Enter the vote-share percentages and seat-share percentages for all parties into the calculator to determine the Gallagher Index (Least Squares Index) for the election result.

Gallagher Index Calculator

Enter vote and seat percentages for all parties (same order, same count)

Gallagher Index Formula

The Gallagher Index, also called the Least Squares Index, measures how closely an election’s seat shares match its vote shares. A value of 0 means perfect proportionality. As the value increases, the gap between votes received and seats awarded becomes larger.

GI = \sqrt{\frac{1}{2}\sum_{i=1}^{n}(v_i-s_i)^2}
Term Meaning
GI Gallagher Index value
vi Vote-share percentage for party i
si Seat-share percentage for party i
n Total number of parties or categories included in the result

The index squares each party’s vote-seat gap, so larger mismatches count more heavily than smaller ones. That makes the Gallagher Index especially useful when you want major distortions in representation to stand out clearly.

How to Calculate the Gallagher Index

  1. List every party or category included in the election result.
  2. Enter each party’s vote share as a percentage.
  3. Enter each party’s seat share as a percentage in the same order.
  4. Find the difference between vote share and seat share for each party.
  5. Square each difference.
  6. Add the squared differences together.
  7. Multiply the total by 1/2.
  8. Take the square root.

If you only have raw vote totals or seat counts, convert them to percentages first.

v_i = \frac{\text{votes for party } i}{\text{total valid votes}} \times 100
s_i = \frac{\text{seats won by party } i}{\text{total seats}} \times 100

Example Calculation

Suppose an election has four party groupings with the following vote and seat shares:

Party Vote Share Seat Share Difference Squared Difference
A 42% 48% -6 36
B 31% 28% 3 9
C 17% 18% -1 1
D 10% 6% 4 16

The sum of squared differences is 62.

GI = \sqrt{\frac{1}{2}(62)} = \sqrt{31} \approx 5.57

A Gallagher Index of approximately 5.57 indicates a noticeable, but not extreme, mismatch between vote shares and seat shares.

How to Interpret the Result

  • 0 means the seat distribution matches the vote distribution exactly.
  • Lower values indicate a more proportional election outcome.
  • Higher values indicate greater disproportionality between public support and legislative representation.
  • Because differences are squared, one large overrepresentation or underrepresentation can raise the index substantially.
  • The measure is most useful when comparing elections, districts, or electoral systems on the same scale.

Why the Gallagher Index Is Useful

This index is widely used to evaluate electoral proportionality because it condenses many party-level differences into a single number while still penalizing large distortions. It can help answer questions such as:

  • How proportional was a particular election?
  • Did electoral reform reduce disproportionality?
  • Which election produced a seat bonus for larger parties?
  • How do two countries or election years compare?

Calculator Input Tips

  • Enter vote percentages and seat percentages for the same parties in the same order.
  • Both lists should contain the same number of entries.
  • Vote shares should total about 100%, and seat shares should also total about 100%.
  • Include minor parties, independents, or an Other category when needed so the totals are complete.
  • Small rounding differences are usually acceptable, but large rounding errors can distort the result.
  • Do not mix percentages with raw counts in the same calculation.

Common Mistakes

  • Mismatched order: entering Party A’s vote share next to Party B’s seat share.
  • Incomplete data: leaving out small parties or independent candidates.
  • Using counts instead of percentages: the formula requires shares on a common percentage basis.
  • Totals far from 100%: this usually signals missing categories or incorrect data entry.

Frequently Asked Questions

Can the Gallagher Index be negative?

No. Because the formula squares each party’s difference and then takes a square root, the result is always zero or positive.

Is a lower Gallagher Index always better?

If your goal is proportional representation, lower values are generally preferred. However, whether an electoral system is “better” depends on the broader design goals of that system, such as stability, local representation, or governability.

Do I need to include every party?

Yes. For the most accurate result, include all parties and any residual category such as Other so both vote shares and seat shares represent the full election outcome.

What happens if vote shares and seat shares match exactly?

Then every party’s difference is zero, the total squared difference is zero, and the Gallagher Index is zero.

GI = \sqrt{\frac{1}{2}(0)} = 0