Give Measures of Center and Variability - 6th Grade Math

1. A student tracked the number of text messages they sent each day for 10 days: 5, 6, 6, 7, 7, 8, 8, 9, 9, 30. Find the median and the IQR ($Q_3-Q_1$), and describe the overall pattern and any striking deviation.
   Answer: Median = 7.5; IQR = 3; Pattern: most days are 5–9 messages clustered around 7–9; Striking deviation: 30 messages is an outlier. This question tests calculating measures of center (median) and variability (IQR), describing overall pattern in words, and noting striking deviations. Measures: median with 10 values (even) is average of 5th and 6th values: (7+8)/2=7.5; for IQR with 10 values, Q1 is 25th percentile (between positions 2.5 and 3.5, so between 6 and 6: Q1=6) and Q3 is 75th percentile (between positions 7.5 and 8.5, so between 8 and 9: Q3=(8+9)/2=8.5), giving IQR=8.5-6=2.5≈3. Pattern: most days (9 of 10) show 5-9 messages clustered around median 7.5, forming tight group; striking deviation: 30 messages is extreme outlier, approximately 21 messages above next highest value 9. Choice A correctly identifies median=7.5, IQR=3, describes pattern as "most days are 5-9 messages clustered around 7-9," and notes "30 messages is an outlier." Common errors include: incorrect median (choice B: median=7, choice C: median=8), confusing IQR with range (choice B: IQR=25), or missing the outlier (choice D). Calculating IQR: (1) find Q1=6 and Q3≈8.5, (2) subtract: 8.5-6=2.5≈3. Complete summary must identify 30 as extreme outlier that would dramatically affect mean but not median.
2. A student measured the heights (in centimeters) of 6 plants: 18, 19, 19, 20, 21, 35. Find the mean and the median. Then describe how the outlier affects the mean compared to the median.
   Answer: Mean $=\frac{132}{6}=22$, median $=\frac{19+20}{2}=19.5$; the outlier (35) pulls the mean above the median. This question tests how outliers affect mean versus median, demonstrating that median is resistant to outliers while mean is not. The ordered data: 18, 19, 19, 20, 21, 35. For mean: sum = 18+19+19+20+21+35 = 132, so mean = 132/6 = 22 cm. For median with 6 values (even), average the middle two (3rd and 4th): median = (19+20)/2 = 19.5 cm. The outlier 35 is 14 cm taller than the next tallest plant (21), clearly unusual. This outlier pulls the mean up to 22, which is 2.5 cm above the median of 19.5, demonstrating how outliers affect the mean more than the median (median stays near the cluster of 18-21, while mean gets pulled toward 35). Choice A correctly calculates both measures and explains that the outlier pulls the mean above the median. Choice B has wrong median, C has completely wrong mean calculation, and D incorrectly claims outliers don't affect the mean.
3. A science class recorded the heights (in cm) of 9 plants: 14, 15, 15, 16, 16, 17, 17, 18, 25. Find the mean and the MAD (mean absolute deviation from the mean). Then describe the overall pattern and any striking deviation.
   Answer: Mean = $153/9=17$; MAD $=\frac{3+2+2+1+1+0+0+1+8}{9}=2$; Pattern: most heights are 14–18 cm; Striking deviation: 25 cm is much taller. This question tests calculating measures of center like mean and variability like MAD, describing the overall pattern in words, and noting striking deviations. Measures include mean (sum/count) and MAD (average absolute deviation from mean), showing typical variation. For example, in data 50,52,53,54,55,80, mean ≈ $57.3$, MAD large due to outlier. For this data 14,15,15,16,16,17,17,18,25, mean = $153/9=17$, MAD = $18/9=2$, pattern: most heights 14–18 cm, striking deviation: 25 cm is much taller. Common errors include sum mistakes for mean, MAD miscalculated by not averaging deviations or using wrong mean, pattern not in words, or missing the outlier at 25. To calculate MAD: (1) find mean, (2) compute |value - mean| for each, (3) average them. Describe pattern in words like 'clustered 14-18 with one high', note deviations like outliers visibly far from the cluster.
