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AP Statistics Practice Quiz: Introducing Statistics: Why Be Normal?

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

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

When two different random samples are taken from the same large population, why are the shapes of their distributions unlikely to be identical?

All Questions (7)

When two different random samples are taken from the same large population, why are the shapes of their distributions unlikely to be identical?

A) Because of inherent random variation in sampling.

B) Because the population's shape changes between samples.

C) Because the sample sizes must have been different.

D) Because the data for one sample was recorded incorrectly.

Correct Answer: A

The provided content states that variation in the shapes of data distributions from the same population can be random. This inherent randomness, often called sampling variability, means that two samples will almost never be perfect replicas of each other or the population.

A researcher observes that the shapes of distributions from two samples, supposedly from the same population, are noticeably different. According to the principles of statistical inquiry, what is the primary question this variation suggests?

A) Which sample is a more accurate representation of the population?

B) Was the sample size large enough to draw any conclusions?

C) Is the observed variation in shape likely due to random chance, or does it suggest a non-random cause?

D) What is the exact mean and standard deviation of the population?

Correct Answer: C

The core idea from the provided content is to determine if the variation is random or not. This is the first question a statistician must consider before drawing further conclusions about the population or the sampling process.

A quality control inspector takes a sample of products from a production line in the morning and another in the afternoon. The distribution of product weights from the morning sample is shaped differently than the distribution from the afternoon sample. This variation in shape naturally leads the inspector to question if...

A) the total number of products made has changed.

B) the variation is simply random or if the production process has changed.

C) the measurement scale is calibrated correctly.

D) the central limit theorem applies to the samples.

Correct Answer: B

The key concept from the content is that variation in sample distributions forces us to ask about its cause. The two main possibilities are random chance (expected variation) or a systematic, non-random cause (like a change in the manufacturing process).

If the variation in the shapes of distributions among multiple samples from a population is determined to be non-random, what could this suggest?

A) The population is perfectly normally distributed.

B) The samples were all perfectly representative of the population.

C) A systematic factor is influencing the samples differently.

D) The law of large numbers has been violated.

Correct Answer: C

The provided content specifies that variation can be random or not. Non-random variation implies that something beyond chance is at play, such as a change in the underlying population or a flaw in the sampling methodology that affects samples in a systematic way.

A statistician analyzes the distributions of ten different samples drawn from the same population. They conclude that the variation in the shapes of these distributions is consistent with what would be expected from random sampling. What is the most appropriate interpretation of this finding?

A) The population's true shape has been definitively identified.

B) There is no evidence to suggest that anything other than chance is causing the differences between the sample distributions.

C) All ten samples must have been collected using a flawed methodology.

D) The population from which the samples were drawn must be uniform in shape.

Correct Answer: B

This scenario describes a situation where the observed variation is attributed to random chance, as mentioned in the content. Therefore, there's no reason to suspect a non-random cause, such as a change in the population or a systematic bias in sampling.

The fundamental reason to analyze the variation in shapes of distributions from different samples of the same population is to...

A) prove that the population is normally distributed.

B) calculate the exact parameters of the population.

C) distinguish between differences due to random chance and those due to a systematic influence.

D) ensure that every sample is perfectly identical to the population.

Correct Answer: C

This question addresses the core purpose outlined in the content. The study of variation in sample distributions is primarily concerned with differentiating between random (chance) variation and non-random (systematic) variation.

Which of the following scenarios best illustrates a question arising from potentially non-random variation in the shapes of sample distributions?

A) Two pollsters survey the same population and get slightly different, but similarly shaped, distributions of voter preferences.

B) A student repeatedly samples from a well-shuffled deck of cards and notes minor fluctuations in the shape of the distribution of card values.

C) A biologist finds that the distribution of fish lengths from a lake is symmetric in the spring but heavily skewed in the fall, suggesting a seasonal effect.

D) A researcher takes two small random samples from a large city and finds their income distributions differ slightly.

Correct Answer: C

Options A, B, and D describe scenarios where slight differences are likely due to random sampling variation. Option C describes a significant, patterned change in shape (symmetric to skewed) that is linked to a systematic factor (season), which is a hallmark of the non-random variation mentioned in the content.