AP Statistics Quiz: Comparing Distributions Of A Quantitative Variable

What this quiz covers

This quiz focuses on Comparing Distributions Of A Quantitative Variable, 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. Express has a higher median and more high outliers than Standard.
2. Standard has a higher median and larger IQR; only Standard shows high outliers. (correct answer)
3. Standard has a lower median but a larger IQR; both show high outliers.
4. Both options have about the same median, but Express has a larger range.
5. Express has a lower median and a larger IQR than Standard.

Explanation: This question examines comparing delivery time distributions with five-number summaries and outliers in AP Statistics. The Standard option has a higher median (55 vs. 42) and larger IQR (35 vs. 25), indicating longer typical times with more variability, plus several high outliers above 140 only in Standard. Express shows no outliers, suggesting more consistency. Choice C distracts by claiming Standard has a lower median, which contradicts the data. In distribution comparisons, evaluate center (median), spread (IQR), and outliers; here, Standard's outliers inflate its max and range. Mini-lesson: Outliers can indicate skewness or anomalies—plot them on boxplots; higher medians with larger IQRs often mean a group is slower but more unpredictable.

Question 2

1. Class B has a higher median and a larger IQR than Class A.
2. Class A has a higher median, but both classes have the same IQR.
3. Both classes have the same median; Class B has a slightly larger IQR. (correct answer)
4. Both classes have the same median; Class A has a larger IQR.
5. Class B has a lower median and a smaller overall range than Class A.

Explanation: This question tests comparing quiz score distributions via five-number summaries in AP Statistics. Both classes share a median of 15, showing similar typical scores, but Class B has a slightly larger IQR (7 vs. 5), indicating more variability in the middle 50%. The ranges are identical (14 for both), and mins/maxs are close. Choice A distracts by claiming Class B has a higher median, which is false. Comparing distributions involves assessing center (same medians here), spread (larger IQR for B suggests more dispersion), and shape (both seem symmetric). Mini-lesson: Compute IQR as $Q_3 - Q_1$ to quantify spread; equal medians don't imply identical distributions—check spreads and extremes for full insights.

Question 3

1. Class 2 has a higher typical score because it has more students total.
2. Class 1 appears to have a higher center than Class 2, with Class 2 having more low scores. (correct answer)
3. Class 2 has a higher center because it has more scores in 3–5 than Class 1.
4. The classes have the same center because both have students scoring 9–10.
5. Class 1 is more variable because it has fewer scores in the middle bin 6–8.

Explanation: This question assesses comparing quiz score distributions between two classes using binned counts, focusing on center and shape. Class 1 has more scores in higher bins (10 in 6-8, 5 in 9-10) versus Class 2 (6 in 6-8, 4 in 9-10), suggesting Class 1's higher center, while Class 2 has more low scores (3 in 0-2, 7 in 3-5) indicating a left-skewed shape with lower typical scores. Both have 20 students, so counts directly compare; spreads are similar across bins. Choice C distracts by claiming Class 2 has higher center due to more in 3-5, overlooking the overall shift toward higher bins in Class 1. In distribution comparisons, estimate center from where data clusters, compare spreads by bin coverage, and note shapes like skewness; binned data helps visualize without individual values.

Question 4

1. Brand Y has a higher median lifetime than Brand X, and Brand Y has a larger IQR.
2. Brand X has a slightly higher median lifetime than Brand Y, and the spreads (IQRs) are the same. (correct answer)
3. Brand X is more variable because it has a smaller range.
4. Brand X and Brand Y have the same median because their IQRs overlap.
5. Brand Y has a larger range and therefore a higher center.

Explanation: This problem focuses on comparing battery lifetime distributions for Brand X and Brand Y using boxplots and five-number summaries. Brand X has a median of 14 hours, slightly higher than Y's 13, showing a marginally higher center, and both have identical IQRs of 8 hours (X: 18-10, Y: 17-9), indicating similar middle 50% spreads. The ranges are close (X: 16, Y: 15), with no outliers, supporting comparable variability. Choice A tempts by reversing the medians and claiming Y has larger IQR, possibly from miscalculating or swapping groups. Remember, in comparing quantitative distributions, prioritize center (median for skewed data), spread (IQR resists outliers), and note any unusual features like outliers; here, the slight difference in medians and equal IQRs are key.

