I'm trying to calculate how polarizing a question is. Let's say I have a question that has 3 possible choices. A certain percentage of people choose a specific answer.
Answer a: $30\%$ Answer b: $50\%$ Answer c: $20\%$
I was thinking I could try to calculate the standard deviation. But since the answers are not assigned to a certain value I'm not sure how to apply the formula. Maybe someone can help me here?
There are a number of 'measures of diversity' (see Wikipedia). Perhaps the simplest is called the Simpson Index (often denoted $\lambda$).
It is the sum of squares of relative frequencies for the categories. In your case: $$\lambda = \sum_{i=1}^3 r_i^2 = .3^2 + .5^2 + .2^2 = .38.$$
Very roughly speaking, this is the probability that two people chosen at random from the population would have matching views as to the correct choice A, B, or C. Notice that $1/K \le \lambda \le 1,$ where $K$ is the number of categories, here three. Smaller values of $\lambda$ indicate greater diversity.
If you want test whether all three categories are equally popular (probability 1/3 each), you might do a chi-squared goodness-of-fit test of that null hypotheses. For that, you would need to know the total number of people involved. (If $n = 100,$ the chi-squared statistic is 14.0 > 5.99, and the null hypothesis is rejected; if $n = 30,$ the chi-squared statistic is 4.2 < 5.99, and it is not rejected. In your case, the chi-squared statistic is $$Q = \sum_{i=1}^3 \frac{(nr_i - n/3)^2}{n/3}.$$ which has approximately a chi-squared distribution with DF = 3-1=2. The 'critical value', separating rejection and not, is the 95th quantile of that distribution, 5.99.