How do you compute distance for a record mixing salary, tenure, department and a remote flag?

## Why the usual metrics do not apply Euclidean, Manhattan and cosine distance all assume every feature is a number whose differences mean something. An HR record breaks that assumption: salary in euros, tenure in years, a department code with a dozen unordered levels, and a remote-working flag. There is no meaningful subtraction between `Finance` and `Logistics`, and no shared unit joining euros to years. ## The naive route and where it hurts The common move is to encode everything numerically — one indicator column per department level, 0/1 for the flag, standardised salary and tenure — and use Euclidean distance. This works, and it is often the right pragmatic choice, but three things happen quietly. First, a department mismatch has a fixed cost: two records in different departments differ by 1 in two indicator columns, contributing 2 to the squared distance, whether the departments are adjacent teams or unrelated functions. Second, **standardising indicator columns is dangerous**: a rare level's column has a tiny standard deviation, so dividing by it can make a mismatch on that rare level swamp every other feature in the record. Third, the implicit weighting between the categorical block and the numeric block is an accident of how many levels each categorical had, not a decision anyone made. ## Gower's construction Gower's coefficient tackles the problem differently: instead of mapping everything into one space and then measuring, it measures each feature *in its own terms* and forces the result onto a common 0-to-1 scale before combining. For a pair of records `i` and `j`, per feature `k`: - **Quantitative** (salary, tenure): `d_k = |x_ik - x_jk| / R_k`, where `R_k` is the feature's range across the data. Identical values give 0, the two extremes of the feature give 1. - **Nominal** (department): `d_k = 0` if the codes match, `1` if they do not. - **Binary** (remote flag): the same match/mismatch rule for a symmetric flag. For an *asymmetric* binary attribute — where presence is informative and absence is not, such as a rare qualification — Gower's original scheme drops the pair from the average when both records are absent, so two people who both lack a rare certificate are not credited with a similarity for it. - **Ordinal**: converted to ranks first, then treated like a quantitative feature on the rank scale. The overall dissimilarity is the weighted average of the per-feature terms, `sum_k w_k d_k / sum_k w_k`, with weights defaulting to 1. Because each term is bounded by 1, so is the result. ## The special cases worth naming If every feature is a symmetric binary indicator — a 20-item symptom checklist, say — every per-feature term is 0 or 1 and the average is simply the count of disagreeing items divided by 20. That count is **Hamming distance**, the number of positions at which two equal-length sequences differ; Gower is its normalised form. Finding the most similar prior patient on such a checklist is a pure Hamming query, and no scaling question arises at all, because every feature already contributes on the same 0/1 scale by construction. If every feature is quantitative, Gower is Manhattan distance on range-normalised features, divided by the feature count. ## Missing values, weights and the caveats Missing data is handled per pair: if either record lacks feature `k`, that feature is excluded from both the numerator and the denominator for that pair. This is convenient but not free — different pairs are then compared on different feature sets, so a pair that agrees on the three features they share can outrank a pair compared on all twelve. Watch it when missingness is heavy or systematic. Three more caveats belong in a senior answer. **Range normalisation is outlier-sensitive**: one executive salary of ten million stretches `R_salary` so that every ordinary salary difference collapses towards zero and salary silently stops mattering. **Equal default weights are an assumption**, and with eight numeric columns and one flag, the flag contributes one ninth of the decision no matter how important it is to the business — set weights deliberately if the domain has an opinion. **The output is a pairwise dissimilarity, not a set of coordinates**, so methods that need points in a vector space rather than a distance matrix need a different treatment. ## Choosing between the two routes Encode-then-Euclidean is faster, composes with anything that expects numeric features, and is a reasonable default when categoricals are few and low-cardinality. A Gower-style mixed dissimilarity is worth it when the categorical and numeric blocks are both substantial, when missingness is common, or when you want the per-feature contributions to be interpretable — being able to say "this pair scored 0.7 because department and remote status both differ" is a real advantage when a human has to sanity-check the neighbours. The strong answer names both routes, the failure mode of each, and the data property that decides.