Question 5

1. Driving has a higher median commute time than transit, and driving has a larger IQR.
2. Transit has a higher median commute time than driving, and transit has a larger IQR. (correct answer)
3. The two groups have similar centers because their minimums are close.
4. Driving is more variable because its maximum is smaller.
5. Transit must have outliers because its maximum is 90.

Explanation: Comparing commute time distributions for driving and transit users involves five-number summaries to evaluate center and variability. Transit's median is 40 minutes, higher than driving's 25, showing longer typical commutes, and transit's IQR of 30 (55-25) exceeds driving's 17 (35-18), indicating greater middle spread; transit's higher maximum (90 vs. 55) also suggests more variability. No outliers are noted, and shapes aren't detailed, but centers and spreads differ clearly. A distractor is choice A, which reverses the medians and IQRs, possibly from confusing group labels. Mini-lesson: for quantitative variables, compare centers (medians), spreads (IQR for robustness), and outliers; boxplots visually aid in seeing overlaps or shifts in distributions.

Question 6

1. West has a higher median car age than East, and West has a smaller range.
2. East has a higher median car age than West, and West shows greater overall spread due to a high outlier. (correct answer)
3. East is more variable because it has no outliers.
4. The medians must be the same because the IQRs overlap.
5. West has a smaller IQR than East because its maximum is larger.

Explanation: This question involves comparing car age distributions in two neighborhoods using five-number summaries to identify differences in center and spread. East has a higher median age of 6 years versus West's 4, indicating older typical cars, while West's high outlier at 20 years increases its overall spread (range 20-0=20) compared to East's (14-1=13). IQRs are similar (East: 9-3=6, West: 7-2=5), but the outlier affects West's total variability. Choice A distracts by reversing the medians and claiming West has smaller range, disregarding the outlier's extension. In comparing distributions, use median for center, IQR for consistent spread, and consider outliers separately as they can inflate range without representing typical variability.

Question 7

1. Beach has a higher median and a larger IQR than Park. (correct answer)
2. Park has a higher median and a larger IQR than Beach.
3. Beach has a lower median, but Park has a smaller IQR.
4. Both events have the same median, but Park has a larger overall range.
5. Park has a higher median, and Beach has a larger overall range.

Explanation: This question assesses comparing trash weight distributions with five-number summaries in AP Statistics. The Beach event has a higher median (14 vs. 12 lbs) and larger IQR (12 vs. 10 lbs), indicating more typical trash per volunteer with greater middle variability. Park has a larger range (39 vs. 33) due to its max. Choice B distracts by assigning higher median and IQR to Park, which is incorrect. Comparing distributions requires examining center (median for typical value), spread (IQR for consistency), and extremes; both seem right-skewed. Mini-lesson: Higher medians suggest overall shifts; larger IQRs mean more dispersion in core data—visualize with boxplots to see overlaps or differences.

Question 8

1. Brand Y has a higher mean and a higher median than Brand X.
2. Brand X and Brand Y have the same center because their IQRs are equal.
3. Brand Y has a slightly higher mean, but Brand X has a higher median; the IQRs are equal. (correct answer)
4. Brand X is right-skewed and Brand Y is left-skewed, so Brand X must have a lower mean.
5. Brand X has a larger IQR, so it must have the larger mean.

Explanation: This question evaluates comparing battery life distributions using means, medians, shapes, and IQRs in AP Statistics. Brand Y has a slightly higher mean (10.1 vs. 9.8 hours) but lower median (9.9 vs. 10.2), with equal IQRs of 1.4, reflecting shape influences—Y is right-skewed, X left-skewed. Skewness explains why Y's mean exceeds its median, unlike X. Choice A distracts by claiming Y has both higher mean and median, ignoring the reversal. In comparisons, note how shape affects center measures; equal IQRs indicate similar variabilities. Mini-lesson: In skewed distributions, medians resist tail pulls unlike means—left-skew pulls mean below median, right-skew above; compare both for insights.

Question 9

1. 12th grade has a higher median and a larger IQR than 9th grade.
2. 9th grade has a higher median and a larger IQR; only 9th grade shows high outliers. (correct answer)
3. Both grades have the same median, but 12th grade has more high outliers.
4. 9th grade has a lower median, but 12th grade has a larger overall range.
5. Both grades have the same IQR, but 9th grade has a higher maximum.

