Calculating the Mean & Standard Deviation of Discrete Random Variables

I. Introduction
When presented with a discrete distribution, one must have a way to measure center and spread, just as is done for continuous distributions. This help page explains what formulas to use to measure both center and spread of a discrete distribution.

*For more information on what discrete variables are & the difference between discrete vs. continuous, click here


II. Calculating the Variance and Standard Deviation of a Discrete Random Variables

Use the following three formulas to calculate either the mean, variance, or standard deviation of a discrete distribution. Note that the mean of a random variable is often called the expected value of the random variable. 



\(E(X) = \mu_x =\displaystyle\sum_{i=1}^{n} x_i p_i \) \(VAR(X) = \displaystyle\sigma^{2}_{x} = \displaystyle\sum_{i=1}^{n} (x_i - \mu_i )^{2}p_{i} \) \(SD(X) = \displaystyle\sigma_{x} =\sqrt{ \displaystyle\sum (x_i - \mu_i )^{2}p_{i} } \)

Note (1): This is true where \(x_i\) is the value of the random variable and \(p_i\) is the probability that x takes the value i.

Note (2): The mean must first be calculated before either the variance or the standard deviation can be calculated.


III. Example

A local baseball league was selling raffle tickets to generate funds for the league. Each ticket costs $1.00. If you bought a raffle ticket, there was a total of three possible outcomes. Participants win nothing with probability of 80%, participants win three dollars with probability of 19%, and participants win 10 dollars with probability of 1%. If X is a random variable that represents the net profit earned, complete the following questions:
a. Construct a probability table for random variable X.

b. Calculate the mean of the distribution.
c. Calculate the variance of the distribution.
d. Calculate the standard deviation of the distribution.

a. Given that $1 dollar was paid to participate in the raffle and our random variable X represents the total profit earned, $1 must be subtracted from the winnings to get the true profit. For example, if you pay $1.00 to participate, and win $10.00, your net profit equals $10.00 minus the $1 you paid to play, or just $9.00. Thus the probability table is:

X $-1.00 $2.00 $9.00
P(X=x) 0.8 0.19


b. Now that we have constructed our probability table, we can calculate the mean of the distribution:

\(E(X) = \mu_x =\displaystyle\sum_{i=1}^{n} x_i p_i \)

\(\mu_{x}\) = ($-1.00)(0.8) + ($2.00)(0.19) + ($9.00)(0.01)

\(\mu_{x}\) = $-0.33

c. Now that we have calculated the mean of the distribution, we can calculate the variance using the following formula:

\(VAR(X) = \displaystyle\sigma^{2}_{x} = \displaystyle\sum_{i=1}^{n} (x_i - \mu_i )^{2}p_{i} \)

\(\sigma^{2} =\) (-$1.00 - (-0.33))2(0.8) + ($2.00 - (-0.33))2(0.19) + ($9.00 - (-0.33))2(0.01)

\(\sigma^{2} =\) (-$1.00 + 0.33))2(0.8) + ($2.00 + 0.33))2(0.19) + ($9.00 + 0.33))2(0.01)

\(\sigma^{2} =\) 2.2611

d. Calculate the standard deviation of the distribution.

\(\sigma = \sqrt{2.2611} = $1.504\)