Comparing levels of fairness among different elections in different countries and electoral systems
For each party, we take the "% of votes" and subtract the "% of seats" to get the "difference" in percentages for each party.
First, all the parties’ scores are squared, and they all become positive numbers.
Second, you add up all of the squares from all of the parties into one score for the entire election. Then you can compare this "score" with other elections' scores. Whichever election has the “least squares” has the least disproportionate electoral outcome. (See below FAQ #1)
Learning from history with the Gallagher Index
The Gallagher Index "scores" for individual elections can also be applied to many elections over longer spans of time in history. As we look back over time, we can see consistent patterns and tendencies. To see Canada's average "score" between 1950 and 2016, compared to eleven other countries (with various electoral systems) see this example. Or look here. Or to see many past elections from many countries (with various electoral systems), click here and look for the subtitle "Values of indices."
FREQUENTLY ASKED QUESTIONS (FAQ)
1. Did Gallagher invent the idea of "least squares?"
No. Gallagher himself said that the idea of “Least Squares” (LSq) has a long history in the natural sciences and social sciences. His index ("LSq = ...") is simply another form of it applied to elections. The method of “least squares” is used to compare an observed value (“what actually happens”) to its data point (“the ideal” or “reference point”). ("Least Squares" is discussed in both a Simple English Wikipedia article, and a regular Wikipedia article.)
2. Why halve (cut in half) the sum of the squares?
The parties which "got more seats than their votes warranted" actually "took" those seats from other losing parties (in a sense). If we would count both those "seats taken" and those same "seats received," then we would be counting that same disproportionality two times. But we only want to count that disproportionality once, so we halve the sum of the squares (divide it by two).
3. Why is there a square root done at the very end?
Because it reverses the exponential effect of all the squaring that was done earlier in the Gallagher calculation. This is done to make the final answer number more similar to the
numbers it was derived from -- namely the percentage
differences in the "differences" column.
The "exponential effect" is shown here: 2x2=4, but 3x3=9. Comparing 4 to 9 is an exponentially larger difference than comparing 2 to 3. To undo that particular
"exponential" effect of all the squaring we did earlier in the calculation, we simply do a square root at the end of the calculation (after the initial squaring has finished serving its purpose of converting all numbers into positive numbers).
4. The rules for federal elections in Canada require that certain provinces always get a certain quantity of seats - on a province by province basis. If so, then the Gallagher index for Canada ought to ALSO reflect that. In other words, the Gallagher data should be collected on a province by province basis; and the Gallagher score should be calculated on a province by province basis. Only after that is done, can we then add up all of those provincial scores and then average them out to get the true national "composite Gallagher index" score. Agree?
Yes. And if we do that, then the above table calculation of 12 for Canada is incorrect. It should instead show a "composite Gallagher index" of 17.1. Byron Weber Becker developed this index.
5. What level of math do I need to understand the Gallagher Index?
Grade 8: Practice with squares and square roots together is here, and here in equations.
6. Why does the Gallagher Index have the word "index" in its name?
A lot of squaring is done in this formula. Look at this example: 82 = 8 × 8 = 64. In that example, the "2" is the "index" of the number 8. The index of a number says how many times to use the number in a multiplication. (Other names for index are "exponent" or "power.") (Learn more at this link.) The Gallagher Index does a lot of squaring and so it relies heavily on the mathematical function of an "index" to arrive at its conclusions.
LEARN MORE at these links (listed from "easy" to "hard" ...sort of):
- What is the Gallagher Index?
- The Gallagher Index: A measure of Disproportionality (a Google Doc)
- Composite Gallagher Index in detail - and much more - at Byron Weber Becker's Elections Modelling website (for Canada)
- English Wikipedia entry for the Gallagher Index
- Michael Gallagher's work on Electoral Systems at Trinity College Dublin (see this subtitle: "Calculate the indices for any election")
- a more general link to Gallagher's work on "least squares index"
Other Useful links (not listed from "easy" to "hard"):
- Fair Vote Canada
- Représentation équitable au Canada
- Wikipédia en français - Indice de Gallagher
- List of Recommendations from the Canadian Parliamentary Committee on Electoral Reform. Recommendation #1 mentions the Gallagher Index