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.

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.

 MEAN/EXPECTED VALUE VARIANCE STANDARD DEVIATION $$E(X) = \mu_x =\displaystyle\sum_{i=1}^{n} x_i p_i$$ $$VAR(X) = \displaystyle\sigma^{2}_{x} = \displaystyle\sum_{i=1}^{n} (x_i - \mu_i )^{2}p_{i}$$ $$SD(X) = \displaystyle\sigma_{x} =\sqrt{ \displaystyle\sum (x_i - \mu_i )^{2}p_{i} }$$

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. Solution: 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 0.01 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.

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