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AP Statistics Help: Describing The Distribution Of Quantitative Variables

Review real example questions for Describing The Distribution Of Quantitative Variables in AP Statistics.

Question 1 / 10

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A local gym records the ages of its members. The data reveal a large number of members between 20 and 35 years old and another large number of members between 55 and 70 years old, with very few members in between these two ranges. Which unusual feature is present in this distribution?

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Question 1

A local gym records the ages of its members. The data reveal a large number of members between 20 and 35 years old and another large number of members between 55 and 70 years old, with very few members in between these two ranges. Which unusual feature is present in this distribution?

  1. A single significant outlier representing an extremely old member.
  2. A gap, representing the range of ages where there are very few members. (correct answer)
  3. A uniform shape, because all age groups are equally represented.
  4. A symmetric shape, because the two groups of members are balanced.

Explanation: The correct answer is B. A gap is a region in a distribution where there are no, or very few, data values. The description of 'very few members in between' two concentrated groups of ages indicates a gap. The distribution is also likely bimodal with two clusters, but the gap is the feature that describes the space between them.

Question 2

The distribution of the number of hours students at a large high school spent on homework last week is unimodal and skewed to the right. Which of the following is a plausible explanation for the shape of this distribution?

  1. The number of hours spent on homework is roughly the same for all students surveyed.
  2. There are two distinct groups of students: those who do a lot of homework and those who do very little.
  3. Most students spend a relatively small to moderate number of hours, but a few students spend an exceptionally large number of hours. (correct answer)
  4. Most students spend many hours on homework, while only a few students spend very little time on it.

Explanation: The correct answer is C. A distribution that is skewed to the right has a main body of data on the left (lower values) and a tail extending to the right (higher values). This shape corresponds to a situation where most students spend a small to moderate amount of time on homework, and a few students spend a very large amount of time, creating the right tail.

Question 3

The following is a set of exam scores for a small class of students: 35, 72, 75, 78, 81, 83, 85, 88.

Which value in this dataset is best described as a potential outlier?

  1. 35, because it is unusually low compared to the other scores which are clustered together. (correct answer)
  2. 88, because it is the maximum value and is therefore an extreme point in the dataset.
  3. 79.5, the mean of the dataset, because it represents the central tendency.
  4. 72, because it is the second lowest score and establishes the beginning of the main cluster.

Explanation: The correct answer is A. The majority of the scores (72, 75, 78, 81, 83, 85, 88) are clustered in the 70s and 80s. The score of 35 is significantly lower than this cluster, making it a potential outlier. Being the maximum or minimum value does not automatically qualify a point as an outlier; the separation from the rest of the data is the key factor.

Question 4

A survey of residents in a large city asked for their daily commute time to work. The data show two distinct peaks: one around 20 minutes and another around 50 minutes. Which term best describes the shape of this distribution?

  1. Unimodal, as commute time is a single variable being measured.
  2. Bimodal, as there are two distinct clusters of data suggesting two common commute times. (correct answer)
  3. Uniform, as all commute times in a large city are generally equally likely.
  4. Skewed, as there will likely be a tail of very long commute times for some residents.

Explanation: The correct answer is B. A distribution with two distinct peaks is called bimodal. This shape often suggests that the data come from two different subgroups within the population, for example, people using different modes of transportation or living in different parts of the city.

Question 5

A botanist measures the number of seeds produced by each of 36 plants grown under the same conditions. The dotplot shows values from 90 to 130 seeds, with the highest concentration around 108–112, and roughly similar frequencies on both sides of that center; there are no isolated points far from the rest.

Which statement best describes the distribution?

  1. The distribution is approximately symmetric and unimodal, centered near about 110 seeds, with no apparent outliers. (correct answer)
  2. The distribution is strongly right-skewed because 130 is larger than 90.
  3. The distribution is strongly left-skewed because the smallest values are below 100.
  4. The distribution is bimodal because there are values on both sides of 110.
  5. The distribution is uniform because the range is about 40 seeds.

Explanation: This question tests describing quantitative distributions like seed counts in a dotplot, focusing on shape, center, spread, and outliers. The distribution is approximately symmetric and unimodal, centered near 110 seeds with balanced frequencies on both sides from 90-130 and no outliers. Distractor B incorrectly calls it strongly right-skewed because 130 > 90, but skew requires an imbalanced tail, not just range asymmetry. Mini-lesson: Symmetry means mirror-image halves; unimodal has one peak. Bimodality shows two peaks, uniformity even spread, left skew tails low. Always verify balance around the center and check for detached points as outliers.

Question 6

A teacher recorded quiz scores (out of 10 points) for 35 students. The dotplot below shows the distribution of scores. Which statement best describes the distribution?

  1. The distribution is left-skewed, with many high scores and a tail toward lower scores. (correct answer)
  2. The distribution is right-skewed, with many low scores and a tail toward higher scores.
  3. The distribution is approximately symmetric because the most common score is 8.
  4. The distribution is uniform because all integer scores from 2 to 10 appear.
  5. The distribution is bimodal because there is a score of 2.

