Describing the Distribution of Quantitative Variables

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AP Statistics › Describing the Distribution of Quantitative Variables

Questions 1 - 10
1

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?

The distribution is strongly right-skewed because 130 is larger than 90.

The distribution is uniform because the range is about 40 seeds.

The distribution is approximately symmetric and unimodal, centered near about 110 seeds, with no apparent outliers.

The distribution is strongly left-skewed because the smallest values are below 100.

The distribution is bimodal because there are values on both sides of 110.

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.

2

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?

The distribution is uniform because all integer scores from 2 to 10 appear.

The distribution is left-skewed, with many high scores and a tail toward lower scores.

The distribution is bimodal because there is a score of 2.

The distribution is approximately symmetric because the most common score is 8.

The distribution is right-skewed, with many low scores and a tail toward higher scores.

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.

3

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?

The distribution is strongly right-skewed because most trays have high counts and a few have much lower counts.

The distribution is strongly left-skewed because most trays have high counts and a few have much lower counts.

The distribution is uniform because counts occur at many different values from 28 to 47.

The distribution is bimodal because there is an outlier at 28.

The distribution is approximately symmetric and unimodal, centered around about 41–43 seeds, with one unusually low tray near 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.

4

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?

The distribution is bimodal with peaks near 24 minutes and 38 minutes.

The distribution is strongly right-skewed because the maximum time is 40 minutes.

The distribution is approximately uniform from 20 to 40 minutes, indicating similar performance each day.

The distribution is strongly left-skewed because the minimum time is 22 minutes.

The distribution is roughly symmetric and unimodal, centered around about 28–30 minutes, with no clear outliers.

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.

5

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?

The distribution is left-skewed, with most athletes near 22 push-ups and a tail toward smaller values.

The distribution is right-skewed, with most athletes near 35 push-ups and a tail toward higher values.

The distribution is approximately symmetric because the minimum and maximum are equally far from the mean.

The distribution is bimodal, with two clusters around about 22–24 and 34–36 push-ups.

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.

6

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?

The distribution is approximately symmetric and unimodal, with no clear outliers.

The distribution is strongly left-skewed because there are a few very large homework times.

The distribution is bimodal because there is one value far from the rest.

The distribution is uniform because the times are spread evenly from 20 to 110 minutes.

The distribution is strongly right-skewed, with a cluster around 30–45 minutes and a few unusually large values near 90–110 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.

7

A track coach recorded the 100-meter dash times (in seconds) for 28 athletes. The dotplot below shows the distribution. Which statement best describes the distribution?

The distribution is approximately symmetric and unimodal, centered around about 13 seconds.

The distribution is uniform because the times cover a range from about 11.5 to 14.5 seconds.

The distribution is bimodal because there is a single value near 14.5 seconds.

The distribution is strongly left-skewed because there are a few very slow times.

The distribution is strongly right-skewed because there are a few very fast times.

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.

8

A gym records the number of minutes 60 members spent on a treadmill during one visit. The dotplot below shows the distribution of times (in minutes).

Which statement best describes the distribution?

Dotplot counts by minute:

  • 10: 1
  • 12: 1
  • 14: 2
  • 16: 3
  • 18: 4
  • 20: 6
  • 22: 7
  • 24: 8
  • 26: 7
  • 28: 6
  • 30: 5
  • 32: 4
  • 34: 3
  • 36: 2
  • 38: 1

(Each count represents how many members had exactly that time.)

The distribution is uniform because the times range from 10 to 38 minutes.

The distribution is left-skewed with a single high outlier near 38 minutes.

The distribution is strongly right-skewed because the largest value is 38 minutes.

The distribution is bimodal because there are values at both 20 and 30 minutes.

The distribution is approximately symmetric and unimodal, centered around about 24–26 minutes, with no clear outliers.

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.

9

A company tracks the number of customer support tickets received each day for 30 days. The dotplot shows most days between 18 and 26 tickets, but there are two days with 55 and 58 tickets, far from the rest.

Which statement best describes the distribution?

The distribution is approximately symmetric because there are values on both sides of 22 tickets.

The distribution is uniform because the tickets vary from day to day.

The distribution is left-skewed because the smallest days are below 18 tickets.

The distribution is bimodal because there are two unusually large values.

The distribution is right-skewed with two high outliers around 55–58 tickets.

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.

10

A bakery tracks the number of cookies sold per hour on a Saturday (12 hours). The counts are:

18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 60

Which statement best describes the distribution of cookies sold per hour?

The distribution is approximately symmetric with center near 21 and no outliers.

The distribution is right-skewed because of a single unusually large value (60) compared with the rest.

The distribution is bimodal because 21 occurs three times.

The distribution is uniform because the values from 18 to 23 all appear.

The distribution is left-skewed because most values are in the low 20s.

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.

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