. Seven sixes are forty-two. Then twenty-one, eight, nine and ten from the last four rows. Add those eight products and the whole class's families come to a hundred and eighty-eight people. That is the same total you would have got by adding thirty-six numbers. It is just the folded way of getting it.

2:35Now what goes underneath. The total is a hundred and eighty-eight. Divided by what? Not by eight. Eight is how many rows the table has, and a row is not a student. A hundred and eighty-eight over eight is twenty-three and a half, which is not the average of anything in this room. Divide by thirty-six - the total frequency, which is how many students there actually are. That is the single most useful sentence in this topic. The divisor is the total frequency, and never the number of rows.

3:14A hundred and eighty-eight over thirty-six. That does not come out exactly. It is five point two two two two, going on for ever. Reported to two places it is five point two two - and two different things are worth saying out loud about that number. The first: no family has five point two two people in it. It is the balance point of thirty-six family sizes, not a family.

3:44The second: five point two two is rounded. The exact answer is not a decimal that stops. A class should always know which of those two it is looking at. Now the middle. Thirty-six values, so the median sits between the eighteenth and the nineteenth of them, in order. You could write out all thirty-six and count along. Do not. Run the frequencies up from the smallest value instead. Three. Then three plus eleven is fourteen. Then plus nine is twenty-three. Then thirty, thirty-three, thirty-four, thirty-five, thirty-six.

4:25And here is what those running totals ARE. Each one is the last position that value occupies in the sorted list. Read them that way and the table answers a question about any position at all. The fourteenth value is a four, because the fours run out at fourteen. The twenty-third is a five, because the fives run out at twenty-three. So every position from the fifteenth to the twenty-third holds a five. That is the whole run.

5:00Eighteen and nineteen are both inside it. Both are fives, so the median is five. Not five point two two, and not the middle row of the table either - the middle row sits between six and seven. Rows are values. Positions are data. Thirty-six numbers can be done by hand. A hundred and thirty-two cannot, comfortably - and that is where a sheet earns its place. Put a table of marks into one: a column of names, then a column for each subject.

5:37Every box has an address made of two parts. A letter names the column, a number names the row, and it takes both to name a box. So E five is the fifth row's mark in column E - that student's maths mark, forty-two. And B seven is not a column, whatever it looks like. B is a column. Seven is a row. B seven is one box, and it holds twenty-seven.

6:07The second idea is the one that does the work. You can name a whole block of boxes instead of listing numbers. Write the first box, a colon, and the last box. B three colon G three is one student's six marks, all the way across. Then ask for the total of that block, and the sheet answers two hundred and fifty. Ask for the average of B seven colon D seven - three of another student's marks - and it answers thirty exactly.

6:40Notice what the sheet did not do. It did not know any statistics. It added up a block, and it added up a block and divided by three. The same two operations you have been doing by hand, with a shorter way of saying which numbers. Two tables to finish on, and a warning in the middle of them. Forty-two students recorded how many bicycle rides they took in a week, and the counts are drawn as a stack of dots over each number.

7:14Same object as a frequency table. Forty-two students, a hundred and ninety-three rides, so the mean is four point six oh - rounded - and the median, read off the running totals, is four. Now the warning. A week has seven days. Anybody who rode more than seven times must have ridden twice on some day, and five students are above seven. But that is AT LEAST five, not exactly five. The plot records weekly totals, so a student with four rides could have taken two of them on one day and nothing on the board rules it out.

7:54The data forces five and permits more. Knowing the difference between those two is the point. The last table is the one that makes the rule unmissable. Sixty-two students threw darts at a target, and the table records how many throws each needed. One student needed one throw. Nobody needed two, and nobody needed three. Ten rows. Two of them empty. So the divisor is not ten. Four hundred and seventy-three throws over sixty-two students is seven point six three, rounded.

8:34And when you run the frequencies up, the empty rows do not move the running total at all. It stands still at one for three rows together. Which is exactly right, because a value nobody reported occupies no positions. The thirty-first and thirty-second are both eight, so the median is eight. The smallest anybody needed was one, and the largest was ten. A frequency table is a list with the repetitions folded up. Weight to average it, accumulate to find the middle, and never once divide by the number of rows.