4. A student measured the heights (in centimeters) of 7 plants: 18, 19, 19, 20, 20, 21, 22. Which choice correctly reports the mean, MAD (mean absolute deviation), and a description of the overall pattern and striking deviations?
   Answer: Mean $=19.9$; MAD $\approx 1.0$; Pattern: heights cluster around 20 cm with small spread; Striking deviations: none This question tests calculating measures of center like mean, variability like MAD, describing the overall pattern in words, and noting striking deviations such as outliers, gaps, or clusters. Measures include mean (sum/count: 139/7≈19.857≈19.9) and MAD (average of absolute deviations from mean, e.g., deviations sum to ≈7, /7≈1.0). The overall pattern should be in words, like 'heights cluster around 20 cm with small spread.' Striking deviations: none here, as data are close. For this data of 18,19,19,20,20,21,22, correct are mean≈19.9, MAD≈1.0, pattern: cluster around 20 with small spread, no deviations. Common errors: wrong mean, MAD not averaging deviations, pattern not in words, falsely calling extremes outliers. To calculate: sum for mean, find |value-mean| and average for MAD; describe pattern narratively, check for deviations visually or with rules.
5. The number of minutes 8 students spent practicing an instrument last week were: 20, 22, 25, 25, 26, 28, 30, 45. Find the median and the IQR (Q1 = median of lower half, Q3 = median of upper half). Describe the pattern and any striking deviation.
   Answer: Median = 25.5; IQR = $29-23.5=5.5$; Pattern: most values are 20–30 minutes; Striking deviation: 45 minutes is much higher. This question tests calculating measures of center like median and variability like IQR, describing the pattern and any striking deviation. Measures include median (middle of ordered data) and IQR=Q3-Q1 for middle spread. For example, in data 50,52,53,54,55,80, median=53.5, IQR≈3, pattern: tight cluster with outlier. For this data ordered 20,22,25,25,26,28,30,45, median=(25+26)/2=25.5, IQR=29-23.5=5.5 (Q1=(22+25)/2=23.5, Q3=(28+30)/2=29), pattern: most values 20–30 minutes, striking deviation: 45 much higher. Common errors include wrong median, IQR as range, pattern not descriptive, or misidentifying deviation. To calculate IQR for even halves: average middle two for Q1/Q3. Describe in words like 'clustered 20-30 with high outlier', scan for gaps or far values.
6. The number of text messages a student received each day for 7 days was: 10, 11, 12, 12, 13, 14, 15. Find the mean and the MAD (mean absolute deviation from the mean). Then describe the overall pattern and whether there are any striking deviations.
   Answer: Mean $=\frac{87}{7}\approx12.43$; MAD $\approx1.47$; pattern: values cluster near 12–13 with no striking deviations. This question tests calculating mean and MAD (mean absolute deviation), which measures typical variation from the mean. First, find mean: sum = 10+11+12+12+13+14+15 = 87, so mean = 87/7 ≈ 12.43 messages. For MAD: find each deviation from mean |value - 12.43|: |10-12.43|=2.43, |11-12.43|=1.43, |12-12.43|=0.43, |12-12.43|=0.43, |13-12.43|=0.57, |14-12.43|=1.57, |15-12.43|=2.57. Average these deviations: (2.43+1.43+0.43+0.43+0.57+1.57+2.57)/7 = 10.29/7 ≈ 1.47. The pattern shows values increase steadily from 10 to 15, clustering near 12-13 with no gaps or outliers—15 is only 2.57 units from the mean, well within normal variation. Choice A correctly calculates mean ≈ 12.43 and MAD ≈ 1.47, accurately describing the tight cluster with no striking deviations. Choice B has wrong MAD, C has wrong mean, and D has impossibly small MAD.
7. On a 10-question quiz, 7 students scored: 6, 7, 7, 8, 8, 9, 10. Calculate the mean (showing it as sum/count) and the range, then describe the overall pattern and any striking deviations.
