Finding the busiest class, then placing the mode inside it

0:00Ten matches, and the number of wickets taken in each one. Two, six, four, five, nought, two, one, three, two, three. The mode is the value that turns up most often, and to find it you do not calculate anything. You count. Nought once. One once. Two three times. Three twice. Four, five and six once each. Ten results in all, and one of those counts is bigger than the others.

0:28So the mode is two. Notice what just happened. The answer was already sitting in the data, and the table only made it easy to see. Now here is the same idea applied to a grouped table. Twenty households, sorted by how many people live in each one. One to three, three to five, five to seven, seven to nine, nine to eleven. And the counts: seven, eight, two, two, one.

0:58Ask for the mode and you immediately hit a wall. Grouping kept the counts and threw away the values. Eight households are somewhere between three and five people, and which sizes they actually were is gone. There is no most-common value left to point at, because there are no values left at all. So the mode can no longer be read off. It has to be built. And the thing that gets built has to come out of the only information that survived — the counts.

1:32Building it is two jobs, and almost every mistake made with the mode comes from running them together. Job one: decide which class the mode is in. Job two: decide where inside that class it sits. Those are different questions and they use different information, which is the whole point of separating them. The first one looks at a single number. The second one looks at three. And the second one is the one people skip, because once you have found the busiest class it feels as though the work is over.

2:08It is not. You are half done, and the half that is left depends on two classes you have not looked at yet. Job one is quick. Find the class with the largest count. In the household table that is eight, so the modal class is three to five. Frequency alone decides it. Nothing else is consulted, and nothing else needs to be. There is one thing to be careful about. If two classes tie for the largest count, there is no modal class.

2:42Not the first of them, not the last of them. There is no answer, and a table like that needs a different question asked of it. But when one class is strictly the busiest, job one is finished, and it has cost you a single glance. Now — and this is the part worth slowing down for — where inside that class does the mode go? Start by ruling something out. The modal class's own count cannot answer that on its own.

3:14Take the class three to five with its eight households, and look at it alone, with the rest of the table covered up. Eight households, somewhere between three and five. Where in there? Nothing in front of you distinguishes one part of that class from another. Eight is one number and it has to place a point along a whole interval. So the answer has to come from outside the class, and there is only one place it can come from — the classes on either side.

3:47Uncover them and the picture changes completely. Below sits a class of seven, almost as crowded. Above sits a class of two, nearly empty. That is not a symmetric situation, and the mode should not be placed as if it were. So three counts do the placing: the modal class's own, and its two neighbours'. And here is the picture that makes it obvious. Draw the three classes as bars. Now draw one line from the top left corner of the tall middle bar across to the top left corner of the bar on its right.

4:26And another from the top right corner of the middle bar back to the top right corner of the bar on its left. Those two lines cross somewhere inside the modal class. That crossing point is the mode. Look at what moves it. Make the bar on the left taller and the crossing slides left. Make the bar on the right taller and it slides right. The mode leans toward whichever neighbour is more crowded. That is the whole behaviour, and it is visible before any arithmetic happens.

5:01And the crossing is a genuine construction — two straight lines and the point they meet at. Everything that follows is just that point written down algebraically. Written down, it comes out as a ratio, and the ratio is the most useful form of it. Take how far the modal count stands above the class below. Call that the first excess. And how far it stands above the class above. That is the second.

5:34Both are measured from the modal class's own count, downward to each neighbour in turn. The mode then divides the modal class in the ratio of those two excesses, first to second. So it is a point on a segment, placed the way you would place any point dividing a segment in a given ratio. Which means you can predict the answer before computing it. Bigger first excess, and the point sits further along. Bigger second excess, and it sits nearer the start.

6:08And notice the ratio depends on all three counts, not two. Change the modal class's own count and both excesses change, so the mode moves even though the neighbours did not. Run it on the households. The modal class is three to five, two units wide, holding eight. Below it, seven. Above it, two. First excess: eight minus seven is one. Second excess: eight minus two is six. One to six. So the mode divides that class one seventh of the way along.

