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AP Statistics Practice Quiz: Mean and Standard Deviation of Random Variables

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

According to its statistical definition, which of the following best describes a parameter?

All Questions (9)

According to its statistical definition, which of the following best describes a parameter?

A) A numerical value measuring a characteristic of a sample taken from a larger group.

B) A numerical value measuring a characteristic of a population or a complete probability distribution.

C) The outcome of a single trial of a random experiment.

D) A chart or graph that displays the results of a study.

Correct Answer: B

A parameter is a numerical value that describes a characteristic of an entire population or a theoretical distribution. A value calculated from a sample is called a statistic.

The mean of a discrete random variable X, also known as its expected value, is calculated by which of the following methods?

A) Finding the most frequently occurring value of X.

B) Calculating the average of the maximum and minimum values of X.

C) Summing the product of each possible value of X and its corresponding probability.

D) Summing all the probabilities for the possible values of X.

Correct Answer: C

The mean (or expected value) of a discrete random variable is a weighted average, calculated as the sum of each value multiplied by its probability: E(X) = Σ[value * probability].

A small business sells 0, 1, 2, or 3 specialty cakes per day. The probability distribution for the number of cakes sold, X, is given below. P(X=0) = 0.1 P(X=1) = 0.2 P(X=2) = 0.4 P(X=3) = 0.3 What is the mean (expected) number of cakes sold per day?

A) 1.5 cakes

B) 1.9 cakes

C) 2.0 cakes

D) 2.5 cakes

Correct Answer: B

The mean is calculated by summing the product of each value and its probability: E(X) = (0)(0.1) + (1)(0.2) + (2)(0.4) + (3)(0.3) = 0 + 0.2 + 0.8 + 0.9 = 1.9 cakes.

The mean number of typos on a randomly selected page of a certain novel is 2.4. Which of the following is the best interpretation of this parameter?

A) Every page in the novel has either 2 or 3 typos.

B) The most common number of typos on a page is 2.4.

C) If one were to count the typos on all pages of the novel and find the average, that average would be 2.4 typos per page.

D) It is impossible for any single page to have more than 2.4 typos.

Correct Answer: C

The mean (expected value) of a random variable should be interpreted as the long-run average over many repetitions. In this context, it represents the average value over the entire population of pages.

The standard deviation of a discrete random variable is a measure of which characteristic of the distribution?

A) The center or long-run average value.

B) The most probable outcome.

C) The typical spread or variability of values around the mean.

D) The symmetry of the distribution.

Correct Answer: C

The standard deviation measures the typical or average distance of the outcomes of the random variable from its mean. It is a measure of spread or variability.

Let X be a random variable with only two outcomes: X=0 with probability 0.8, and X=5 with probability 0.2. The mean of X is 1.0. What is the standard deviation of X?

A) 1.0

B) 2.0

C) 4.0

D) 16.0

Correct Answer: B

First, calculate the variance: Var(X) = (0 - 1.0)^2 * (0.8) + (5 - 1.0)^2 * (0.2) = (-1)^2 * (0.8) + (4)^2 * (0.2) = 1 * 0.8 + 16 * 0.2 = 0.8 + 3.2 = 4.0. The standard deviation is the square root of the variance, which is sqrt(4.0) = 2.0.

A local charity is running a raffle. They sell 500 tickets at $10 each. There is one grand prize of $1,000, two second-place prizes of $250, and five third-place prizes of $50. Let X be the net winnings for a person who buys one ticket. The mean of X is -$4.50 and the standard deviation is $49.75. Which is the best interpretation of the standard deviation?

A) The most a person can lose is $49.75.

B) On average, a person who buys a ticket will lose $49.75.

C) The net winnings for a ticket holder will typically differ from the mean of -$4.50 by about $49.75.

D) The average prize value awarded by the charity is $49.75.

Correct Answer: C

The standard deviation measures the typical distance of the outcomes from the mean. In this context, the actual outcome (a large prize or a loss of $10) is typically far from the average outcome of -$4.50, and the standard deviation of $49.75 quantifies this spread.

The random variable X represents the number of days a student is absent from a statistics class during a semester. If the mean of X is 3.1 days and the standard deviation is 1.2 days, which of the following provides the correct units for these parameters?

A) The mean is 3.1 days, and the standard deviation is 1.44 squared days.

B) The mean is 3.1 days, and the standard deviation is 1.2 days.

C) The mean is 3.1, and the standard deviation is 1.2; both are unitless.

D) The mean is 3.1 days, and the standard deviation is a probability of 1.2.

Correct Answer: B

Parameters for a discrete random variable, such as the mean and standard deviation, should be interpreted in context with appropriate units. Both the mean and the standard deviation have the same units as the random variable itself. The variance would have squared units.

A discrete random variable X can take on the values 10, 20, and 40. The probabilities are P(X=10)=0.5 and P(X=20)=0.2. What is the mean of X?

A) 17.0

B) 21.0

C) 23.3

D) 26.0

Correct Answer: B

First, find the missing probability. The sum of probabilities must be 1. So, P(X=40) = 1 - 0.5 - 0.2 = 0.3. Then, calculate the mean: E(X) = (10)(0.5) + (20)(0.2) + (40)(0.3) = 5 + 4 + 12 = 21.0.