Representing Data with Number Line Plots - Statistics and Probability

A class recorded the number of pages read over a weekend by 15 students: 18, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 28, 30, 32. Which plot best represents the data as a histogram using bins of width 5: 15–19, 20–24, 25–29, 30–34?
Answer: 15–19: 1, 20–24: 7, 25–29: 5, 30–34: 2 This question requires creating a histogram with specific bins. A histogram groups data into intervals (bins) and shows the frequency with bars. For bins 15-19, 20-24, 25-29, 30-34, we count how many values fall in each range. From our data: 18 falls in 15-19 (1 value); 20, 21, 22, 22, 23, 24, 24 fall in 20-24 (7 values); 25, 26, 27, 28, 28 fall in 25-29 (5 values); 30, 32 fall in 30-34 (2 values). Answer A correctly shows these frequencies: 15-19: 1, 20-24: 7, 25-29: 5, 30-34: 2. A common error is including boundary values in the wrong bin - remember that 24 goes in the 20-24 bin, not 25-29. Always check that your total frequency equals the number of data points (1+7+5+2=15).
The quiz scores (out of 20) for 16 students are: 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 15, 16, 17, 18, 19. Which statement correctly describes the distribution shown by a histogram with equal-width bins?
Answer: The distribution is roughly symmetric, with most scores in the middle (about 12–15). This question asks about a histogram of quiz scores. A histogram groups data into bins (intervals) and shows counts as bar heights. Looking at the 16 scores from 8 to 19, they spread fairly evenly across the range with slight clustering in the middle values (12-15). This creates a roughly symmetric shape, not skewed to either side. There's no large gap between 13 and 14—both values appear in the data. A common mistake is confusing gaps in the data with gaps between consecutive integers. To verify symmetry: compare counts in bins equidistant from the center—they should be similar for a symmetric distribution.
The quiz scores (out of 20) for 13 students were: 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17. Which plot best represents the data as a dot plot on a number line?
Answer: Dots at 9(1), 10(1), 11(2), 12(3), 13(1), 14(2), 15(1), 16(1), 17(1) This question asks for a dot plot of quiz scores. From the data: 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, we count frequencies for each value. Score 11 appears twice, 12 appears three times, and 14 appears twice, while all others appear once. Answer A correctly shows dots at 9(1), 10(1), 11(2), 12(3), 13(1), 14(2), 15(1), 16(1), 17(1). The distribution shows a concentration around scores 11-14 with single values extending to both ends. A common plotting error is losing track when counting repeated values - use tally marks or cross off values as you count them. To verify, ensure your total dot count equals 13 (1+1+2+3+1+2+1+1+1=13).
A PE teacher recorded the number of push-ups completed by 11 students in one minute: 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 21. Which plot best represents the data as a dot plot on a number line?
Answer: Dots at 14(1), 15(2), 16(3), 17(1), 18(2), 19(1), 21(1). A dot plot shows push-up counts as dots on a number line, with stacks like three at 16 and two at 15, 18. The data maps to dots at 14 (one), 15 (two), 16 (three), 17 (one), 18 (two), 19 (one), 21 (one), creating a cluster around 16 with tails. The plot features a peak at 16 and slight right skew to 21, spanning 14 to 21. Choice A correctly lists these frequencies for the visual representation. A common misconception is overlooking stacked dots as single points, but stacks indicate multiples. Check the number line scale from 14 to 21. Then, count dots per value to match the 11 students.
A cafeteria recorded the prices (in dollars) of 12 lunch items: 1.25, 1.50, 1.50, 1.75, 2.00, 2.00, 2.25, 2.50, 2.50, 2.75, 3.00, 3.25. Which plot best represents the data as a dot plot (each value shown as one dot above its price)?
Answer: Dots at: 1.25(1), 1.50(2), 1.75(1), 2.00(2), 2.25(1), 2.50(2), 2.75(1), 3.00(1), 3.25(1). The plot is a dot plot, displaying prices on a number line. Each price is mapped by placing a dot above its value, stacking for duplicates. Key features include clusters at 1.50, 2.00, 2.50 with 2 dots each, spread from 1.25 to 3.25, and a fairly uniform shape with increasing prices. The correct answer A matches because it accurately lists the positions and counts of dots for each price in the data. A common misconception is to forget to stack dots for repeated values, leading to incorrect counts. To analyze, first check the scale on the number line for the price range. Then, count the dots at each price to match the frequency in the data set.