Explanation: This question tests comparing absence distributions with five-number summaries and outliers in AP Statistics. The 9th grade has a higher median (3 vs. 2) and larger IQR (5 vs. 3), showing more typical absences with greater variability, plus several high outliers above 15 only in 9th. 12th has a smaller range (12 vs. 25). Choice A is a distractor, wrongly giving higher median and IQR to 12th. Comparing involves center (median), spread (IQR), and outliers; 9th appears more right-skewed due to outliers. Mini-lesson: Outliers can extend ranges—identify them using 1.5*IQR rule; higher medians with larger IQRs suggest a group has more issues overall.

Question 10

1. Low-flow has a higher center (mean and median) and the same IQR as Standard.
2. Standard has a higher center (mean and median), and the IQRs are the same. (correct answer)
3. Standard has a higher center, and Low-flow has a smaller range.
4. Both groups have the same center, but Standard has a larger IQR.
5. Low-flow has a lower median but a larger IQR than Standard.

Explanation: This question focuses on comparing water use distributions using means, medians, IQRs, and ranges in AP Statistics. The Standard group has higher center measures (mean 230 vs. 210, median 225 vs. 205), suggesting greater typical usage, with identical IQRs of 60 indicating similar middle spreads. The ranges differ slightly (240 vs. 260), but this isn't emphasized in the supported comparison. Choice A is a distractor, incorrectly giving Low-flow the higher center. To compare distributions, use mean and median for center (noting skewness if they differ), IQR for spread resistant to outliers, and range for overall variability; here, both groups have comparable spreads. Mini-lesson: When means and medians align in direction, it supports consistent center differences; equal IQRs mean similar variability in the central data, regardless of extremes.

Question 11

1. Weekend has a higher median and a larger IQR than Weekday. (correct answer)
2. Weekend has a higher median but a smaller IQR than Weekday.
3. Weekday has a higher median, and Weekend has the smaller overall range.
4. Both have the same median, but Weekend has a larger overall range.
5. Weekday has a larger IQR, so it must have a higher maximum.

Explanation: This question focuses on comparing customer count distributions using five-number summaries in AP Statistics. Weekends have a higher median (30 vs. 25) and larger IQR (18 vs. 14), indicating more typical customers with greater hourly variability. Weekends also have a larger range (60 vs. 33). Choice B distracts by claiming Weekends have smaller IQR, which is false. To compare, assess center (median for typical), spread (IQR for middle), and range; both seem right-skewed. Mini-lesson: Higher Q1, median, Q3 suggest an overall shift upward; larger IQRs mean less predictability—use these to inform business decisions.

Question 12

1. Adults have a higher center because their highest relative frequency is in 30–59.
2. Teens tend to have higher sugar intake than adults, with more mass in the higher bins. (correct answer)
3. Adults are more variable because they have more in the lowest bin 0–29.
4. The two groups have the same center because both have some intake in 120–149.
5. Teens have a lower center because they have less in 30–59 than adults.

Explanation: Comparing sugar intake distributions for teens and adults uses relative-frequency histograms to assess center and shape differences. Teens show higher center with more mass in upper bins (0.35 in 60-89, 0.25 in 90-119, 0.15 in 120-149) compared to adults (0.25, 0.08, 0.02), while adults have more in lower bins (0.25 in 0-29, 0.40 in 30-59), suggesting teens' higher typical intake. Spreads appear similar, but teens' distribution is right-skewed toward higher values. Choice E is a distractor, claiming teens have lower center due to less in 30-59, but this ignores their concentration in higher bins overall. A mini-lesson: in histogram comparisons, estimate center from weighted bin middles, compare spreads by bin range, and note shape; relative frequencies allow fair comparisons across group sizes.

Question 13

1. Honors has a lower median homework time than Regular, and Honors has a larger IQR.
2. Honors has a higher median homework time than Regular, and Regular shows greater overall spread due to a high outlier. (correct answer)
3. The two groups have about the same center because their IQRs overlap.
4. Regular has a higher median homework time than Honors, and neither group has outliers.
5. Honors and Regular have the same IQR, but Regular has a lower maximum.