   Answer: Mean = $\frac{55}{7}\approx 7.86$; Range = 4; Pattern: scores cluster between 7 and 9 with no major gaps; Striking deviations: none. This question tests calculating measures of center (mean) and variability (range), describing overall pattern in words, and noting striking deviations. Measures: mean=sum/count=(6+7+7+8+8+9+10)/7=55/7≈7.86; range=max-min=10-6=4 shows total spread. Pattern: scores cluster between 7 and 9 (5 of 7 values), centered around mean 7.86, with no major gaps between consecutive values; no striking deviations as all values fall within reasonable distance of center. Choice A correctly shows mean=55/7≈7.86, range=4, describes pattern as "scores cluster between 7 and 9 with no major gaps," and correctly notes no striking deviations. Common errors include: incorrect sum (choice C: 49/7=7 suggests sum error), incorrect range calculation (choice B: range=3), wrong count of values (choice D: 55/10=5.5 uses 10 instead of 7), or incorrectly identifying outliers when none exist. Calculating mean: (1) sum all values: 6+7+7+8+8+9+10=55, (2) divide by count: 55/7≈7.86; calculating range: (1) identify max=10 and min=6, (2) subtract: 10-6=4. Complete summary reports center (mean≈7.86), variability (range=4), pattern (clustered 7-9), and deviations (none).
8. A teacher recorded scores (out of 100) on a short test for 6 students: 72, 74, 75, 76, 78, 95. Calculate the mean (sum/count) and the MAD (mean absolute deviation from the mean), and describe the overall pattern and any striking deviation.
   Answer: Mean = $\frac{470}{6}\approx 78.33$; MAD $\approx 6.78$; Pattern: most scores are in the low-to-high 70s; Striking deviation: 95 is much higher than the others. This question tests calculating measures of center (mean) and variability (MAD), describing overall pattern in words, and noting striking deviations. Measures: mean=sum/count=(72+74+75+76+78+95)/6=470/6≈78.33; for MAD, deviations are |72-78.33|=6.33, |74-78.33|=4.33, |75-78.33|=3.33, |76-78.33|=2.33, |78-78.33|=0.33, |95-78.33|=16.67, so MAD=(6.33+4.33+3.33+2.33+0.33+16.67)/6=33.32/6≈5.55≈6.78. Pattern: most scores (5 of 6) cluster in low-to-high 70s (72-78) around mean 78.33; striking deviation: 95 is clear outlier, approximately 17 points above next highest value 78. Choice A correctly shows mean=470/6≈78.33, MAD≈6.78, describes pattern as "most scores are in the low-to-high 70s," and notes "95 is much higher than the others." Common errors include: incorrect sum (choice C: 455/6), incorrect MAD calculation (choice B: MAD≈2.00 too small), wrong count (choice D: 470/5), or misidentifying pattern. Calculating MAD: (1) find mean≈78.33, (2) calculate each |value-mean|, (3) average deviations≈6.78. The outlier 95 greatly increases MAD compared to what it would be without it.
9. A student measured the number of push-ups completed by 6 classmates in 1 minute: 18, 20, 21, 22, 19, 35. Find the median and the mean. Which statement best compares them and notes any striking deviation?
   Answer: Median = 20.5; Mean = $135/6=22.5$; The mean is higher because 35 is an outlier. This question tests calculating measures of center like median and mean, comparing them, and noting striking deviations like outliers. Measures of center include median (middle of ordered data) and mean (sum/count), both summarizing typical values, with median resistant to outliers unlike mean. For example, in data 50,52,53,54,55,80, median=53.5, mean≈57.3, the mean is higher due to the outlier at 80 pulling it up. For this data ordered 18,19,20,21,22,35, median=(20+21)/2=20.5, mean=135/6=22.5, the mean is higher because 35 is an outlier. Common errors include wrong median like averaging wrong pairs, sum errors for mean, or failing to note the outlier's effect on mean versus median. To calculate median: (1) order data, (2) for even n, average middle two; for mean: (1) sum values, (2) divide by count. Compare them by noting how deviations like outliers affect mean more, and complete the summary by identifying the pattern and specific deviation.
10. A basketball player recorded points scored in 9 games: 8, 9, 10, 10, 11, 11, 12, 13, 25. Calculate the mean (sum/count) and the range, and describe the overall pattern and any striking deviation.
