AP Statistics · Topic 2.9 · Unit 2

AP Stats 2.9: Mean & SD of Random Variables

By Jude Wallis · Published July 28, 2026

Topic 2.9 in the Fall 2026 AP Statistics course finds parameters of a discrete random variable. The mean, or expected value, is the sum of each value times its probability. The standard deviation is the square root of the sum of squared deviations from the mean, each weighted by its probability.

AP Statistics: Unit 2 (topics 2.9). In the Fall 2026 AP Statistics course, parameters of random variables are Unit 2 Topic 2.9, aligned to skills 3.B and 4.D.

What topic 2.9 covers

Topic 2.9 summarizes a probability distribution with two numbers. The objective is to calculate and interpret the mean and standard deviation of a discrete random variable.

Both are parameters. A parameter is a numerical value measuring a characteristic of a probability distribution or a population, and its value is a single fixed number, in contrast to a statistic that varies from sample to sample as covered in parameter vs statistic.

Expected value (the mean)

The expected value, or mean, of a probability distribution is denoted $E(X)$ or $\mu_X$ , read as 'mu of X'. For a discrete random variable it is calculated as:

\mu_X = \sum x_i \, P(x_i)

Here $x_i$ is a possible value of the random variable and $P(x_i)$ is the probability of that value, so you multiply each value by its probability and add the products. The expected value is the long-run average outcome of the random variable, not necessarily a value the variable can actually take. A fair die, for instance, has expected value 3.5.

Standard deviation and variance

The standard deviation of the distribution is denoted $SD(X)$ or $\sigma_X$ , read as 'sigma of X'. For a discrete random variable it is:

\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \, P(x_i)}

You take each value's deviation from the mean, square it, weight it by that value's probability, add the results, and take the square root. The standard deviation is the typical distance of the values from the mean over the long run. Its square, without the square root, is the variance, written $V(X)$ or $\sigma_X^2$ . Interpret both in the context of the specific population, using appropriate units.

Interpreting the parameters in context

The mean and standard deviation earn full credit only when you interpret them for the specific population and attach units. The mean is the long-run average value of the random variable, so a mean of 1.15 pets means that over many households the average number of pets is about 1.15.

The standard deviation is the typical distance of values from that mean, so a standard deviation of about 0.96 pets means individual households usually fall within roughly one pet of the average. Describing features of a graph without tying them to the situation is the common way students lose these points, so name the variable, the population, and the units every time you report a parameter.

Mean and standard deviation of pet counts

Let X be the number of pets owned, with P(0) = 0.30, P(1) = 0.35, P(2) = 0.25, and P(3) = 0.10. Find the mean and the standard deviation.

Mean: $\mu_X = 0(0.30) + 1(0.35) + 2(0.25) + 3(0.10) = 0 + 0.35 + 0.50 + 0.30 = 1.15$ .
Squared deviations from 1.15: $(0-1.15)^2 = 1.3225$ , $(1-1.15)^2 = 0.0225$ , $(2-1.15)^2 = 0.7225$ , $(3-1.15)^2 = 3.4225$ .
Weight each by its probability: $1.3225(0.30) = 0.39675$ , $0.0225(0.35) = 0.007875$ , $0.7225(0.25) = 0.180625$ , $3.4225(0.10) = 0.34225$ .
Sum to get the variance: $0.39675 + 0.007875 + 0.180625 + 0.34225 = 0.9275$ , then $\sigma_X = \sqrt{0.9275} = 0.9631$ .

The mean is $\mu_X = 1.15$ pets and the standard deviation is $\sigma_X = \sqrt{0.9275} \approx 0.963$ pets.

Frequently asked questions

How do you find the expected value of a discrete random variable?

Multiply each possible value by its probability and add the products: the mean mu of X equals the sum of x times P(x). The expected value is the long-run average outcome, so it can be a number the variable never actually takes, such as 3.5 for a fair die.

How do you find the standard deviation of a discrete random variable?

Find each value's deviation from the mean, square it, multiply by that value's probability, and add the results to get the variance. The standard deviation is the square root of that sum. It measures the typical distance of the values from the mean over the long run.