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

By Consumer Dummies

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

    image0.jpg

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

    image1.jpg

    Here, n = 18 and p = 0.4, so

    image2.jpg

  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

    image3.jpg

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

    image4.jpg

    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

    image5.jpg

  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

    image6.jpg

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

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

    image7.jpg

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

    image8.jpg

    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

    image9.jpg

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

    image10.jpg

    In this case, p = 0.14 and

    image11.jpg

    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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