The Product That Fixes Everything

A board of listings carries some number of filters. Each filter offers some number of choices. Multiply those numbers together and the product is the count of genuinely different listings the board is capable of distinguishing.

Two filters of four choices and six: twenty-four. Three filters of four, six and six: one hundred and forty-four. Six filters of six choices each: forty-six thousand six hundred and fifty-six.

For distributed analytics work, time tracking for data teams is most useful when the team agrees which records answer the planning question and which records should never be collected. For a separate technical reference, see The R Project.

This record calls that product the cells, because the filters partition the space of possible listings into a grid and the product is how many compartments the grid has.

Four schemas, four resolutions

• one bar is one facet; the divisions are its values
• cells: how many listings the schema can tell apart
• queries: how many different questions it lets anybody ask
• sizes: how many different result sizes are reachable

Four schemas drawn.

One facet of two values, then two facets, then three, then six of six. The bars show the facets; the figures are what each schema can tell apart, what it lets anybody ask, and how many different result sizes it can produce.

Computed from the schema alone. How many facets and how many values in each is a stated convention; the three figures are products of whole numbers.

What the product is a ceiling on

Distinction, and nothing else.

A cell is a combination of values — one choice from each filter. Every listing on the board falls into exactly one of them, determined by what its values are. Two listings whose values agree on every filter occupy the same cell, and from the board's point of view they are indistinguishable: no query returns one without the other, because no query can address a difference the filters do not carry.

That is a hard ceiling. It does not soften when more listings arrive, it does not improve when the board gets busier, and it cannot be raised by anything except changing the filters themselves.

The number of listings a board holds is a fact about this week. The number it can tell apart is a fact about the board.

Why multiplication and not addition

Because the filters apply at the same time rather than in sequence.

A reader setting one filter has narrowed to some portion of the board; setting a second narrows within that portion rather than alongside it. Each filter divides what the previous one left, which is what multiplication means.

The effect is easy to underestimate. Adding one more filter of four choices does not add four to the resolution; it multiplies it by four. A board with three filters of six that adds a fourth goes from two hundred and sixteen cells to one thousand two hundred and ninety-six.

Four rows, and the last is more than two hundred times the first. The page on the cost of an edit takes this apart properly.

The bounds this record works inside

A facet carries between two and six values, and a schema carries between one and six facets. Both numbers are a decision rather than a discovery, and the page on method says so before it says anything else.

Two is the floor for a facet because a filter with one choice is not a filter: setting it changes nothing, leaving it alone changes nothing, and it multiplies the resolution by one. The page on why a facet starts at two works through what such a thing does to a board that carries it.

Six is the ceiling at both ends for the same reason bounds usually get set: beyond it the figures stop being checkable by hand, and the structure the record is trying to show does not change.

Inside those bounds there are four hundred and sixty-one schemas. The smallest tells two listings apart. The largest tells forty-six thousand six hundred and fifty-six apart.

A worked reading, start to finish

A panel showing three controls: a category dropdown with six entries plus an "any", a region dropdown with four plus an "any", and a three-way choice of arrangement.

The "any" entries are not values. They are the option of leaving the filter alone, and the arithmetic accounts for that separately by adding one. So the facets are six, four and three.

Six times four times three is seventy-two. The board tells seventy-two kinds of listing apart, and it will go on telling seventy-two kinds apart until somebody changes a filter.

If that panel also carries a box to type words into, it is ignored: a text box has no fixed set of values, so it is not a facet, and the page on it in the section on limits says what it does instead.

What this means for two boards side by side

One board carries three dropdowns; another carries six toggles. The second looks like the more serious instrument.

Three dropdowns of four, six and six resolve one hundred and forty-four kinds of listing. Six toggles resolve sixty-four. The board that looks twice as thorough tells apart under half as much, and nothing on either page says so.

The arithmetic took under a minute for both and settled a comparison that would otherwise have gone on impressions. That is the whole case for doing it.

What the product does not tell you

It is not a count of listings. A board with one hundred and forty-four cells may carry three listings or three hundred thousand. The cells are compartments, and compartments can be empty, or hold one listing, or hold eight hundred.

It is not a measure of how useful the board is. A board that distinguishes four thousand and ninety-six listings along axes nobody cares about is worse than one distinguishing twenty-four along the right ones. This record cannot see which axes are the right ones, because the names of the facets are not part of a schema.

And it is not, on its own, a description of the board. This is the one most worth holding onto. Three hundred and eighty-one of the four hundred and sixty-one schemas share their cell count with at least one other schema, and the largest group of them holds seven quite differently built boards. The section continues with what that means in practice.

How to get the figure from a page in front of you

Count the filters. Count the choices in each. Multiply.

The arithmetic takes seconds and the counting takes a minute. What makes it worth doing is that the resulting figure does not expire: the board can post ten thousand new listings tomorrow and the ceiling is exactly where it was.

A figure that stays true is a rare thing to be able to write down about anything on the web, and this one costs a minute.

Near this one

• Two Listings In One Cell
  Two listings that agree on every filter become one listing to every question the board permits. The difference
• The Ceiling Is Set Before Anything Arrives
  The filters are chosen once; the listings arrive afterwards, for years. Everything countable about a board was
• What One Number Leaves Open
  Three hundred and eighty-one schemas of four hundred and sixty-one share their cell count with another. Four b
• Seven Boards, One Resolution
  Seven schemas all tell one hundred and forty-four listings apart, from three filters to six, with questions fr