# Understanding the Shape of a Binomial Distribution

The binomial distribution describes the probability of obtaining k successes in n binomial experiments.

If a random variable X follows a binomial distribution, then the probability that X = k successes can be found by the following formula:

P(X=k) = nCk * pk * (1-p)n-k

where:

• n: number of trials
• k: number of successes
• p: probability of success on a given trial
• nCkthe number of ways to obtain k successes in n trials

The binomial probability distribution tends to be bell-shaped when one or more of the following two conditions occur:

1. The sample size (n) is large.

2. The probability of success on a given trial (p) is close to 0.5.

However, the binomial probability distribution tends to be skewed when neither of these conditions occur. To illustrate this, consider the following examples:

### Example 1: Sample Size (n) is Large

The following chart displays the probability distribution for when n = 200 and p = 0.5. The x-axis displays the number of successes during 200 trials and the y-axis displays the probability of that number of successes occurring.

Since both (1) the sample size is large and (2) the probability of success on a given trial is close to 0.5, the probability distribution is bell-shaped.

Even when the probability of success on a given trial (p) is not close to 0.5, the probability distribution will still be bell-shaped as long as the sample size (n) is large. To illustrate this, consider the following two scenarios when p = 0.2 and p = 0.8.  Notice how the probability distribution is bell-shaped in both scenarios.

### Example 2: Probability of Success (p) is Close to 0.5

The following chart displays the probability distribution for when n = 10 and p = 0.4. Although the sample size (n = 10) is small, the probability distribution is still bell-shaped because the probability of success on a given trial (p = 0.4) is close to 0.5

### Example 3: Skewed Binomial Distributions

When neither (1) the sample size is large nor (2) the probability of success on a given trial is close to 0.5, the binomial probability distribution will be skewed to the left or right.

For example, the following plot shows the probability distribution when n = 20 and p = 0.1. Notice how the distribution is skewed to the right.

And the following plot shows the probability distribution when n = 20 and p = 0.9. Notice how the distribution is skewed to the left.

### Ending Notes

Each of the charts in this post were created using the statistical programming language R. Learn how to plot your own binomial probability distributions in R using this tutorial.