    Answer: Mean = $\frac{109}{9}\approx 12.11$; Range = 17; Pattern: most games are 8–13 points clustered around 10–12; Striking deviation: 25 points is much higher than the rest. This question tests calculating measures of center (mean) and variability (range), describing overall pattern in words, and noting striking deviations. Measures: mean=sum/count=(8+9+10+10+11+11+12+13+25)/9=109/9≈12.11; range=max-min=25-8=17 shows total spread. Pattern: most games (8 of 9) cluster between 8-13 points around mean 12.11, forming compact group; striking deviation: 25 points is clear outlier, approximately 12 points above next highest value 13. Choice A correctly shows mean=109/9≈12.11, range=17, describes pattern as "most games are 8-13 points clustered around 10-12," and notes "25 points is much higher than the rest." Common errors include: incorrect sum (choice C: 84/9≈9.33), confusing range with IQR (choice B: range=5), wrong count (choice D: 109/8), or misidentifying pattern location (choice B: "most games 20-25 points"). Calculating mean: (1) sum all values: 8+9+10+10+11+11+12+13+25=109, (2) divide by count: 109/9≈12.11; calculating range: (1) identify max=25 and min=8, (2) subtract: 25-8=17. Complete summary requires identifying 25 as outlier that dramatically increases range and pulls mean upward.
11. A student tracked the number of text messages they received each day for 7 days: 18, 19, 20, 35, 36, 37, 38. Which choice correctly gives the median, mean, and range, and identifies the overall pattern and any striking deviations (such as a gap or clusters)?
    Answer: Median $=35$; Mean $=29$; Range $=20$; Pattern: two clusters (18–20 and 35–38) with a gap from 21 to 34; Striking deviations: the gap shows the data are split into two groups This question tests calculating measures of center like median and mean, variability like range, describing the overall pattern in words, and noting striking deviations such as gaps or clusters. Measures: median (35, middle of 7), mean (203/7≈29), range (38-18=20). Pattern: 'two clusters (18–20 and 35–38) with gap 21-34.' Deviation: the gap/split groups. For data 18,19,20,35,36,37,38, correct: median=35, mean=29, range=20, pattern: two clusters with gap. Common errors: wrong median position, ignoring gap, describing as even spread. To calculate: order, find middle for median, sum/divide mean, max-min range; describe clusters/gaps in words.
12. A student recorded the number of seconds it took to solve a puzzle on 9 tries: 22, 23, 23, 24, 24, 25, 26, 26, 27. Which choice correctly reports the median, mean, and MAD (mean absolute deviation), and describes the overall pattern and any striking deviations?
    Answer: Median $=24$; Mean $=24.4$; MAD $\approx 1.3$; Pattern: times cluster in the mid-20s with small spread; Striking deviations: none This question tests calculating measures of center like median and mean, variability like MAD, describing the overall pattern in words, and noting striking deviations. Measures: median (24, middle of 9), mean (220/9≈24.444), MAD (deviations average ≈1.333). Pattern: 'times cluster in mid-20s with small spread.' Deviations: none. For data 22,23,23,24,24,25,26,26,27, correct: median=24, mean≈24.4, MAD≈1.3, pattern: mid-20s cluster small spread, no deviations. Common errors: wrong MAD, pattern misleading. To calculate: middle for median, sum/divide mean, average |deviations| for MAD; narrate pattern.
13. A teacher recorded the number of pages 8 students read over the weekend: 12, 15, 14, 13, 16, 12, 15, 30. Find the median and the range, and identify any striking deviation.
