Mean, Variance, and Standard Deviation of the Binomial Distribution

As with most probability distributions, examining the different properties of binomial distributions is important to truly understanding the implications of them. The mean, variance, and standard deviation are three of the most useful and informative properties to explore. In this next section we'll take a look at these different properties and how they are helpful in establishing the usefulness of statistical distributions. The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.

Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads. If you performed this experiment over and over again, what would the mean number of heads be? On average, you would expect half the coin tosses to come up heads. Therefore, the mean number of heads would be 6. In general, the mean of a binomial distribution with parameters $N$ (the number of trials) and $p$ (the probability of success for each trial) is:

$m=Np$

Where $m$ is the mean of the binomial distribution.

The variance of the binomial distribution is:

$s^2 = Np(1-p)$, where $s^2$ is the variance of the binomial distribution.

The coin was tossed 12 times, so $N=12$. A coin has a probability of 0.5 of coming up heads. Therefore, $p=0.5$. The mean and standard deviation can therefore be computed as follows:

$m=Np=12\cdot0.5 = 6$

$s^2=Np(1-p)=12\cdot0.5\cdot(1.0-0.5)=3.0$

Naturally, the standard deviation ($s$) is the square root of the variance ($s^2$).

Coin Flip

Coin flip experiments are a great way to understand the properties of binomial distributions.