Standard deviation for probability (multiple choice) In a multiple choice test with 4 answer options, the probability of guessing correct is 25% (assuming you have no knowledge about the materials at all). Thus, regardless of sample size, my estimated mean would be 25 (dependent variable = percentage correct).
What would be the standard deviation here?
I have found this formula:
sqrt(N * p * (1-p))

So for N = 40, the standard deviation would be:
sqrt(40 * 0,25 * (1-0,25)) = 0,74.

Is this correct? If so, why is the standard deviation dependent on the sample size?
 A: The formula you have is the standard deviation of the number correct (under binomial sampling), not the proportion correct.
However, because the proportion correct is the number correct multiplied by $\frac{1}{N}$, we can use basic properties of variance to figure out that the standard deviation of the sample proportion is $\sqrt{p(1-p)/N}$ (also called the standard error of a proportion). 
[Note that here $p$ is the population proportion not the sample proportion]
On the "why" question:  it's the usual situation - standard deviations of means of independent, identically distributed random variables relate to sample size.
Again from basic properties of the variance,
$\text{Var}(X_1+X_2+...+X_N)=\text{Var}(X_1)+\text{Var}(X_2)+...+\text{Var}(X_N)$
$\qquad=N\text{Var}(X_1)$
Hence $\text{Var}(\bar{X})=\frac{1}{N^2}.N\text{Var}(X_1)=\frac{1}{N}\text{Var}(X_1)$
So the standard deviation of the mean is the standard deviation (square root of variance) of an individual observation, divided by $\sqrt{N}$, .
(I tend to call it the "sigma on root n effect".)
This is the standard error of the mean, and in the case of a binomial, the individual Bernoulli-trial standard deviation is $\sigma=\sqrt{p(1-p)}$
--
In addition, you probably don't want to calculate with the percentage correct (25); better to stick with the proportion correct, or the number correct. (The calculations can be converted to work with percentages, but I recommend you stick with the proportion.)
