Summary Statistics for a Quantitative Variable - AP Statistics

1. A hospital summarized the waiting time (in minutes) for 75 patients in an urgent care clinic. The summaries were: mean $=58.0$, median $=49.0$, SD $=30.0$, $Q_1=32.0$, $Q_3=70.0$, minimum $=8$, maximum $=160$ (five-number summary: $8, 32, 49, 70, 160$). Which interpretation is correct?
   Answer: The distribution is likely right-skewed, with a typical wait near 49 minutes, because the mean exceeds the median and the upper tail is much longer. In AP Statistics, this tests understanding quantitative variable summaries, emphasizing shape and center. Choice B correctly describes right-skewness with typical wait near 49 (median), as mean (58) > median and upper tail (70 to 160) much longer than lower (8 to 32). Distractor A confuses middle 50% with min-max (8 to 160) instead of Q1-Q3. Key: For skew, median is robust center; compare tails in five-number summary. IQR (70-32=38) measures central spread. SD (30) shows high variability. Indicates right-skewed waits with long upper extremes.
2. A delivery service recorded the number of packages delivered per route in one day for 55 routes. The summaries were: mean $=98$, median $=95$, standard deviation $=15$, five-number summary (min $=60$, $Q_1=88$, median $=95$, $Q_3=106$, max $=140$), and IQR $=18$. Which interpretation is correct?
   Answer: The distribution is likely approximately symmetric because the mean and median are close and the quartiles are fairly balanced around the median. This question tests shape assessment from summary statistics for packages delivered. Mean (98) close to median (95), with quartiles somewhat balanced (Q1 to median = 7, median to Q3 = 11), suggests approximate symmetry, as in choice B. Choice A distracts by equating IQR (18) to range (80), but range is max - min. Mini-lesson: For symmetry, check if mean ≈ median and quartiles mirror around median; slight differences allow 'approximately symmetric.' Standard deviation (15) measures variability, not typical value, and five-number summary helps visualize tails (both ~30 units long here).
3. A nutritionist measured the sodium content (in mg) of 40 frozen meals sold at a grocery store. The summaries were: mean $=720$, median $=710$, SD $=95$, five-number summary $(520,\ 650,\ 710,\ 780,\ 900)$, and IQR $=130$. Which interpretation is correct?
   Answer: The middle 50% of meals have between 650 and 780 mg of sodium, and the distribution appears roughly symmetric because the mean and median are close and the tails (520 to 650 and 780 to 900) are similar in length. This question tests recognition of approximately symmetric distributions. The five-number summary (520, 650, 710, 780, 900) shows Q1=650 and Q3=780, placing the middle 50% of meals between 650 and 780 mg of sodium. The distribution appears roughly symmetric because the mean (720) and median (710) are very close, and the tails have similar lengths (520 to 650 = 130 mg, 780 to 900 = 120 mg). Choice A misinterprets SD as a typical sodium value. Choice B confuses IQR with a specific range. Choice D incorrectly claims half the data falls between median and Q3. Choice E reverses the interpretation of Q3.
4. A university summarized the number of credits earned in the first year by a random sample of 80 first-year students. The summaries were: mean $=28.5$ credits, median $=30$ credits, standard deviation $=4.2$ credits, five-number summary (min $=12$, $Q_1=26$, median $=30$, $Q_3=32$, max $=36$), and IQR $=6$ credits. Which interpretation is correct?
   Answer: The middle 50% of students earned between $26$ and $32$ credits in their first year. The skill is interpreting quartiles and IQR for credits earned, a quantitative variable. Q1 = 26 and Q3 = 32 mean the middle 50% of students earned between 26 and 32 credits, as choice A correctly interprets. Choice E is a distractor, wrongly claiming 75% earned fewer than Q1 (26), but Q1 is the 25th percentile. In a mini-lesson, quartiles split data: 25% below Q1, 50% below median, 75% below Q3; IQR (6 credits) is the spread of this middle half. Mean (28.5) < median (30) suggests left skew, and standard deviation (4.2) quantifies spread, not a 'typical' value.
5. An environmental scientist recorded daily rainfall (in mm) on 90 days during a season. The summaries were: mean = 6.8 mm, median = 2.1 mm, SD = 12.4 mm, five-number summary = (min 0.0, $Q_1$ 0.0, median 2.1, $Q_3$ 7.0, max 85.0), and IQR = 7.0 mm. Which interpretation is correct?
