Mean and standard deviation of a discrete random variable

What the exam asks

• Calculate the mean (expected value) $\mu_X = E(X)$, variance $\sigma_X^2$, and standard deviation $\sigma_X$ of a discrete random variable from a table, histogram, or word description.
• Interpret $\mu_X$ and $\sigma_X$ in context, including reference to the long run (many repetitions) and correct units.
• Determine a missing probability or value $x_i$ before or by using the expected value formula.

Core ideas

Parameters of a probability distribution

Because a probability distribution describes the theoretical behavior of a random variable over all possible outcomes, its summary measures are parameters, denoted by Greek letters ($\mu_X$ and $\sigma_X$), rather than sample statistics ($\bar{x}$ and ).

Expected value (mean)

The mean, or expected value, of a discrete random variable $X$ is a weighted average of all possible values, where each value $x_i$ is weighted by its probability $P(x_i)$. On the AP formula sheet, this appears as:

• $\mu_X$ is the balance point of the probability histogram.
• $\mu_X$ does not have to be a possible value of $X$ (nor an integer). Never round to a whole number to make it look like a realistic single outcome.

Variance and standard deviation

The variance $\sigma_X^2$ is the weighted average of the squared deviations from the mean, and the standard deviation $\sigma_X$ is the square root of the variance:

Interpretations in context

AP scoring guidelines usually look for three elements when you interpret $\mu_X$ or $\sigma_X$: long-run repetition, context, and units.