I know that for 50%-50% situations like coin toss the standard deviation for N number of toss is given by $$\sigma = \sqrt{.25N} = \sqrt{N}/2$$
What is the type though that I can calculate the standard deviation in situations that 3 results are equal to come out? (33,33% - 33,33% - 33,33%)
Or what is the type to calculate standard deviation for 4-way results (25% - 25% - 25% - 25%),5-way results, 6-way results, 12-way results etc
Is it any more generalised rule?
If not i would only like to know the type for 3,6 and 12-way results
Thank you!
First you have a sequence of i.i.d. Bernoulli trials $X_i \sim \text{Bernoulli}(p)$, then the sum of them (the total number of "success") follows a Binomial distribution: $$ \sum_{i=1}^n X_i \sim \text{Binomial}(n, p)$$ and thus the variance is given by $np(1 - p)$. In particular when $p = 0.5$, you have your stated result.
When you generalized to more than two outcomes per trial, say $k$ outcome per trial, then the frequencies of each outcome jointly follows a multinomial distribution, and each individual frequency will marginally follow a Binomial distribution.
So it depends on what you want - and if you really want to find out the standard deviation. The standard deviation is the same, except that the $p$ is now changed to $\frac {1} {k}$.