A band director recorded the lengths (in minutes) of 13 practice sessions: 18, 20, 20, 22, 24, 25, 25, 25, 26, 28, 30, 30, 32. Which statement correctly describes the distribution shown by a dot plot?
Answer: The distribution has a cluster at 25 minutes and is otherwise spread without large gaps. This question analyzes a dot plot of practice session lengths. Looking at the 13 values, three sessions were 25 minutes (creating a cluster), two were 20 minutes, two were 30 minutes, with other values spread between. The data shows no large gaps—values progress fairly steadily from 18 to 32 minutes. There's no outlier at 40 minutes, and values exist between 24 and 28 (specifically 24, 25, 26, 28). A common error is overinterpreting small gaps as significant breaks in the distribution. To identify true gaps: look for intervals where multiple consecutive values are missing, not just single missing integers.
Daily high temperatures (in $^\circ$ F) for 14 days were: 62, 63, 63, 64, 65, 66, 66, 67, 68, 69, 70, 71, 71, 72. Which statement correctly describes the distribution shown by a histogram using 2-degree bins (62–63, 64–65, 66–67, 68–69, 70–71, 72–73)?
Answer: The histogram would show moderate counts across bins, with a slight cluster in the middle (66–69). This question asks about a histogram with specific 2-degree bins. Sorting the 14 temperatures into bins: 62-63 has 3 values, 64-65 has 2 values, 66-67 has 4 values, 68-69 has 2 values, 70-71 has 2 values, and 72-73 has 1 value. This shows moderate counts across bins with slight clustering in the middle (66-69 range). The histogram would have bars at all intervals with no gaps. A common mistake is thinking bin boundaries exclude certain values, but 66-67 includes both 66 and 67. To create a histogram: count how many data points fall into each bin, then draw bars with heights matching these counts.
A counselor recorded the number of colleges each of 20 seniors applied to: 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 8. Which statement correctly describes the distribution when shown in a box plot?
Answer: The distribution is right-skewed because the upper whisker extends farther (up to 8) than the lower whisker (down to 1). A box plot displays college applications on a number line with min 1, Q1 2.5, median 4, Q3 5.5, max 8, showing a box from 2.5 to 5.5 and longer upper whisker. Data values map to this summary, highlighting more spread in the upper half. The shape is right-skewed due to the extended upper whisker and tail to 8. Choice A accurately describes this skew from the farther upper extension. A common misconception is assuming symmetry if the box is even, but whisker lengths indicate skew. Ensure the scale accommodates integers from 1 to 8. Then, count values in each quartile section to validate the summary.
A student measured the number of pages read each day for 14 days: 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 18, 19, 20. Which plot best represents the data as a dot plot on a number line?
Answer: Dots at 12(2), 13(2), 14(3), 15(2), 16(2), 18(1), 19(1), 20(1). A dot plot displays each page count as a dot on a number line, stacking dots for repeats like three at 14. The data maps to dots at 12 (two), 13 (two), 14 (three), 15 (two), 16 (two), 18 (one), 19 (one), and 20 (one), showing clusters around 14-16 with tails on both sides. The plot reveals a slight right skew with more spread on the higher end and overall range from 12 to 20. Choice A accurately lists these frequencies, matching the plotted dots. A common misconception is miscounting stacks as separate values, but each stack represents repeats at one point. Begin by verifying the number line scale fits the min 12 to max 20. Then, tally the dots per value to confirm total data points equal 14.
A cafeteria tracked the waiting times (in minutes) for 15 students: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9. Which plot best represents the data as a histogram using bins of width 2: 2–3, 4–5, 6–7, 8–9?
Answer: 2–3: 3, 4–5: 5, 6–7: 4, 8–9: 3 This question requires creating a histogram with bins of width 2. For waiting times data: 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, we count values in each bin. For 2-3: values 2, 3, 3 give frequency 3. For 4-5: values 4, 4, 4, 5, 5 give frequency 5. For 6-7: values 6, 6, 6, 7 give frequency 4. For 8-9: values 8, 8, 9 give frequency 3. Answer A correctly shows 2-3: 3, 4-5: 5, 6-7: 4, 8-9: 3. The distribution appears roughly symmetric with a slight peak in the middle bin. A common error is forgetting that bin boundaries include both endpoints - the bin 2-3 includes both 2 and 3. Always verify your total: 3+5+4+3=15 matches our 15 data points.