   Answer: Because the mean is much larger than the median and the maximum is far above $Q_3$, the distribution is strongly right-skewed with occasional very rainy days. This question illustrates extreme right skewness in environmental data. The mean (6.8 mm) being much larger than the median (2.1 mm) indicates strong right skewness, dramatically confirmed by the maximum (85.0 mm) being far above Q3 (7.0 mm). With Q1=0.0 mm, at least 25% of days had no rainfall, typical for seasonal precipitation patterns. The IQR of 7.0 mm represents the spread of the middle 50% of data, not the total range. The large standard deviation (12.4 mm) reflects high variability due to occasional heavy rain days. This distribution shows how a few extreme values can greatly affect the mean while the median better represents typical daily rainfall.
6. A fitness app summarized the number of steps taken in one day for a random sample of 120 adult users. The summaries were: mean $=7800$ steps, median $=8200$ steps, SD $=2100$ steps, $Q_1=6500$, $Q_3=9300$, minimum $=1200$, maximum $=14{,}500$ (five-number summary: $1200, 6500, 8200, 9300, 14500$). Which interpretation is correct?
   Answer: The distribution is likely left-skewed because the mean is less than the median, and the lower tail extends farther below $Q_1$ than the upper tail extends above $Q_3$. This AP Statistics question evaluates interpreting summary statistics for a quantitative variable, emphasizing shape, center, and spread. Choice A is correct, identifying left-skewness because the mean (7800) is less than the median (8200), with the lower tail extending farther (1200 to 6500) than the upper (9300 to 14500). A distractor like choice B errs by using the range (14500-1200=13300) instead of the proper IQR (9300-6500=2800) to describe the middle 50% spread. When analyzing summaries, use the median as the center for skewed data, as it's resistant to extremes. The five-number summary helps visualize shape: a longer lower whisker suggests left-skew. Standard deviation (2100) quantifies overall variability but isn't the typical value. This setup indicates a left-skewed distribution with a long tail of low step counts.
7. An environmental scientist measured ozone concentration (in parts per billion, ppb) at noon on 75 summer days in one city. The summaries were: mean $=54.3$ ppb, median $=50$ ppb, SD $=18.7$ ppb, five-number summary $(22,\ 39,\ 50,\ 62,\ 118)$, and IQR $=23$ ppb. Which interpretation is correct?
   Answer: Because the mean is greater than the median and the maximum is far above $Q_3$, the distribution is likely right-skewed with some unusually high ozone days. This question evaluates understanding of skewness indicators in environmental data. The mean (54.3) exceeds the median (50), suggesting right skewness. The maximum (118) is extremely far from Q3 (62), creating a difference of 56 ppb, while the minimum (22) is only 17 ppb below Q1 (39). This asymmetry confirms right skewness with outliers on the high end. Choice A correctly identifies the IQR property. Choice B misinterprets SD as a percentage. Choice C confuses IQR with range. Choice E makes an absolute claim contradicted by the data spread.
8. A school district recorded the number of minutes 120 high school students spent on homework last night. The summaries were: mean = 62.4 min, median = 55 min, SD = 28.1 min, five-number summary = (min 10, $Q_1$ 40, median 55, $Q_3$ 75, max 160), and IQR = 35 min. Which interpretation is correct?
   Answer: Because the mean is greater than the median and the maximum is far above $Q_3$, the distribution is likely right-skewed, with typical middle 50% between 40 and 75 minutes. This question tests understanding of how summary statistics reveal distribution shape and typical values. The mean (62.4) exceeding the median (55) indicates right skewness, as the mean is pulled toward higher values by extreme observations like the maximum of 160 minutes. The IQR of 35 minutes tells us the middle 50% of students did homework between Q1=40 and Q3=75 minutes, not that the total range is 35 minutes. The standard deviation of 28.1 minutes measures typical deviation from the mean, not a typical homework time. When interpreting medians, remember that 50% fall below and 50% fall above, not that half fall between the median and maximum.
9. A library recorded the number of books checked out per visit for 75 patrons on a weekday. The summaries were: mean = 3.6 books, median = 3 books, SD = 2.2 books, five-number summary = (min 0, $Q_1$ 2, median 3, $Q_3$ 5, max 12), and IQR = 3. Which interpretation is correct?