Explanation: This question tests identifying distribution shape from a dotplot of quiz scores. The dotplot shows many students scored high (8-10 points), with progressively fewer students earning lower scores, creating a tail extending toward the left (lower scores). This pattern indicates left-skewness, where the tail points toward smaller values. The distribution is not right-skewed (choice B) because the tail extends left not right, not symmetric (choice C) despite having a mode at 8, not uniform (choice D) because frequencies vary greatly, and not bimodal (choice E) since one low score doesn't create a second mode. Quiz scores often show left-skewness when most students perform well but a few struggle.

Question 7

A biology teacher recorded the number of seeds germinated (out of 50) for each of 32 trays under the same conditions. The dotplot below shows the distribution of germinated-seed counts. Which statement best describes the distribution?

  1. The distribution is strongly right-skewed because most trays have high counts and a few have much lower counts.
  2. The distribution is strongly left-skewed because most trays have high counts and a few have much lower counts.
  3. The distribution is approximately symmetric and unimodal, centered around about 41–43 seeds, with one unusually low tray near 28. (correct answer)
  4. The distribution is uniform because counts occur at many different values from 28 to 47.
  5. The distribution is bimodal because there is an outlier at 28.

Explanation: This question asks about describing a distribution of seed germination counts from a dotplot. The data shows most trays had 41-43 seeds germinate out of 50, forming a symmetric bell shape around this center, with one unusually low value at 28 that stands apart as an outlier. This creates an approximately symmetric, unimodal distribution with an outlier. The distribution is not right-skewed (choice A) or left-skewed (choice B) because the main cluster is symmetric, not uniform (choice D) because values cluster rather than spread evenly, and not bimodal (choice E) because an outlier doesn't create a second mode. When describing distributions, identify outliers separately from the overall shape of the main data cluster.

Question 8

A runner tracked the time (in minutes) it took to complete a particular 5K route on 26 different days. The dotplot below shows the distribution of times.

Which statement best describes the distribution?

  1. The distribution is approximately uniform from 20 to 40 minutes, indicating similar performance each day.
  2. The distribution is roughly symmetric and unimodal, centered around about 28–30 minutes, with no clear outliers. (correct answer)
  3. The distribution is strongly right-skewed because the maximum time is 40 minutes.
  4. The distribution is strongly left-skewed because the minimum time is 22 minutes.
  5. The distribution is bimodal with peaks near 24 minutes and 38 minutes.

Explanation: This question asks you to describe a distribution from a dotplot of running times. The dotplot shows times clustered around 28-30 minutes, with values spreading out fairly evenly on both sides (from 22 to about 36 minutes). The data appears to have a single peak and decreases relatively symmetrically on both sides, with no values standing far apart from the main cluster. This creates a roughly symmetric, unimodal distribution with no clear outliers. When assessing symmetry, imagine folding the distribution at its center - symmetric distributions look similar on both sides.

Question 9

A coach recorded the number of push-ups completed in one minute by 35 athletes. The dotplot below shows the distribution.

Which statement best describes the distribution?

  1. The distribution is bimodal, with two clusters around about 22–24 and 34–36 push-ups. (correct answer)
  2. The distribution is approximately symmetric because the minimum and maximum are equally far from the mean.
  3. The distribution is right-skewed, with most athletes near 35 push-ups and a tail toward higher values.
  4. The distribution is left-skewed, with most athletes near 22 push-ups and a tail toward smaller values.
  5. The distribution is uniform from 20 to 40 push-ups, indicating equal frequencies at all values.

Explanation: This question asks you to identify a bimodal distribution from a dotplot. The dotplot shows two distinct clusters of data: one group of athletes completed around 22-24 push-ups, and another separate group completed around 34-36 push-ups, with fewer athletes in between these clusters. This creates a bimodal distribution with two clear peaks. Bimodal distributions often suggest two different groups within the data - perhaps these athletes represent different fitness levels or training programs. When looking for modes, identify peaks or clusters in the data; bimodal means two distinct peaks.

Question 10

A school counselor recorded the number of minutes 36 students spent on homework last night. The dotplot below shows the distribution of times (in minutes). Which statement best describes the distribution?

  1. The distribution is approximately symmetric and unimodal, with no clear outliers.
  2. The distribution is strongly left-skewed because there are a few very large homework times.
  3. The distribution is bimodal because there is one value far from the rest.
  4. The distribution is strongly right-skewed, with a cluster around 30–45 minutes and a few unusually large values near 90–110 minutes. (correct answer)
  5. The distribution is uniform because the times are spread evenly from 20 to 110 minutes.

Explanation: This question tests your ability to describe the shape of a distribution from a dotplot. The data shows most students spent between 30-45 minutes on homework, with a few students spending much longer times (90-110 minutes). When most data clusters on the left with a tail extending to the right, we call this right-skewed. The distribution is not symmetric (choice A) because the tail extends only to the right, not bimodal (choice C) since there's only one main cluster, and not uniform (choice E) because values aren't evenly spread. When describing distributions, always look for the overall shape, center, spread, and any unusual features like outliers or gaps.