AP Statistics Quiz: Describing The Distribution Of Quantitative Variables

What this quiz covers

This quiz focuses on Describing The Distribution Of Quantitative Variables, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

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

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

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

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

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

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

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

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

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

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.

Question 11

1. The distribution is approximately symmetric and unimodal, centered around about 13 seconds. (correct answer)
2. The distribution is strongly right-skewed because there are a few very fast times.
3. The distribution is strongly left-skewed because there are a few very slow times.
4. The distribution is uniform because the times cover a range from about 11.5 to 14.5 seconds.
5. The distribution is bimodal because there is a single value near 14.5 seconds.

Explanation: This question asks about describing a distribution of sprint times from a dotplot. The data appears to cluster around 13 seconds with roughly equal numbers of times slightly faster and slightly slower, creating a bell-shaped pattern. This indicates an approximately symmetric, unimodal distribution. The distribution is not right-skewed (choice B) or left-skewed (choice C) because there's no pronounced tail in either direction, not uniform (choice D) because values cluster in the middle rather than spreading evenly, and not bimodal (choice E) which would require two distinct peaks. Symmetric distributions have similar shapes on both sides of the center and are common for athletic performance data.

Question 12

1. The distribution is strongly right-skewed because the largest value is 38 minutes.
2. The distribution is approximately symmetric and unimodal, centered around about 24–26 minutes, with no clear outliers. (correct answer)
3. The distribution is bimodal because there are values at both 20 and 30 minutes.
4. The distribution is left-skewed with a single high outlier near 38 minutes.
5. The distribution is uniform because the times range from 10 to 38 minutes.

Explanation: This question assesses the skill of describing the distribution of a quantitative variable, specifically treadmill times, by examining shape, center, spread, and outliers in a dotplot. The distribution is approximately symmetric and unimodal, with the peak around 24-26 minutes and no clear outliers, as the data mirror evenly on both sides without isolated points. A common distractor is choice A, which incorrectly identifies the distribution as strongly right-skewed solely based on the largest value being 38 minutes, ignoring the overall symmetry. In describing distributions, remember to look for symmetry where the left and right sides are mirror images, skewness where there's a tail on one side, and modality based on the number of peaks. Outliers are values that stand apart from the main body of data, which aren't present here. Always consider the entire pattern rather than isolated points.

Question 13

1. The distribution is approximately symmetric because there are values on both sides of 22 tickets.
2. The distribution is right-skewed with two high outliers around 55–58 tickets. (correct answer)
3. The distribution is left-skewed because the smallest days are below 18 tickets.
4. The distribution is uniform because the tickets vary from day to day.
5. The distribution is bimodal because there are two unusually large values.

Explanation: Describing distributions in AP Statistics for variables like daily tickets requires evaluating dotplots for shape, center, spread, and outliers. The distribution is right-skewed, with most values between 18-26 tickets but two high outliers at 55 and 58, creating a tail to the right. Choice C distracts by calling it left-skewed based on low values below 18, but the tail is actually on the high end. Mini-lesson: Outliers are extreme values that don't fit the pattern; right skew means the mean is pulled rightward by high extremes. Symmetry has balanced tails, while bimodality shows two modes—here, the two high points aren't a separate peak but outliers. Uniformity implies even spread without clustering.

Question 14

1. The distribution is approximately symmetric with center near 21 and no outliers.
2. The distribution is left-skewed because most values are in the low 20s.
3. The distribution is right-skewed because of a single unusually large value (60) compared with the rest. (correct answer)
4. The distribution is bimodal because 21 occurs three times.
5. The distribution is uniform because the values from 18 to 23 all appear.

Explanation: This problem evaluates describing distributions for quantitative data like cookies sold per hour, using listed values to determine shape, center, spread, and outliers. The distribution is right-skewed due to a single high outlier at 60, while most values cluster in the low 20s, creating a tail to the right. Distractor D calls it bimodal because 21 occurs three times, but bimodality requires two separate peaks, not repeated values in one cluster. Mini-lesson: Skewness is identified by the longer tail's direction; outliers like 60 stand out and affect measures like the mean more than the median. Symmetry has mirrored sides, uniformity equal frequencies across ranges, and left skew tails low. Always check for isolated extremes.

Question 15

1. The distribution is left-skewed because there are no unusually fast times.
2. The distribution is right-skewed, with a few possible high outliers among the slowest times. (correct answer)
3. The distribution is symmetric because most times are between 430 and 470 seconds.
4. The distribution is uniform because the times cover a range from 430 to 565 seconds.
5. The distribution is bimodal because there are three slow times.