   Answer: Because the mean is slightly larger than the median and the maximum is well above $Q_3$, the distribution is likely right-skewed, with a few patrons checking out many books. This question examines mild right skewness in count data. The mean (3.6 books) being slightly larger than the median (3 books) suggests right skewness, confirmed by the maximum (12 books) being well above Q3 (5 books). This indicates a few patrons checking out many books pull the mean upward. The IQR of 3 correctly measures the spread between Q1=2 and Q3=5, representing the middle 50% of the data. The standard deviation of 2.2 books measures variability, not that patrons typically checked out 2.2 books. The range is actually 12 books (from 0 to 12), not 3 books. Understanding these distinctions helps correctly interpret summary statistics for count data.
10. A city measured the one-way commute time (in minutes) for 60 randomly selected workers. The summary statistics were: mean $=31.2$, median $=30.5$, SD $=6.1$, $Q_1=27.0$, $Q_3=35.0$, minimum $=18$, maximum $=45$ (five-number summary: $18, 27, 30.5, 35, 45$). Which interpretation is correct?
    Answer: The IQR is $35-27=8$ minutes, meaning the middle 50% of commute times span about 8 minutes. In AP Statistics, this question tests knowledge of summary statistics for quantitative variables, including measures of spread like the IQR. The correct choice B properly calculates IQR as 35-27=8 minutes, correctly noting it spans the middle 50% of commute times. Choice A is a distractor, claiming strong right-skewness based solely on max > min, which is always true and ignores the close mean (31.2) and median (30.5) suggesting mild skew if any. A mini-lesson: IQR measures central spread, resistant to outliers, unlike range (45-18=27). Compare mean and median for shape; here they're similar, indicating near-symmetry. SD (6.1) shows average deviation from mean, not the typical value. These stats suggest a fairly symmetric distribution with low variability.
11. A school district summarized the number of absences per student during a semester for 200 randomly selected high school students. The summaries were: mean $=6.8$ days, median $=4$ days, standard deviation $=7.5$ days, five-number summary (min $=0$, $Q_1=1$, median $=4$, $Q_3=9$, max $=45$), and IQR $=8$ days. Which interpretation is correct?
    Answer: The distribution is likely right-skewed, with some students having very large numbers of absences. This question tests the interpretation of summary statistics for absences, a quantitative variable, emphasizing shape inference from center measures and extremes. The mean of 6.8 days is greater than the median of 4 days, and the maximum of 45 days is far beyond Q3 (9 days), indicating right skewness with outliers in absences. Choice B accurately captures this likely right-skewed distribution. A distractor like choice A confuses the IQR of 8 days with the range of most values, but IQR only covers the middle 50%. For a mini-lesson, compare mean and median to assess skewness: mean > median suggests right skew due to high outliers pulling the mean up. The five-number summary reveals the spread, with the standard deviation of 7.5 days showing overall variability, not a 'typical' value as sometimes misinterpreted.
12. A teacher summarized scores (out of 100) on a quiz for 95 students. The statistics were: mean $=78$, median $=82$, SD $=10$, five-number summary $(35,\ 72,\ 82,\ 88,\ 96)$, and IQR $=16$. Which interpretation is correct?
    Answer: The distribution is likely left-skewed because the mean is less than the median and the lower tail (down to 35) extends farther than the upper tail (up to 96). This question tests recognition of left-skewed distributions. The distribution is left-skewed because the mean (78) is less than the median (82), indicating the tail pulls the mean downward. Additionally, the lower tail extends much farther (35 to 72 = 37 points) than the upper tail (88 to 96 = 8 points). Choice B confuses IQR with range. Choice C misinterprets SD as a typical score. Choice D treats Q1 as a frequency rather than a position. Choice E incorrectly claims half the data falls between the median and maximum. When mean < median and the lower tail is longer, the distribution exhibits left skewness.
13. A realtor tracked the selling prices (in thousands of dollars) of 50 homes in a neighborhood. The summary statistics were: mean $= 365$, median $= 340$, standard deviation $= 110$, five-number summary $(180, 260, 340, 420, 820)$, and IQR $= 160$. Which interpretation is correct?
