A grouped result emits a category that has no rows in it — what do a count, a sum and a mean each return?

## Getting to an empty group in the first place Most grouped results never contain an empty group, because a group is formed from rows and no rows means no group. An empty group exists only when something outside the data supplies the key value: the key column carries a **declared set of allowed values** and the tool emits the members of that set that no row used. The question therefore only arises in that setting — but once you have asked for a complete shape, every reduction in the result has to produce something for a set with nothing in it, and that is where the surprises are. ## What each reduction has to do with nothing | Reduction | Result for a group with no rows | Why | |---|---|---| | The size of the group | zero, everywhere | there are no rows, and counting nothing is unambiguous | | The count of present values in a column | zero, everywhere | no values exist to be present | | A sum | zero in some designs, absent in others | a fold over an empty set can return its identity, or decline to answer | | A mean | absent, or an error | the denominator is zero and no honest number exists | | The largest or smallest value | absent, an error, or the neutral end of the range | there is no candidate value to return | | A distinct-value count | zero, everywhere | the set of distinct values is itself empty | The pattern underneath the table is worth naming, because it makes the whole row set memorable: - **Reductions that count** have a true answer for nothing. Zero rows is a fact, not a guess. - **Reductions that fold with an identity** (a sum being the obvious one) *could* return that identity, and designers disagree about whether they should. Returning zero says "the total is nothing"; returning an absent marker says "there was nothing to total". Both are coherent positions, so treat the behaviour as something to check rather than something to know. - **Reductions that must select or divide** (a mean, a largest value, a median) have no honest answer. There is no denominator and no candidate value, so the only truthful outputs are an absent marker or a refusal. A design that returns the neutral end of a numeric range for the largest value of nothing is returning a number that was never in the data, and that number will propagate into anything downstream that reads it. ## Why the zero in a report is the dangerous one A reader looking at a row that says `0` cannot tell, from that cell alone, which of two things happened: 1. Rows existed for that category and their values genuinely added to zero. 2. No rows existed for that category and the reduction folded to its identity. Those two mean completely different things to a business — one is "we did this and it netted out", the other is "we did not do this" — and no amount of staring at the total distinguishes them. The distinguishing column is **the size of the group**, which is zero in the second case and non-zero in the first. This is the concrete reason to carry the group's size beside the aggregates in any result whose empty groups are emitted: it is the only field that separates the two readings, and it costs almost nothing to produce alongside the reduction that was already running. ## A mean over an empty group is not zero This is the single most common wrong answer, and it comes from treating a mean as "a sum with a divide bolted on". Over an empty group there is a sum (possibly), but the divisor is zero, and no convention makes zero over zero into a number. Designs therefore return an absent marker, or they signal an error. If you see zero in that cell, either the tool returned the sum's identity and something downstream rounded the story, or a later step filled the absent marker in with zero — and that later fill is a decision somebody made, not a property of the mean. ## What a good answer sounds like Separate the three classes rather than reciting outcomes: counting reductions have a true answer, folding reductions have a defensible answer that designs disagree about, and selecting or dividing reductions have no answer at all. Then close on the consumer: state that you would carry the size of the group next to the aggregates, so that a zero in the report can be read correctly, and say that you would not fill an absent mean with zero without a stated reason, because that turns "we do not know" into a measurement.