The gap

Neither The Best Nor The Worst Case

Across the fifteen gaps that leave anything, the median shared time is four hours. The mean over all twenty-five would describe no arrangement at all.

Fifteen whole-hour gaps leave some shared time. Lined up by how much, the middle one leaves four hours. That is the median of this record, and it is the number worth quoting when somebody wants a single figure.

The mean of the same fifteen is a little over four and a quarter hours, which is close enough to agree and far enough to be worth separating.

When shared hours become part of an operating policy, teams comparing employee monitoring software should define the purpose, access rules and correction process before collecting any records. A useful external reference for the surrounding topic is IANA.

The fifteen that leave something

outlined: the day where the person is
hatched: the other end
solid: the hours they share
a reversed cell counts none

Nine gaps drawn.

Nine of the fifteen are drawn; the sequence runs one, two, three, four, five, six, seven, eight, seven, six, five, four, three, two, one, and its middle value is four.

Computed from the gap. The working day is a stated convention, not a measurement; everything else here is arithmetic.

Why the median and not the mean

The mean is dangerous here for a reason that has nothing to do with this particular set of numbers: it invites the inclusion of the ten dead gaps. Average the shared time across all twenty-five gaps and the answer drops to a little over two and a half hours, a figure that describes no arrangement anybody has and sits below the amount of overlap at which the arrangement is usually considered viable at all.

That average is arithmetically correct and practically meaningless, because it mixes two populations that behave differently: arrangements that work to some degree and arrangements that do not work at all. Mixing them produces a number that belongs to neither.

An average over a set containing a floor is an average over two different questions. The honest move is to answer them separately: how many leave nothing, and among those that leave something, how much.

The two-part answer

Ten of twenty-five leave nothing. That is the first figure and the damaging one.

Among the fifteen that leave something, the median is four hours, the range runs from one to eight, and nine of the fifteen leave four hours or more.

Those three sentences describe the record completely. Any single number that tries to do the same job either hides the floor or exaggerates it.

What four hours means in context

Four hours is exactly half the eight-hour day, and at a gap of four it sits entirely on one side of the middle: four shared afternoon hours and no shared morning, or the mirror image.

So the median case is not "half the day, spread out". It is "half the day, all of it in one block, at one end". Those are quite different things to plan around, and the total alone does not distinguish them.

The distribution behind the median

The fifteen workable gaps produce a tidy shape: the shared time runs one, two, three, four, five, six, seven, eight, seven, six, five, four, three, two, one as the gap sweeps from seven hours behind to seven ahead.

That is a symmetric triangle with a single peak. It has no tail, no cluster and no second mode, which is why a single summary figure is defensible at all. Many distributions do not permit one.

It also means every value from one to eight occurs, and most of them occur twice — once on each side of zero. Only the peak value of eight occurs once. Anybody picking a gap at random from the fifteen is about twice as likely to land on a value in the middle of the range as on either extreme, which is another way of saying the median and the typical case genuinely coincide here.

Shared hours	How many gaps produce it
8	1
7	2
4	2
1	2

What this record is not claiming about likelihood

The paragraph above said "picking a gap at random from the fifteen", and that phrase is doing careful work.

Nothing in this record establishes that the twenty-five whole-hour gaps occur equally often in the world. They almost certainly do not: arrangements are made for reasons, and the reasons favour some distances over others. Gaps of one or two hours presumably arise far more often than gaps of eleven.

The record computes all twenty-five because computing all of them costs nothing and because any weighting would require evidence it does not have. The median of four is therefore the median of this table, not the median of real arrangements, and it should never be quoted as the latter.

A complete enumeration is not a sample. It answers "what is possible" exactly and "what is common" not at all, and the two get confused whenever a summary statistic is attached to one.

When the median is the wrong tool too

If the question is whether a specific arrangement works, the median is irrelevant. The gap is known, the table gives the answer directly, and the distribution across all possible gaps has nothing to say about it.

The median earns its keep in exactly one situation: somebody is describing the general shape of the problem and needs one number that is neither the cheerful extreme nor the discouraging one. For that, four hours — stated alongside the ten dead gaps — is the honest summary.