Many impressions

What the Median of This Table Means

The median arrangements figure is two hundred and ten — a fact about how this table was built, not about sets of pages in the world.

at most 151200 arrangements

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Across all three hundred and thirty-eight shapes, the median arrangements figure is two hundred and ten. The mean is a little over nine thousand four hundred and sixty-two.

Those two numbers describe the same column and disagree by a factor of forty-five. Before either is quoted, it is worth being clear about what sort of fact they are, because it is not the sort most medians are.

Smallest, median, largest

dashed: a statement a reader can check from outside
solid: published, but nothing to check it against
short and pale: never reached by this reader

Three shapes drawn; the left-to-right order is the drawing's own and carries no meaning.

The first row is the smallest shape in the record, the last is the largest, and the middle one carries the median arrangements figure. The span between the outer rows is a factor of seventy-five thousand six hundred.

Computed from the three numbers alone. How many statements exist, how many are met and how many can be checked is a stated convention; the counts are arrangements and combinations of whole numbers.

This median is a property of the table

The record covers two to ten published statements, one to six met, and every count of checkable statements from none to all. Those ranges were chosen. Nothing sampled them, nothing surveyed them, and no claim is made that real sets of pages are distributed this way. If a team already has strong project data, it can still see the service directly and compare the setup without committing to a broader deployment.

So the median says: of the shapes this record happens to tabulate, half have an arrangements figure at or below two hundred and ten. Change the ranges and the median moves. Stop at eight published instead of ten and it drops; allow eight met instead of six and it rises steeply. Another useful point of reference for the wider question is U.S. Bureau of Labor Statistics.

A median over a constructed table is a summary of the construction. It is a legitimate thing to report and an illegitimate thing to generalise from, and the line between those is where most misuse of a median happens.

The weighting nobody would choose

There is a further wrinkle, and it is the kind that only shows up when the table is counted rather than eyeballed.

The shapes are not spread evenly across the pages. A page with two statements published carries three shapes, because the checkable count can be none, one or two. A page with ten published carries eleven. So the larger pages contribute nearly four times as many shapes to the median as the smallest ones, purely because there are more ways for their third number to come out.

Nobody decided that larger pages should count four times. It falls out of tabulating every shape once, which is the only neutral-looking choice available and is not neutral.

Every table has a weighting. A table that does not declare one has the weighting its construction gave it, and nobody chose that.

Why the mean is worse here

The mean is larger than the arrangements figure of two hundred and eighty-seven of the three hundred and thirty-eight shapes in the record. Fifty-one shapes exceed ten thousand, and they carry it on their own.

That is the standard behaviour of an average over a quantity that grows multiplicatively. Arrangements is a product of descending integers; the top of the range is enormous compared with the middle, and an average sits where almost nothing is.

The record reports the median and the range together — two to one hundred and fifty-one thousand two hundred — and prints the mean nowhere else, because the mean of this column answers no question anybody has.

What the drawing is for

Three rows, chosen as the smallest shape, a shape at the median, and the largest.

Two published with one met: two arrangements. Seven published with three met: two hundred and ten. Ten published with six met: one hundred and fifty-one thousand two hundred.

The outer rows differ by a factor of seventy-five thousand six hundred. The drawing cannot show that to scale, and does not try: the strokes count statements, and the figures are printed as figures. A bar chart of this column would be a single tall bar and three hundred and thirty-seven invisible ones, which is why the record draws statements rather than magnitudes.

Two hundred and ten, again

The median arrangements figure is two hundred and ten. The arrangements figure for seven published with three met is two hundred and ten — which is why that shape could be used in the drawing. The contents figure for ten published with six met is also two hundred and ten.

Three different quantities. The first is a summary of the table, the second is a count of orderings on one page, and the third is a count of subsets on a different page. Only the first of them would change if the record's ranges changed.

The range is the better summary

If one figure has to stand for this column, the range does it with less distortion than either average: two to one hundred and fifty-one thousand two hundred.

A range cannot be mistaken for a typical value. It states the two edges and leaves the shape of what lies between them to the table, which is where the information actually is. Quoted alongside the published and met bounds that produced it, it says everything a single summary can honestly say about this column.

The record prints the range in its totals beside the median for exactly that reason, and the two together are the only form in which either figure appears. A median without its range invites the reading that most of the table is near it, and in this column almost none of it is.

What to do with the figure

Report it as what it is. "Half the shapes in this table are at or below two hundred and ten arrangements" is true and checkable. "A typical set of pages admits about two hundred arrangements" is not supported by anything here, and nothing here could support it: the record contains no information about which shapes occur in the world or how often.

That limitation is permanent rather than provisional. It is not a gap waiting for data; it is a consequence of the record being arithmetic rather than observation.