The mean and standard deviation of a binomial random variable with parameters \(n\) and \(p\) are

\[ \mu = np \] and \[ \sigma = np(1 - p)\text{,}\] respectively.

Let us recall the example, from the previous section. The probability to pass the final statistic exam is 0.3. We consider a class of 25 students. The random variable \(X\), which corresponds to success in the exam is a binomial random variable, and follows a binomial distribution with parameters \(n=25\) and \(p=0.3\). Thus the mean, \(\mu\), and the standard deviation, \(\sigma\), can be computed as follows:

\[\mu = np = 25 \times 0.3 = 7.5\]


\[ \sigma = np(1 - p) = 25 \times 0.3 \times (1-0.3) = 5.25 \]

The following plot visualizes the binomial distribution with parameters \(n=25\) and \(p=0.3\), and its mean \(\mu\) and standard deviation \(\sigma\).