__Comparing levels of fairness among different elections in different countries and electoral systems__
When you vote you want fairness. So it’s important that the

__proportion____of seats__that your chosen party gets is going to come very close to matching the__proportion____of votes__they got. That’s called proportional representation.
The Gallagher Index measures

**those two match in past elections -- both in our country and others.**__how closely__
This measurement or
“score” enables us to accurately

__various levels of proportionality among various past elections from various electoral systems of various countries. As we look back over time, we can see consistent__**compare**__patterns and tendencies__.
Once that comparing is
done from

__past__elections, then people can better choose, design, and improve their electoral system to fit__their__particular needs in__future__elections.**Regardless of which party we are connected to, the Gallagher index provides us with a**

__Non partisan:____common__way to assess things. It can’t be swayed by bias because it's purely mathematical and therefore purely

__objective__. That’s why it’s worth learning about how the math works.

__Understanding it in chewable chunks in a table form__
The summary of the Gallagher Index is expressed in an
algebra formula, but when you actually use it, you need to list all the parties
in a table format anyway…so we might as well
just use a table format.

Shown here is the
example of the 2015 Canadian federal election (thanks to Raellerby via Wikimedia), but you can use the same table
for any election from any country and from any electoral system.

For each party, we take the

__"% of votes"__and subtract the "

__% of seats__" to get the "

__difference__" in percentages for each party.

__About the last column: Finding a way to compare the disproportionality of entire elections__
In order to compare how
disproportionate this

__entire__election is in comparison with other__entire__elections, we need to somehow create__one single score__for this entire election. We need to gather all of the__"parties__’ scores of disproportionality" into one score of__"election__disproportionality."**Before we can add up all of the parties’ scores, we need to convert them all into positive numbers**.

**We can do that by squaring them all**(multiplying the number by itself)

**.**

Why does squaring them
all convert them all into positive numbers? This works because you’ll get a
positive number as a result -- regardless of whether you multiply a positive
number by itself, or whether you multiply a negative number by itself. (Learn more about multiplying two negatives here.)

**Summary of the last column:****, all the parties’ scores are squared, and they all become positive numbers.**

__First____, 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)__

**Second**

**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__).

__same__

__same__

__two__

__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.

**L**

**E**

**ARN MO**__at these links (listed from "easy" to "hard" ...sort of):__

**RE**- Measuring Unfairness - Calculating Canada's Gallagher Index (interactive)

- 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

I am not a mathematician, that is certain, but I am trying to comprehend this Gallagher Index. My one question after first reading the explanation is, "What would have been the end result in regards to the final number of seats in the last Canadian election if this Index method had been in place"? I think it would be easier for someone like myself to grasp the ultimate consequences of implementing such a system if I knew the answer to that. Thank you.

ReplyDeleteThe Gallagher Index is not itself an election system, it is one way to measure the fairness of an election. If a "Proportional Representation" voting system had been in place in 2015, each party would have had seats in close proportion to its share of the vote. For example, the Liberals with 39.47% of the vote would have had closer to 133 seats instead of the 184 which they obtained. (Of course, it goes without saying that if a different voting system had been in place, then the vote percentages might have changed too.)

DeleteTo put things in perspective: Generally, countries with a long-term average Gallagher Index under 9.0 have what is described as Proportional Representation. Countries without PR usually have a Gallagher Index over 9.0. Therefore, a Gallagher Index is usually a good measure of whether a country gives voters representation (seats in a legislature) in reasonable proportion to how they voted. (There are a few exceptions. For example the Gallagher Index for the whole United States House of Representatives is misleadingly low as a result of the Primary election process that precedes the main election. However, Byron Weber Becker's improved "composite Gallagher Index" mentioned above would show a much higher Index value, demonstrating the inherent dis-proportionality of these U.S. elections within each State, caused by their First-Past-The-Post voting system.)

ReplyDelete