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!
 A: 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.