A science class measured temperatures (in $^\circ$ C) of a liquid at 10 different times: 18.2, 18.4, 18.5, 18.5, 18.7, 18.8, 18.8, 19.0, 19.1, 19.3. Which plot best represents the data as a histogram with bin width 0.5: 18.0–18.4, 18.5–18.9, 19.0–19.4?
Answer: Bars: 18.0–18.4 has 2 values; 18.5–18.9 has 5 values; 19.0–19.4 has 3 values. A histogram bins temperatures on a number line with 0.5 intervals, placing values like 18.2 and 18.4 in 18.0–18.4 (two), five in 18.5–18.9, and three in 19.0–19.4. Data maps to bars showing a peak in the middle bin and shorter adjacent bars, with spread from 18.2 to 19.3. Key features include a cluster around 18.5–18.9 and slight asymmetry. Choice A matches the precise counts per bin for accurate representation. A common misconception is misplacing decimal values across bin boundaries, but here 18.5 goes into 18.5–18.9 correctly. Check the decimal scale on the number line for bin accuracy. Then, count temperatures in each bin to confirm total of 10 measurements.
A track coach recorded the times (in seconds) for 12 students to run a 100-meter sprint: 12.1, 12.4, 12.4, 12.6, 12.8, 12.8, 12.9, 13.0, 13.2, 13.2, 13.5, 13.7. Which plot best represents the data as a dot plot on a number line?
Answer: Dots at 12.1(1), 12.4(2), 12.6(1), 12.8(2), 12.9(1), 13.0(1), 13.2(2), 13.5(1), 13.7(1) This question asks us to create a dot plot for sprint times. In a dot plot, each data value gets its own dot placed above that value on a number line. Looking at our data: 12.1, 12.4, 12.4, 12.6, 12.8, 12.8, 12.9, 13.0, 13.2, 13.2, 13.5, 13.7, we need to count how many times each value appears. The values 12.4, 12.8, and 13.2 each appear twice, while all others appear once. Answer A correctly shows dots at 12.1(1), 12.4(2), 12.6(1), 12.8(2), 12.9(1), 13.0(1), 13.2(2), 13.5(1), 13.7(1), where the number in parentheses indicates how many dots. A common mistake is miscounting repeated values - always list your data first and mark off each value as you plot it. To verify, count that you have exactly 12 dots total, matching our 12 data points.
The number of text messages received in a day by 20 students were: 0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 14, 16, 18. Which statement correctly describes the distribution shown by a histogram of these counts?
Answer: The distribution is right-skewed because many students received few messages and fewer students received many messages. This question asks about a histogram of text message counts. Looking at the 20 values, many students (about 12) received 0-7 messages, while fewer students received higher counts, with values spreading out to 18. This creates a right-skewed distribution—the bulk of data is on the left (low values) with a tail extending right (high values). The distribution is continuous from 0 to 18 with no gap between 4 and 12. A common misconception is confusing the direction of skew: right-skewed means the tail points right, not that most data is on the right. To identify skew: look where most data clusters versus where the tail extends.
A PE teacher recorded the number of push-ups completed by 12 students in one minute: 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26. Which plot best represents the data as a box plot (using the five-number summary)?
Answer: Min 18, $Q_1$ 20, Median 22, $Q_3$ 24, Max 26 This question requires finding the five-number summary for push-up data. With 12 values: 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, we find: Min=18, Max=26, Median=(6th+7th)/2=(22+22)/2=22. For quartiles with 12 values, Q1 is the median of the lower 6 values=(3rd+4th)/2=(20+20)/2=20, and Q3 is the median of the upper 6 values=(9th+10th)/2=(24+24)/2=24. Answer A correctly shows Min 18, Q1 20, Median 22, Q3 24, Max 26. The box plot would show the data is fairly symmetric with the median centered in the box. A common error is using the wrong positions for quartiles - with an even number of values, carefully identify which values belong to each half. Always verify that Q1 < Median < Q3.
A science class measured the pH of 14 water samples: 6.4, 6.5, 6.6, 6.6, 6.7, 6.8, 6.8, 6.9, 7.0, 7.0, 7.1, 7.2, 7.3, 7.4. Which plot best represents the data as a dot plot on a number line marked in tenths from 6.4 to 7.4?
