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The statistics you actually need to read voting research

Voting papers lean on a small, specific set of statistical ideas — and in this field several of them aren't background maths at all: they are the thing being argued about. Whether you total scores or take their median isn't a technical detail, it's the difference between two voting methods. This page covers only the concepts that change a voting answer, and points at where each one bites. It is not a statistics course; every entry is here because you'll misread a paper without it.

Level: 201 → 301 · deep dive Companion: election simulation models (the model-building subset) · simulate utilities, not ballots.

1. Mean, sum, and median — a method-defining choice

The big one. These are not three ways of saying "average"; picking between them picks a voting method:

Rule Method Behaviour
Sum (≡ mean, when everyone scores everyone) Score / Range, and STAR's scoring round rewards intensity — a 5 counts five times a 1
Median Majority Judgment ignores how extreme the extremes are

Why it matters: the mean is dragged by outliers, the median isn't. So under a summing method, exaggerating your scores (5s and 0s instead of 4s and 1s) genuinely moves the total — which is exactly the residual strategic incentive PVSI measures for STAR. Under a median method that exaggeration does almost nothing… but neither does sincere intensity, so the method can't tell "everyone's decent second choice" from "one faction's passion." Robustness and expressiveness are the trade, and mean-vs-median is where you make it.

Watch for this: an argument that quietly assumes the mean is the right summary is assuming a cardinal, intensity-counting view of democracy. That's a defensible position, not a neutral one.

2. Sum vs mean — they diverge exactly when the denominator does

If every voter scores every candidate, ranking by total and ranking by average give the same order — the denominator is identical, so it cancels. They come apart the moment some ballots don't count for some candidates: abstentions, blanks, and skipped races.

That's not academic. It's precisely the runoff-denominator question — should the runoff percentage be out of all ballots, or only those expressing a preference (excluding Equal Support)? Any paper reporting a percentage owes you its denominator.

3. Distribution shape — consensus, polarization, and why the average hides it

A single number destroys the shape of the thing it summarizes. Three electorates can share an identical mean score:

  • Consensus — scores bunched together. Nearly everyone agrees.
  • Polarized (bimodal) — two clumps at opposite ends, nobody in between.
  • Uniform — scores spread evenly. No structure at all.

A rated ballot preserves this and a ranked ballot destroys it — with rankings you can see that voters split, never how far apart they are. That is the substance behind the majority-of-consensus critique, and it's why this repo's simulations report the model alongside every rate.

4. Variance — the statistical name for "divisive"

Variance (and its square root, standard deviation) measures spread. It's the most under-used idea in this literature, because it's what "polarizing candidate" actually means:

Two candidates with the same average score can be completely different politicians. One got 3s from everybody. The other got 5s from three voters in every five and 0s from the other two. Same mean, 3.0 — wildly different variance, 0.0 against 6.0.

Mind the arithmetic, because the stock version of this example doesn't tie: half the electorate at 5 and half at 0 averages 2.5, and no flat integer score on a 0–5 ballot matches that. Getting a real tie means moving the split (as above) or moving the flat score — see the worked election, where Score can't separate the two at all, Approval takes the consensus candidate and STAR's runoff takes the polarizing one.

That single sentence is the majority criterion debate in miniature: the broadly-liked candidate and the polarizing one, arithmetically tied on the mean, separated only by the spread. Whenever a method is accused of electing a "divisive" winner, the claim is about variance.

One catch worth carrying: on a bounded ballot, variance isn't independent of the mean — a candidate averaging 4.5 can't have a variance above 2.25 no matter how split the electorate is, so raw variance partly ranks candidates by how mediocre their average is. The bound, the fix, and the full argument: variance — the statistical name for "divisive".

5. Normalization — turning feelings into a 0–5 ballot

Min-max normalization rescales a voter's private utilities so their favourite gets the top of the scale and their least favourite the bottom. It's how simulations derive a "sincere" STAR ballot, and roughly what real voters do.

The consequence to remember: heavy strategic normalization — everything pushed to 5 or 0 — collapses a rated ballot into an essentially ordinal one, which is why STAR's escape from Arrow is cleanest under reasonably sincere scoring.

6. Independence vs correlation — why "random ballots" mislead

Two events are independent if one tells you nothing about the other; correlated if it does. Real electorates are heavily correlated — someone who likes candidate A usually likes ideologically similar candidate B.

This is the single most important modelling fact in the field. Impartial culture assumes independence, which is why it manufactures Condorcet cycles and paradoxes far more often than reality does. It's a stress test, not a prediction — and any paper quoting a paradox rate from impartial culture without saying so is overstating it.

7. Expected value, sampling error, and Monte Carlo

  • Expected value — the long-run average. VSE is an expected value: average voter satisfaction across thousands of simulated elections.
  • Sampling error — real polls are wrong by a few points, and that uncertainty is what makes strategy risky. Papers model it explicitly (Wolk–Quinn–Ogren use 5% noise) — which is why a live running tally, having no error at all, raises the strategic incentive above the published figures.
  • Monte Carlo — answer a hard question by simulating it many times and counting. Every simulation in this repo is Monte Carlo, which is why results come with a seed and a trial count: without them the number isn't reproducible.

8. Two formulas worth recognizing on sight

  • Sum of squares → Effective Number of Parties: ENP = 1 / Σ(sᵢ²). Squaring makes big parties dominate, so tiny parties barely register — that's the point.
  • Quota → seats: the Droop quota 1/(M+1) sets the vote share needed for one seat at district magnitude M. Three seats ≈ 25%, ten seats ≈ 9%.

The habit that matters more than any formula

Never quote a rate without its model. Every number in this literature is conditional on assumptions — the electorate model, the strategy model, the candidate count, the tie rule. "STAR and Approval disagree 12% of the time" is meaningless until you add under which model, with which approval cutoff. That standing caution (301.7) is worth more than any single statistic, and it's the fastest way to spot an overclaim — including your own.