Explanation: Describing quantitative distributions, such as mile times in a dotplot, involves assessing shape, center, spread, and outliers for training runs. The distribution is right-skewed with a dense cluster from 430-470 seconds and a few high outliers at slower times like 520-565, but no low outliers. Choice A distracts by claiming left skew due to no fast times, but left skew requires a tail toward lower values, which is absent. Lesson: Right skew pulls the mean higher with high extremes; outliers are isolated from the cluster. Symmetry balances both sides, uniformity has even density, and bimodality needs two peaks—not just a few extremes. Focus on tail direction for skew.

Question 16

1. The distribution is approximately symmetric because the commutes range from 2 to 40 miles.
2. The distribution is right-skewed because there are many short commutes and fewer long commutes, creating a tail to the right. (correct answer)
3. The distribution is left-skewed because the smallest commute distances are close to 0.
4. The distribution is uniform because the histogram uses the same bin width for all intervals.
5. The distribution is bimodal because there are both short and long commutes.

Explanation: In AP Statistics, describing commute distances via histograms requires analyzing shape, center, spread, and outliers for driving data. The distribution is right-skewed with most commutes short (2-12 miles) and a decreasing tail to longer distances up to 40 miles. Choice C is a distractor, labeling it left-skewed because small distances are near 0, but left skew has the tail on the low end, not high. Key lesson: Right skew is common in distances or times with minimums near zero and long tails; bimodality needs two peaks, not just varied lengths. Uniformity requires flat bars, symmetry balanced tails, and histograms' bin widths don't determine shape alone. Assess frequency patterns for skew.

Question 17

1. The distribution is bimodal, suggesting two clusters of heights. (correct answer)
2. The distribution is approximately symmetric because heights are measured on a continuous scale.
3. The distribution is right-skewed because the tallest students are far from the shortest.
4. The distribution is left-skewed because the histogram has a dip in the middle.
5. The distribution is uniform because there are values across many height intervals.

Explanation: AP Statistics skills include describing distributions of variables like heights via histograms, noting shape, center, spread, and outliers. The distribution is bimodal with two distinct peaks at 61-63 and 67-69 inches, suggesting subgroups like genders, and a dip in between. Distractor C mislabels it as right-skewed by noting the range from short to tall, but bimodality is about multiple peaks, not just spread. Mini-lesson: Bimodal distributions often indicate mixed populations; symmetry lacks peaks and tails imbalance. Left skew would tail toward lower values, uniformity means flat frequencies, and continuous scales don't inherently imply symmetry. Look for clusters to identify modality.

Question 18

1. The distribution is approximately symmetric with center near 80 cm and no outliers because most data are near 80 cm.
2. The distribution is left-skewed because the smallest wingspans are far from the center.
3. The distribution is uniform because the histogram uses equal-width bins.
4. The distribution is right-skewed, and the observation in 104–108 cm appears to be a possible high outlier. (correct answer)
5. The distribution is bimodal because there is a bar at 104–108 cm.

Explanation: This problem tests describing quantitative distributions, here wingspans, using a histogram to identify shape, center, spread, and outliers. The distribution is right-skewed with a main peak around 78-82 cm and a long tail to the right, including a possible high outlier in the 104-108 cm bin. Distractor A wrongly claims symmetry because most data are near 80 cm, but ignores the asymmetric tail. Key lesson: Skewness is determined by the direction of the longer tail—right for higher values—and outliers are points detached from the main cluster, often checked with rules like 1.5 times the interquartile range. Uniform distributions have flat, equal bars, not a peak and tail. Bimodality requires two clear peaks, not a single bar far out.

Question 19

1. Approximately symmetric, because the mean and median are both measures of center.
2. Skewed to the right, because the mean is larger than the median.
3. Skewed to the left, because the mean is smaller than the median. (correct answer)
4. Approximately uniform, because the difference between the mean and median is small.

Explanation: The correct answer is C. The mean is sensitive to extreme values (outliers) and is pulled in the direction of the tail of a skewed distribution. Since the mean (55) is substantially less than the median (68), it indicates that there are low values pulling the mean down. This pattern is characteristic of a distribution that is skewed to the left.

Question 20

1. The mean will be substantially greater than the median.
2. The mean will be substantially less than the median.
3. The mean and the median will be approximately equal. (correct answer)
4. The mean will be equal to the mode, but the median may be different.

Explanation: The correct answer is C. In a symmetric distribution, the data are balanced around the center. As a result, the mean (the balance point) and the median (the middle value) are located at the same central position and will be approximately equal.