Answer: A dot plot with two dots at 6.6, two dots at 6.8, two dots at 7.0, and one dot at each of the other listed values. This question tests understanding of dot plot construction. A dot plot places one dot above each data value on the number line. Counting the pH values: 6.6 appears twice, 6.8 appears twice, 7.0 appears twice, and all other values appear once. This matches option A exactly. Option B incorrectly shows single dots where there should be two. A common error is adding extra dots for emphasis or rounding values to whole numbers. To create a dot plot: mark the scale on the number line, then place one dot above the line for each occurrence of that value, stacking dots vertically when values repeat.
A store recorded 18 customer wait times (in minutes): 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 14. Which plot best represents the data as a histogram using bin width 5 minutes with bins 0–4, 5–9, 10–14?
Answer: Histogram counts: 0–4: 8, 5–9: 7, 10–14: 3. The plot is a histogram of wait times with bin width 5. Data values are counted into bins 0–4, 5–9, 10–14. Key features include high frequency in 0–4 with 8, decreasing to 3 in 10–14, spread from 1 to 14, right-skewed shape. The correct answer A matches because the counts are exactly 8,7,3 for the bins. A common misconception is to include boundary values in the wrong bin, like putting 5 in 0–4 instead of 5–9. To analyze, first check the scale and bin definitions. Then, count the points in each bin to confirm the heights.
A cafeteria tracked the waiting times (in minutes) for 16 students: 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10, 11. Which statement correctly describes the distribution when shown in a histogram with 2-minute bins: 2–3, 4–5, 6–7, 8–9, 10–11?
Answer: The histogram would show a cluster in the 6–7 bin and a smaller right tail toward 10–11. A histogram groups waiting times into 2-minute bins on a number line, with bar heights showing counts like one in 2–3, five in 4–5, five in 6–7, three in 8–9, and two in 10–11. Data values fall into bins, creating a visual of higher bars in the middle and tapering off, especially to the right. Key features include a cluster around 6–7 minutes with a right tail extending to 10–11, indicating slight right skew and spread from 3 to 11. Choice A correctly describes this cluster and tail, aligning with the bin frequencies. A common misconception is thinking empty bins mean gaps in data, but here all bins have values except possibly unused ones outside the range. First, check the bin scale to ensure intervals like 2–3 capture values correctly. Then, count entries per bin to match the total of 16 times.
A student measured the mass (in grams) of 10 apples: 148, 150, 151, 152, 152, 153, 154, 156, 158, 160. Which statement correctly describes the distribution shown by a box plot of these data?
Answer: The median is 152.5 g, and the range is 12 g. This question involves a box plot of apple masses. With 10 ordered values, the median is between the 5th and 6th values: (152+153)/2 = 152.5 grams. The range is maximum minus minimum: 160-148 = 12 grams. Box plots show the five-number summary: minimum (148), Q1, median (152.5), Q3, and maximum (160). The box extends from Q1 to Q3 with a line at the median, while whiskers reach to the extremes. A common mistake is using a single middle value as the median with even sample sizes. To find the median: order all values, then average the two middle ones for even-sized datasets.
A band director recorded the 20 practice times (in minutes) for students over a week: 5, 10, 10, 15, 15, 15, 20, 20, 20, 20, 25, 25, 30, 30, 35, 40, 45, 50, 55, 60. Which statement correctly describes the distribution shown by a histogram of these times?
Answer: The distribution is right-skewed because most times are near 10–30 with a tail of larger times up to 60. The plot is a histogram of practice times. Data values are binned, with bars showing frequencies. Key features include cluster in lower times 10-30, spread from 5 to 60, right-skewed shape with long tail to higher times. The correct answer C matches because most are near 10–30 with a tail up to 60. A common misconception is to see it as uniform when there is clear skew due to the tail. To analyze, first check the scale and bin sizes. Then, count points per bin to see the distribution shape.
Daily high temperatures (°F) over 14 days were: 62, 64, 65, 65, 66, 66, 67, 68, 68, 69, 70, 71, 72, 74. Which statement correctly describes the distribution shown by a box plot of these temperatures?
Answer: The median is 67.5°F, and the range is 12°F. The plot is a box plot, summarizing the temperature data. The data values map to the plot by calculating quartiles, median, and extremes for the box and whiskers. Key features include the median at 67.5°F, box from Q1 to Q3 showing middle 50%, and whiskers to min 62°F and max 74°F, with no outliers. The correct answer A matches because the median is the average of 7th and 8th values (67+68)/2=67.5, and range is 74-62=12. A common misconception is to take the median as one of the middle values instead of averaging for even number of data points. To analyze, first check the scale to see the range of temperatures. Then, count the data points and sort them to verify the median and range.

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