6:46One seventh of two units is nought point two eight six, so the mode is three point two eight six. Look where that lands. The class runs from three to five, and the answer is barely a quarter of a unit past three. It is pressed hard against the lower boundary — the boundary it shares with the class of seven households. The crowd was below it, and the mode has sunk toward the crowd.

7:14Now the same method on the thirty exam scripts, and watch it do the opposite. The counts are two, three, seven, six, six, six across six classes fifteen marks wide. The largest is seven, so the modal class is forty to fifty-five. Below it, three. Above it, six. First excess: seven minus three is four. Second excess: seven minus six is one. Four to one. The mode divides the class four fifths of the way along.

7:48Four fifths of fifteen marks is twelve, and forty plus twelve is fifty-two. Exactly fifty-two. So this time the mode is pressed against the UPPER boundary, hard up against the busy class of six above it. Set the two examples side by side. Same formula, opposite ends of the modal class — one a seventh of the way in, the other four fifths. And the reason is right there in the counts: in one the crowd was below, in the other the crowd was above.

8:24Which raises the obvious question. What if the neighbours are equally busy? Then the two excesses are equal, the ratio is one to one, and the mode divides the class exactly in half. Half way along a class is its class mark. So in that case, and only in that case, the mode is the class mark. Take a class of twelve with five on each side. The mode sits dead centre.

8:53And that is worth stating the other way round, because it kills a common shortcut. The mode is NOT the modal class's class mark. It happens to be, when the neighbourhood is balanced, and it is not otherwise. The exam scripts make the point: the class mark of forty to fifty-five is forty-seven point five, and the mode is fifty-two. Four and a half marks apart. The households make it too: the class mark is four and the mode is three point two eight six.

9:28So the class mark is a guess that is right when the data happens to be symmetric, and the formula is what tells you when it is not. Three things the formula quietly assumes, and it is worth knowing all three. First, the classes must all be one width. The excesses place a point along a class, and if the classes are different sizes the comparison between neighbours is no longer a fair one.

9:58Second, the classes must be continuous — each one starting where the last one ended. A table whose classes run fifty to fifty-two, then fifty-three to fifty-five, has gaps in it. Those have to be closed before any of this runs. Third, the data should have a single peak. The formula will cheerfully return a number for a table with several bumps in it, but the number will not mean what you think.

10:28That third condition fails more often than you would guess. Of nine tables checked here, three have more than one local peak. One of them, a table of traffic counts, has three separate humps, and its mode of forty-four point seven one describes only the tallest of them. None of these conditions is exotic. But a formula that returns a number for any input at all is a formula you have to check the input of yourself.

11:02Here is a property of the mode that the mean does not share, and it follows directly from what we have built. The mode looks at exactly three numbers. Every other count in the table is invisible to it. Take a table of thirty-five regions, sorted by how many students there are per teacher. Two of its classes are completely empty, and two regions sit far out at the top end.

11:30The mode of that table is thirty point six two five. Now delete those two far-out regions and recompute. The mean moves. It falls from twenty-nine point two one to twenty-seven point eight — a shift of one point four one. The mode does not move at all. Not a little. Not at all. It is thirty point six two five before and thirty point six two five after. Because the three counts it uses — nine, ten and three — are exactly the same in both tables. The regions that were deleted were never part of the calculation.

12:11That is not a defect. It is what makes the mode worth having when a few extreme observations would drag a mean somewhere unrepresentative. Which brings us to the two answers side by side. On the thirty exam scripts, the mode is fifty-two and the mean is sixty-two. They are not competing. They answer different questions. Fifty-two says: this is where the scripts piled up. Sixty-two says: this is what a script scored on average.

12:45Both are true of the same thirty papers, and neither is a correction of the other. And do not learn the order. It is tempting to remember that the mode came out smaller, and it is not a rule. On a table of eighty patients by age, the mode is thirty-six point eight two and the mean is thirty-five point three seven five. The mode is the larger one there.

13:12On a table of two hundred households by monthly spending, they are more than eight hundred apart, because a long tail of high spenders pulls the mean up and leaves the mode where the crowd is. So the gap between them is itself information — about how lopsided the data is. Two jobs to find the mode. Find the class, then place the point. And the point leans toward the crowd.