My problem is the following:

Let say that there is n persons voting for some values "a", "b", "c", …

For example, consider that 10 persons have voted for values "a", "b", "c", "d" and that the result is :

a: 5 b: 3 c: 2 d: 0

The majority is "a". But I would like to know if the vote for "a" is significant, given the number of persons n and the numbers of votes for each value.

Is there any statistical test or something else from statistics that can be used to solve this problem?

Thanks for your help!

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#### Best Answer

Conceptually, I think the easy way to tackle this problem is by simulation. You could use either binomial or multinomial distributions to calculate confidence intervals, and I could hassle you to define more clearly whether you want to test whether `a`

got more votes than expected, or if you care about pair-wise comparisons etc. But simulation can be quick and we can make assumptions very explicit.

Let's assume a null hypothesis that the votes are random, with equal probability for `a`

, `b`

, `c`

, and `d`

. In R, we can simulate 10,000 elections with 10 votes each:

`votes = replicate(n = 10000, sample(LETTERS[1:4], size = 10, replace = T)) `

And then look at the distribution of total votes for `A`

in each election.

`a_count = colSums(votes == "A") `

Then we can calculate the proportion of simulations in which A got at least 5 votes (as in your data):

`mean(a_count >= 5) # [1] 0.0814 `

So, there's about an 8% chance that random voting would produce as many votes for A as you observed, which isn't significant at the traditional 5% level.

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