When working with binomial random variables and experiments, it is important to understand the mean and standard deviation. In this lesson, you will learn how to analyze binomial experiments using the mean and standard deviation of a binomial random variable.

## What Is a Binomial Random Variable?

Jennifer is working on a project for her science class.

She wants to find out how many people in her college play video games, so she decides to conduct an experiment. Jennifer asks 20 of her college friends if they play video games.First, Jennifer will conduct a **binomial experiment**, which is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure. When conducting a binomial experiment, you will need to become familiar with a few variables. These are *n*, *P*, and *x*.

You will use these variables in finding the mean and standard deviation of the binomial experiment.The *x* represents the number of successes, which is also the **binomial random variable**, the number of successes in a binomial experiment. The binomial random variable in this experiment will be the number of people who say they do play video games. The *n* represents the number of trials and the *P* represents the probability of success on an individual trial.In this scenario, Jennifer has two possible outcomes: either the friend plays video games or they don’t. In binomial experiments, this is known as success or failure because there are only two possible outcomes.

There is a 50/50 chance that the friend will either say yes or no. Therefore, *P* would equal .50 because the probability of success on an individual trial is 50%. Jennifer is asking 20 of her friends if they play video games. Therefore, *n* would equal 20 because the number of friends represents the number of trials in Jennifer’s experiment.Until Jennifer conducts her experiment, we will not know what *x* equals. Let’s take a look at how to analyze experiments using mean and standard deviation based on what we know.

## Mean

Remember, the binomial random variable is the number of successes in a binomial experiment. We can find the mean, or the average expected number, of successes in Jennifer’s experiment.Jennifer presents her project proposal to her teacher. Before she conducts her experiment, her teacher wants to know the expected value and the standard deviation of her experiment. You’ve probably heard the term ‘mean’ or ‘average’.

For example, if the average grade in a class is a 75%, then any person you met in the class could possibly have a 75% or near this grade. In statistics, the mean is also called the **expected value**, the number of successful outcomes expected in an experiment. For example, if the probability of a person that plays video games is 50%, then you would expect that half of the people you meet have played video games.The formula for expected value, or the mean, of a binomial random variable is *n* * *p*. Let’s use Jennifer’s experiment and find the mean.We know the number of trials is 20 because she is asking 20 of her friends.

Remember *n* = 20 and the probability is 50% so *P* = .50. Therefore, our equation would be (20) * (.50). To solve the equation, simply multiply, and you get 10 people. The expected value, or mean, for this experiment is 10, meaning that we expect at least 10 people to answer ‘yes’, that they do play video games.

## Standard Deviation

Now let’s look at **standard deviation**, which is the degree in which the variables are different from the mean. In other words, this formula examines the spread of the probability. If the binomial random variable is the number of successes in an experiment, then before we conduct the experiment, we need to know the range of data to expect. We already know that 10 is the expected value, but is 11 people unusual data? We can use standard deviation to find out.

The standard deviation formula for binomial random variables is the sqrt(*n* * *P* * ( 1 – *P* )). You can use the numbers from our first formula to solve this problem.sqrt((20) * (.50) * (1 – .50))Pause the video here and see if you can get the answer. Don’t forget to use the order of operations properly when solving this formula.

How did you do?The correct answer is approximately 2.24. So what does 2.

24 mean? Standard deviation gives us an understanding of the range of numbers to expect when conducting a binomial experiment. Think of this answer as plus or minus 2.24 people away from the mean. So if the mean is 10, then we can subtract 2.24 from 10 to get 7.

76, and add to 10 to get 12.24. Therefore, we can expect to have anywhere from 7.76 to 12.24 people say that they have played video games. More than 12.

24 and less than 7.76 will be unusual data.

## Lesson Summary

Remember, a **binomial experiment** is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure. When analyzing the data of binomial experiments, you will use three variables: *x*, *n*, and *P*.The *x* represents the number of successes, which is also the **binomial random variable**, the number of successes in a binomial experiment. The *n* represents the number of trials and the *P* represents the probability of success on an individual trial. When analyzing your experiment, you will need to determine the mean and the standard deviation.

In statistics, the mean is also called the **expected value**, the number of successful outcomes expected in an experiment. The formula for expected value, or the mean, of a binomial random variable is *n* * *p*.The **standard deviation** is the degree in which the variables are different from the mean. In other words, this formula examines the spread of the probability. The standard deviation formula for binomial random variables is the sqrt(*n* * *P* * ( 1 – *P* )).

You can use the numbers from our first formula to solve this problem. Standard deviation gives us an understanding of the range of numbers to expect when conducting a binomial experiment.

## Learning Outcomes

When you are finished, you should be able to:

- State the variables used to analyze a binomial experiment
- Recall what the binomial random variable refers to
- Define the terms expected value and standard deviation
- Write the standard deviation and expected value formulas for a binomial random variable