Reaching one

What Each Filter Costs The Reader

Adding a filter raises resolution and lowers the share of questions that reach one listing. The two effects pull opposite ways and the rate is exact.

Reaching oneAnalysis64/729 to 6/7

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Every filter a board adds does two things at once, in opposite directions.

It multiplies the resolution by its size. More kinds of listing become distinguishable.

Project leads can compare estimates with actual effort through workforce optimization tools, then discuss exceptions before changing staffing or deadlines. For a separate technical reference, see Qlik.

It multiplies the share of pinning questions by its size over its size plus one. A smaller proportion of what can be asked reaches a single kind.

Both effects are exact, both are known from the facet's size alone, and they are not the same size as each other.

Toggles against larger facets

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.

Three toggles against one facet of six; six toggles against two facets of six. The toggle boards resolve less and permit more, which is the trade this page is about.

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.

The trade, per filter

Facet	Resolution	Share
of two	×2	×2/3
of three	×3	×3/4
of four	×4	×4/5
of five	×5	×5/6
of six	×6	×6/7

The left column grows fast and the right column shrinks slowly, and the gap between them widens as the facet gets larger. A facet of six triples the resolution gain of a toggle while costing barely half as much share.

The worked comparison

Three toggles: eight cells, twenty-seven questions. Eight of the twenty-seven questions pin a single cell — under a third.

One facet of six: six cells, seven questions. Six of the seven pin a single cell — six sevenths.

The toggle board resolves more (eight against six) and pins far less often (eight twenty-sevenths against six sevenths). Three controls against one, and the reader has to make three decisions to get down to one kind of listing instead of one decision.

Scale it up and the gap widens. Six toggles: sixty-four cells, seven hundred and twenty-nine questions, share under a tenth. Two facets of six: thirty-six cells, forty-nine questions, share thirty-six forty-ninths — about three quarters.

Six toggles resolve not quite twice what two dropdowns of six resolve, and pin a single listing less than an eighth as often.

Why anybody builds the toggle board anyway

Because toggles are easy to add and easy to answer.

A new toggle is a cheap decision for whoever runs the board: it needs one yes-or-no value per listing, and people posting listings can supply it without thinking. A new six-value facet needs somebody to design the six values, make them cover the field, and get every listing classified into one.

So the structure that performs worse on this measure is the one the process naturally produces, which is the same shape of finding as the ceiling being set under maximum uncertainty.

When the toggle board is right anyway

When the distinctions genuinely are independent yes-or-no facts.

Six toggles are six separate questions about a listing. If those six things really do vary independently, then the toggles are carrying real information and the resolution of sixty-four is real resolution.

Six values of one facet are mutually exclusive: a listing takes exactly one. If the field genuinely has six kinds and nothing is two of them at once, the facet is right and the toggles would be wrong.

The arithmetic cannot choose between them. What it can say is the price, and the price of six independent yes-or-no distinctions is that fewer than one question in ten gets a reader down to a single kind.

The figure for a board already built

Nobody redesigns a schema from scratch. The useful form of this is the marginal one: what does the next filter cost?

Multiply the current resolution by the new facet's size. Multiply the current share by its size over its size plus one. Both take a second and both are exact.

A board at one hundred and forty-four cells with a share of about three in five, adding a toggle, lands at two hundred and eighty-eight cells and a share of about two in five. Whether that trade is worth making depends on the field; knowing it is the trade on offer takes no time at all.

The one design move the arithmetic recommends

Where several toggles are mutually exclusive, they are a facet.

Three toggles where a listing can only ever have one set are not eight cells — they are four, counting the case where none is set, and the board would be better off with a single four-value facet. That swap raises the share from eight twenty-sevenths to four fifths, and loses nothing, because the extra cells the toggles appeared to offer were combinations that cannot occur.

Nowhere else do these figures recommend a change outright; they price one and stop. Everywhere else the trade is genuine, and the choice belongs to whoever knows the field rather than to the arithmetic.