Solved – How to determine if a value is significantly larger than other values

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!

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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