    Answer: Because the mean exceeds the median and the maximum is much larger than $Q_3$, the distribution is likely skewed right due to a few very expensive homes. This question assesses understanding of right-skewed distributions in real estate data. The mean ($365,000) substantially exceeding the median ($340,000) is a strong indicator of right skew, where expensive homes pull the mean upward. The maximum value ($820,000) being much larger than Q3 ($420,000) confirms the presence of outliers on the high end - luxury homes that are much more expensive than typical. Common misinterpretations in other options include: IQR represents the spread of the middle 50%, not a typical price; standard deviation doesn't define absolute boundaries for all data; the middle 50% (not 75%) of homes sold between Q1 and Q3; and the median indicates that half sold for less than and half for more than $340,000, not that half sold for exactly that amount.
14. A market researcher surveyed 100 households and recorded monthly spending on streaming services (in dollars). The summaries were: mean $=27.4$, median $=22.0$, standard deviation $=18.6$, five-number summary (min $=0$, $Q_1=12$, median $=22$, $Q_3=35$, max $=95$), and IQR $=23$. Which interpretation is correct?
    Answer: The distribution is likely right-skewed because the mean is greater than the median and the upper tail extends far beyond $Q_3$. Interpreting summaries for spending involves skewness from centers and tails. Mean 27.4 > median 22.0, with upper tail (max - Q3 = 60) longer than lower (Q1 - min = 12), indicates right skewness, correctly in choice B. Choice E distracts, saying about 12% spend ≤ Q1 (12), but it's 25%. Mini-lesson: Right skew when mean > median due to high values; five-number summary shows asymmetry. IQR (23) is robust spread of middle 50%, unlike standard deviation (18.6) affected by extremes.
15. A university summarized the number of credits earned by 200 first-year students in their first semester. The summaries were: mean = $14.2$ credits, median = $15.0$ credits, SD = $3.1$ credits, $Q_1=12.0$, $Q_3=16.0$, minimum = $3$, maximum = $19$ (five-number summary: $3, 12, 15, 16, 19$). Which interpretation is correct?
    Answer: The distribution is likely left-skewed because the mean is less than the median and there is a long lower tail down to $3$ credits. This question in AP Statistics examines summary statistics interpretation for a quantitative variable, focusing on shape. Correct choice A states left-skewness as mean ($14.2$) < median ($15.0$) with longer lower tail ($3$ to $12$) than upper ($16$ to $19$). Distractor D errs on Q3 ($16$), claiming 75% earned more than 16, but actually 75% earned ≤16. Lesson: Mean pulled toward skew tail; use median for center in asymmetry. IQR ($16-12=4$) shows tight central spread. SD ($3.1$) indicates variability, not typical credits. Distribution shows left-skew with low-credit outliers.
16. A company recorded the number of emails received per day by 40 employees. The summary statistics were: mean $= 58$, median $= 57$, standard deviation $= 11$, five-number summary $(32, 50, 57, 65, 83)$, and IQR $= 15$. Which interpretation is correct?
    Answer: Because the mean and median are nearly equal and the five-number summary is fairly balanced, the distribution is likely roughly symmetric. This question tests understanding of symmetric distributions and their characteristics. The mean (58) and median (57) being nearly equal is a key indicator of a roughly symmetric distribution. Additionally, the five-number summary shows reasonable balance - the distances from the median to Q1 and Q3 are similar (7 and 8 respectively), and the extremes are roughly equidistant from the quartiles. Common misconceptions in other options include: IQR measures the spread of the middle 50%, not the total range; standard deviation doesn't define exact boundaries for data; Q1=50 means 25% (not 75%) received fewer than 50 emails; and the median indicates the middle value when data is ordered, not that all employees received that exact number.
17. A streaming service recorded the length (in minutes) of 110 movies watched by a random sample of users during one weekend. The summaries were: mean $=104.8$ min, median $=98$ min, SD $=29.5$ min, five-number summary $(41,\ 83,\ 98,\ 120,\ 210)$, and IQR $=37$ min. Which interpretation is correct?
