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How would one set up a Difference of Means test to test the relationship between type of electoral system (IV) and type of party system (or number...

A Difference of Means test, also called a t-test, is one of the simplest statistical procedures, yet also one of the most widely used because of its broad applicability. It's especially useful in controlled experiments, where the two means being compared are the control group and the experimental group.

Difference of Means requires a categorical variable for the independent variable, and a quantitative variable for the dependent variable. The categorical variable should have only two options (for more categories, you can use a chi-square or F-test).

Here our independent variable is the electoral system, which is clearly categorical; there are multiple types to consider, but the two big ones are "first-past-the-post" plurality vote (like the US) and proportional representation (like Sweden). So we would make those our two categories, and sort countries into each group. If a country is hard to fit into either category (such as "mixed member proportional representation" in Germany), we might want to just leave it out of this analysis.

Then we need a variable to measure the party system, which must be quantitative---so that we can calculate a mean. The number of major parties would probably be a good one to use, where we could define "major" somehow, perhaps as having won at least three seats in the legislature or at least one election of the executive. (This would give the US 2 parties and Sweden 8 parties.) As a robustness check, we might want to see what expanding or contracting the definition of major parties does to our results.

Then, we just compute the mean number of parties among all countries in each category, as well as the standard deviation in each; the basic t-test is just to take the difference of the two means and divide it by the standard error, where the standard error is the average of the standard deviations, divided by the square root of the sample size.

This value we call t, and we can look it up on tables of the t-distribution to see how probable it is that a difference of this size would occur by chance.

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