Explanation: This question tests the skill of comparing distributions of a quantitative variable, specifically homework times between Honors and Regular students using five-number summaries. The median for Honors is 50 minutes, higher than Regular's 30 minutes, indicating a higher center for Honors, while Regular's distribution shows greater overall spread due to the high outlier at 120 minutes, which extends its range beyond Honors' maximum of 90. The interquartile ranges are 30 minutes for Honors (65-35) and 25 minutes for Regular (45-20), but the outlier in Regular increases its overall variability. A common distractor is choice A, which incorrectly states Honors has a lower median, perhaps from misreading the summaries or confusing groups. In comparing distributions, always examine center (median), spread (IQR and range), shape, and outliers to draw supported conclusions; here, the data supports Honors having a higher typical homework time with Regular's spread influenced by an outlier.

Question 14

1. Section A has a higher median than Section B, and Section A has no outliers.
2. Section B has a slightly higher median than Section A, and Section A shows a low outlier. (correct answer)
3. Section A and Section B have the same center because their medians are close, so no other differences are supported.
4. Section A has a larger IQR than Section B because its minimum is smaller.
5. Section B is more variable because it has a higher maximum.

Explanation: This problem compares exam score distributions between two course sections using five-number summaries, emphasizing center and outliers. Section B's median is 75, slightly higher than A's 74, indicating a marginally higher typical score, while Section A has a low outlier at 52, which may pull its minimum down without affecting the IQR much (A: 82-68=14, B: 84-66=18). Ranges are similar (A: 95-52=43, B: 96-60=36), but the outlier highlights A's greater low-end spread. Choice A distracts by claiming A has higher median and no outliers, ignoring the given low outlier. When comparing distributions, focus on median for center, IQR for spread, and identify outliers as they can indicate unusual variability or skewness in the data.

Question 15

1. Farm 2 has a higher median apple weight than Farm 1, and Farm 2 has a larger IQR.
2. Farm 1 has a higher median apple weight than Farm 2 because Farm 1 has a higher $Q_3$.
3. The farms have about the same median apple weight, but Farm 2 has greater overall spread due to a high outlier. (correct answer)
4. Farm 1 is more variable because it has no outliers.
5. Farm 2 has a smaller range, so its distribution is less spread out.

Explanation: Comparing apple weight distributions from two farms uses five-number summaries to evaluate center and spread, noting any outliers. Both farms have the same median of 0.32 pounds, suggesting similar typical weights, but Farm 2 shows greater overall spread due to its high outlier at 0.60 pounds, extending the range to 0.42 (0.60-0.18) versus Farm 1's 0.25 (0.45-0.20). IQRs are close (Farm 1: 0.36-0.28=0.08, Farm 2: 0.35-0.27=0.08), so the middle spreads are comparable. Choice A is a distractor, claiming Farm 2 has higher median and larger IQR, possibly from misreading summaries or confusing Q3 with median. Mini-lesson: for quantitative distributions, compare centers first (medians equal here), then spreads (IQR similar, range differs due to outlier), and highlight outliers; equal medians don't imply identical distributions if spreads vary.

Question 16

1. The No warm-up group has a higher median and a smaller IQR, and it has a high outlier.
2. The Warm-up group has a higher median and a smaller IQR; only the No warm-up group shows a high outlier. (correct answer)
3. Both groups have about the same median, but the Warm-up group has a larger IQR.
4. The Warm-up group has a lower median, and both groups show high outliers.
5. The Warm-up group has a higher median, but the No warm-up group has a smaller IQR.

Explanation: This question evaluates comparing distributions of push-up counts using boxplot features like medians, IQRs, and outliers in AP Statistics. The Warm-up group has a higher median (42 vs. 36), indicating better typical performance, and a smaller IQR (10 vs. 14), showing less variability in the middle half. Only the No warm-up group has a noted high outlier around 70, which could skew its distribution. A distractor like choice A reverses the groups, incorrectly assigning higher median and smaller IQR to No warm-up. When comparing distributions, examine center (median for robustness), spread (IQR for middle variability), and unusual features like outliers; here, the Warm-up group appears more consistent without extremes. Mini-lesson: Boxplots visually compare groups—shorter boxes mean smaller IQRs, and dots indicate outliers; always check if outliers affect overall interpretations.