Figuring Out the Mean, Variance, and Standard Deviation of a Binomial Random Variable

Solving statistics problems can involve finding probabilities, mean, and standard deviation for a specific random variable, in this case the binomial. Solve the following problems about the mean, standard deviation, and variance of binomial random variables.

Sample questions

1. What is the mean of a binomial random variable with n = 18 and p = 0.4?

   Answer: 7.2

   The mean of a binomial random variable X is represented by the symbol

   

   A binomial distribution has a special formula for the mean, which is

   

   Here, n = 18 and p = 0.4, so

   

2. What is the standard deviation of a binomial distribution with n = 18 and p = 0.4? Round your answer to two decimal places.

   Answer: 2.08

   The standard deviation of X is represented by

   

   and represents the square root of the variance. If X has a binomial distribution, the formula for the standard deviation is

   

   where n is the number of trials and p is the probability of success on each trial. For this situation, n = 18 and p = 0.4, so

   

3. What is the variance of a binomial distribution with n = 25 and p = 0.35? Round your answer to two decimal places.

   Answer: 5.69

   The variance is represented by

   

   and represents the typical squared distance from the mean for all values of X.

   For a binomial distribution, the variance has its own formula:

   

   In this case, n = 25 and p = 0.35, so

   

   Rounded to two decimal places, the answer is 5.69.

4. A binomial distribution with p = 0.14 has a mean of 18.2. What is n?

   Answer: 130

   The mean of a random variable X is denoted

   

   For a binomial distribution, the mean has a special formula:

   

   In this case, p = 0.14 and

   

   is 18.2, so you need to find n. Plug the known values into the formula for the mean, so 18.2 = n(0.14), and then divide both sides by 0.14 to get n = 18.2/0.14 = 130.

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