    Answer: Because the mean is greater than the median and the maximum is much larger than $Q_3$, the distribution of movie lengths is likely right-skewed. This question evaluates understanding of right-skewed distributions in entertainment data. The mean (104.8) substantially exceeds the median (98), indicating right skewness. The maximum (210) is extremely far from Q3 (120), a difference of 90 minutes, while the minimum (41) is only 42 minutes below Q1 (83). This strong asymmetry confirms right skewness with long movies as outliers. Choice A incorrectly equates IQR with range. Choice B misinterprets SD as a typical length. Choice D reverses the quartile interpretation - Q3 means 75% were shorter than 120 minutes. Choice E makes an absolute claim contradicted by the wide spread.
18. A teacher recorded quiz scores (out of 20 points) for 30 students. The summary statistics were: mean $= 14.2$, median $= 15.5$, standard deviation $= 3.6$, five-number summary $(2, 13, 15.5, 18, 20)$, and IQR $= 5$. Which interpretation is correct?
    Answer: Because the mean is less than the median and the minimum is far below $Q_1$, the distribution is likely skewed left due to a few very low scores. This question tests recognition of left-skewed distributions in quiz score data. The mean (14.2) being less than the median (15.5) indicates left skew, where the distribution has a tail extending toward lower scores. The minimum value (2) being far below Q1 (13) confirms this - there are a few students with unusually low scores pulling the mean downward. The distractors contain typical misconceptions: IQR measures the spread of the middle 50% of scores, not the number of students; standard deviation indicates typical deviation from the mean, not actual scores; Q3=18 means 75% scored 18 or lower, not 18 or higher; and the range is calculated as max-min = 20-2 = 18, not 5.
19. A fitness researcher recorded the number of minutes of moderate-to-vigorous exercise completed in one week by a random sample of 60 adults in a city wellness program. The summary statistics were: mean $=172$ min, median $=160$ min, standard deviation $=80$ min, five-number summary (min $=20$, $Q_1=110$, median $=160$, $Q_3=210$, max $=420$), and IQR $=100$ min. Which interpretation is correct?
    Answer: The distribution is likely right-skewed because the mean exceeds the median and the maximum is far above $Q_3$. This question assesses the skill of interpreting summary statistics for a quantitative variable, focusing on shape, center, and spread in the context of exercise minutes. The mean of 172 minutes exceeds the median of 160 minutes, indicating a right-skewed distribution, and the maximum of 420 minutes is much farther from Q3 (210 minutes) than the minimum (20 minutes) is from Q1 (110 minutes), supporting this skewness. Choice A correctly identifies this right-skewed shape. A common distractor is choice E, which mistakenly claims left skewness because the median appears closer to Q3, but the distances from the median to Q1 and Q3 are equal (50 minutes each), and the mean-median relationship is key. In a mini-lesson, remember that in right-skewed distributions, the mean is pulled toward the longer right tail, making it larger than the median, while the five-number summary helps visualize the spread and potential outliers. The IQR of 100 minutes describes the spread of the middle 50% of data, resistant to extremes unlike the standard deviation of 80 minutes.
20. A school district recorded the number of minutes it took each of 80 students to finish a standardized reading assessment. The summary statistics were: mean $=42.8$ min, median $=40.0$ min, SD $=9.6$ min, $Q_1=35.0$ min, $Q_3=47.0$ min, minimum $=22$ min, maximum $=78$ min (so the five-number summary is $22, 35, 40, 47, 78$). Which interpretation is correct about the distribution's center, spread, and shape?
    Answer: The typical completion time is about $40$ minutes, and the distribution is likely right-skewed since the mean is greater than the median and the upper tail extends farther. This question assesses understanding of summary statistics for a quantitative variable, focusing on center, spread, and shape in AP Statistics. The correct interpretation is choice B, which accurately states that the typical completion time is about 40 minutes (the median) and infers a right-skewed distribution since the mean (42.8) exceeds the median (40.0), with the upper tail extending farther (from 47 to 78) compared to the lower tail (from 22 to 35). A common distractor is choice C, which mistakenly calculates the IQR as 78-22=56 instead of the correct Q3-Q1=47-35=12, confusing range with interquartile range. In interpreting summaries, remember that for skewed distributions, the median better represents the center than the mean, as the mean is pulled toward the tail. The standard deviation (9.6) measures variability around the mean, but doesn't indicate typical values directly. Shape can be inferred by comparing mean and median, and examining whisker lengths in the five-number summary. Overall, these statistics paint a picture of a right-skewed distribution with moderate spread.

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