Question

What is the standard deviation of a binomial distribution with, $n=10$ and $p=0.70$ ?

Answer 1

To answer our question, we will first understand the meaning of, Binomial distribution experiment. A statistical experiment that consists of ‘n’ repeated trials such that each trial only has two outcomes, where the probability of success “p” is the same in every trial is known as a binomial experiment.

Complete step by step solution:
In a binomial experiment, any trial can result in only two possible outcomes. This outcomes are therefore termed as ‘success’ and ‘failure’. The probability of success and failure is constant in each successive trial and independent, that is, the outcome of one trial does not affect the outcome of other trials.
For example: when we toss a coin 5 times. Let us say that “Heads” is the criteria for success and “Tails” is the criteria of failure. Then, we can see that the probability of success and failure is constant in each successive trial and is equal to 0.5 . Also, , the outcome of one toss does not affect the outcome of other tosses.
Now, there are certain terms associated with a binomial distribution experiment. These are:
‘n’ is the total number of trials.
‘x’ is the total number of successful trials.
‘p’ is the probability of a success in each trial.
‘q’ is the probability of failure in each trial which is equal to ‘$1-p$’.
Also,
The mean of the distribution is represented by $\left( {{\mu }_{x}} \right)$, and is equal to $\left( n\times p \right)$.
The variance is represented by $\left( \sigma _{x}^{2} \right)$, and is equal to $\left[ np\left( 1-p \right) \right]$.
The standard deviation is represented by $\left( {{\sigma }_{x}} \right)$, and is equal to $\sqrt{\left[ np\left( 1-p \right) \right]}$
In our problem, we have:
$\begin{aligned} & \Rightarrow n=10 \\\ & \Rightarrow p=0.7 \\\ & \Rightarrow \left( 1-p \right)=0.3 \\\ \end{aligned}$
Putting these values in the equation of standard deviation, we get:
$\begin{aligned} & \Rightarrow {{\sigma }_{x}}=\sqrt{10\times 0.7\times \left( 1-0.7 \right)} \\\ & \Rightarrow {{\sigma }_{x}}=\sqrt{10\times 0.7\times 0.3} \\\ & \Rightarrow {{\sigma }_{x}}=\sqrt{2.1} \\\ & \therefore {{\sigma }_{x}}=1.449 \\\ \end{aligned}$
Hence, the standard deviation of a binomial distribution with, $n=10$ and $p=0.70$ comes out to be 1.449.

Note: While calculating the standard deviation and variance of a binomial distribution, we should take care of both the formulas and not confuse the one with another. A simple way to remember them is that the square of standard deviation is variance.