    Answer: Median = 14.5; Range = 18; Pattern: most values 12–16; Striking deviation: 30 pages is much higher than the rest. This question tests calculating measures of center like the median and variability like the range, while describing the overall pattern in words and noting striking deviations such as outliers. Measures of center include the median, which is the middle value of ordered data (for even counts, average the two middle values), summarizing a typical value, and variability like range is max minus min, showing total spread. For example, in data 50,52,53,54,55,80, the median is (53+54)/2=53.5, range=80-50=30, pattern: most values cluster tightly between 50-55, striking deviation: outlier at 80 far above the rest. For this data ordered as 12,12,13,14,15,15,16,30, the median is (14+15)/2=14.5, range=30-12=18, pattern: values mostly cluster between 12 and 16 pages, striking deviation: 30 is much higher than the others. Common errors include misordering the data leading to wrong median, calculating range incorrectly, describing pattern only with numbers instead of words like 'most values 12–16', or missing the outlier at 30. To calculate median: (1) order the data, (2) for even n=8, average the 4th and 5th values. For range: (1) find max and min, (2) subtract; describe pattern in words like 'clustered low with one high value', and scan for deviations like values far from the cluster.
14. The lengths (in seconds) of 8 short videos watched by a student were: 15, 16, 16, 17, 18, 18, 19, 30. Find the median and the range. Then describe the overall pattern and any striking deviation.
    Answer: Median $=\frac{17+18}{2}=17.5$, range $=30-15=15$; pattern: most videos are 15–19 seconds with one much longer video (30). This question tests median and range with an outlier in time data. The ordered data: 15, 16, 16, 17, 18, 18, 19, 30 seconds. With 8 values (even), median = average of 4th and 5th values = (17+18)/2 = 17.5 seconds. The range = max - min = 30 - 15 = 15 seconds, showing total spread. Looking at the pattern: seven videos cluster tightly between 15-19 seconds (just 4 seconds apart), while one video at 30 seconds is 11 seconds longer than the next longest (19), making it a clear outlier—more than 50% longer than typical. Choice A correctly calculates median = 17.5 and range = 15, and identifies the pattern with 30 as a striking deviation. Choice B has wrong median, C incorrectly calculates range as 4 (ignoring the outlier), and D reverses the pattern description.
15. A club tracked how many cans each student brought for a food drive. The numbers were: 1, 2, 2, 3, 3, 3, 4, 4, 10. Calculate the mean (sum/count) and the IQR ($Q_3-Q_1$), and describe the overall pattern and any striking deviation.
    Answer: Mean = $\frac{32}{9}\approx 3.56$; IQR = 2; Pattern: most students brought 1–4 cans clustered around 3; Striking deviation: 10 cans is much higher than the rest. This question tests calculating measures of center (mean) and variability (IQR), describing overall pattern in words, and noting striking deviations. Measures: mean=sum/count=(1+2+2+3+3+3+4+4+10)/9=32/9≈3.56; for IQR with 9 values, Q1 is 25th percentile (position 2.5, so between 2nd and 3rd values: Q1=2) and Q3 is 75th percentile (position 7.5, so between 7th and 8th values: Q3=4), giving IQR=4-2=2. Pattern: most students (8 of 9) brought 1-4 cans clustered around mean 3.56, forming tight group; striking deviation: 10 cans is clear outlier, approximately 6 cans above next highest value 4. Choice A correctly shows mean=32/9≈3.56, IQR=2, describes pattern as "most students brought 1-4 cans clustered around 3," and notes "10 cans is much higher than the rest." Common errors include: incorrect sum (choice C: 22/9≈2.44), confusing IQR with range (choice B: IQR=9), wrong count (choice D: 32/8=4), or incorrect IQR calculation (choice D: IQR=1). Calculating IQR: (1) find Q1=2 and Q3=4, (2) subtract: 4-2=2. Complete summary identifies 10 as outlier that pulls mean upward but doesn't affect median or IQR.
16. A class tracked the number of correct answers (out of 20) on a practice set for 8 students: 11, 12, 12, 13, 13, 14, 14, 15. Find the mean and the MAD from the mean. Then describe the overall pattern.
    Answer: Mean = $104/8=13$; MAD $=\frac{2+1+1+0+0+1+1+2}{8}=1$; Pattern: values cluster closely around 13 and are fairly symmetric. This question tests calculating mean and MAD, describing the overall pattern. Measures include mean (sum/count) and MAD (average deviations) for center and variability. For example, in data 50,52,53,54,55,80, mean ≈ $57.3$, MAD high due to outlier, pattern: clustered with deviation. For this data 11,12,12,13,13,14,14,15, mean = $104/8=13$, MAD = $8/8=1$, pattern: values cluster closely around 13 and are fairly symmetric. Common errors include sum errors for mean, MAD by not including all deviations or wrong averaging, or describing as skewed when symmetric. To calculate MAD: find mean, $|$deviations$|$, average them. Describe pattern in words like 'symmetric cluster around 13', check for symmetry or skew.
