Question

In a Binomial Distribution, if 'n' is the number of trials and 'p' is the probability of success, then the standard deviation is given by?

• A. np
• B. $(np)^2$
• C. np(1-p)
• D. $\sqrt{np(1-p)}$

Hint:

The variance of a Binomial distribution is $np(1-p)$. Always remember to take the square root of the variance to obtain the standard deviation.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The standard deviation measures the dispersion of a probability distribution and is defined as the positive square root of its variance.

Step 2: Detailed Explanation:
For a Binomial distribution with parameters $n$ (number of independent trials) and $p$ (probability of success in each trial):
The probability of failure is defined as $q = 1-p$.
The variance ($\sigma^2$) of the distribution represents the expected squared deviation from the mean, calculated as: \[ \sigma^2 = npq = np(1-p) \]
The standard deviation ($\sigma$) is the positive square root of the variance: \[ \sigma = \sqrt{\text{Variance}} = \sqrt{npq} = \sqrt{np(1-p)} \]
Therefore, the standard deviation is expressed as $\sqrt{np(1-p)}$.

Step 3: Final Answer:
The standard deviation of a Binomial distribution is $\sqrt{np(1-p)}$.