To determine the standard deviation of a binomial distribution, let's recall the properties of the binomial distribution.
A binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. The key parameters for a binomial distribution are:
- [tex]\( n \)[/tex]: the number of trials
- [tex]\( p \)[/tex]: the probability of success on each trial
- [tex]\( q \)[/tex]: the probability of failure on each trial, where [tex]\( q = 1 - p \)[/tex]
The standard deviation [tex]\( \sigma \)[/tex] of a binomial distribution can be found using the formula:
[tex]\[ \sigma = \sqrt{n \cdot p \cdot q} \][/tex]
Let’s break down this formula:
1. [tex]\( n \)[/tex] is the number of trials.
2. [tex]\( p \)[/tex] is the probability of success.
3. [tex]\( q \)[/tex] is the probability of failure. Since [tex]\( q = 1 - p \)[/tex], this ensures that [tex]\( p + q = 1 \)[/tex].
So, the standard deviation is calculated by taking the square root of the product of the number of trials [tex]\( n \)[/tex], the probability of success [tex]\( p \)[/tex], and the probability of failure [tex]\( q \)[/tex].
Given the choices:
1. [tex]\( np \)[/tex]: This represents the expected number of successes (mean) in a binomial distribution.
2. [tex]\( n p q \)[/tex]: This represents the variance; it is not the standard deviation.
3. [tex]\( \sqrt{n p q} \)[/tex]: This represents the standard deviation.
4. None of these: An incorrect option in this context.
Therefore, the standard deviation of a binomial distribution is given by option:
(3) [tex]\( \sqrt{n p q} \)[/tex].
So the correct answer is:
[tex]\[ (3) \sqrt{n p q} \][/tex]