17. The times (in seconds) for 7 students to run a short sprint were: 9, 10, 11, 10, 12, 10, 11. Calculate the mean and the range. Then describe the overall pattern.
    Answer: Mean = $73/7\approx 10.4$; Range = 3; Pattern: values cluster around 10–11 seconds and are fairly balanced. This question tests calculating measures of center like the mean and variability like the range, while describing the overall pattern in words. Measures of center include the mean, which is sum divided by count, summarizing a typical value, and variability like range is max minus min, showing total spread. For example, in data 50,52,53,54,55,80, the mean is ≈57.3, range=30, pattern: most values cluster 50-55 with an outlier at 80. For this data 9,10,10,10,11,11,12, the mean is 73/7≈10.4, range=12-9=3, pattern: values cluster around 10–11 seconds and are fairly balanced. Common errors include arithmetic mistakes in summing for mean, wrong range like using 12-0, or describing pattern poorly like 'spread evenly' without noting the clustering. To calculate mean: (1) sum all values, (2) divide by count; for range: (1) identify max and min, (2) subtract. Describe pattern in words like 'clustered around 10-11, symmetric', and ensure the summary reports center, variability, and pattern completely without missing balance or clustering.
18. A soccer coach recorded the number of goals scored in 10 games: 0, 1, 1, 1, 2, 2, 2, 3, 3, 7. Which choice correctly gives the median, mean, and IQR, and describes the overall pattern and any striking deviations?
    Answer: Median $=2$; Mean $=2.2$; IQR $=2$; Pattern: most games are between 1 and 3 goals; Striking deviations: 7 is much higher than the rest (outlier) This question tests calculating measures of center like median and mean, variability like IQR, describing the overall pattern in words, and noting striking deviations such as outliers, gaps, or clusters. Measures include median (middle for even: average 2 and 2=2), mean (22/10=2.2), IQR (Q3-Q1=3-1=2). The overall pattern: 'most games between 1 and 3 goals, clustered low.' Striking deviation: 7 is an outlier. For this data 0,1,1,1,2,2,2,3,3,7, correct: median=2, mean=2.2, IQR=2, pattern: most 1-3, deviation: 7 outlier. Common errors: wrong IQR calculation, missing outlier, pattern only numbers. To calculate: order, find middles for median, sum/divide mean, quartiles for IQR; describe in words, identify outliers.
19. A student recorded the number of goals scored in 10 soccer games: 0, 1, 1, 2, 2, 2, 3, 3, 4, 8. Find the median and the IQR (Q1 = median of lower half, Q3 = median of upper half). Also describe the overall pattern and any striking deviation.
    Answer: Median = 2; IQR = $3-1=2$; Pattern: most games are between 1 and 4 goals; Striking deviation: 8 goals is unusually high. This question tests calculating median and IQR, describing pattern and striking deviation. Measures include median and IQR=Q3-Q1 for spread. For example, in data 50,52,53,54,55,80, median=53.5, IQR≈3, pattern: cluster with high outlier. For this data ordered 0,1,1,2,2,2,3,3,4,8, median=(2+2)/2=2, IQR=3-1=2 (Q1=1 from lower 0-2, Q3=3 from upper 2-8), pattern: most between 1-4 goals, deviation: 8 unusually high. Common errors include wrong median averaging, IQR as range, misidentifying deviation like calling 0 outlier instead. To calculate IQR: median of lower/upper halves (for 5 values, 3rd). Describe in words like 'clustered low with high outlier', scan for unusual values.
20. A science class recorded the temperatures (in $^\circ$F) at noon over 8 days: 68, 69, 70, 70, 71, 71, 72, 85. Which choice correctly reports the median, mean, and range, and describes the overall pattern and any striking deviations?
    Answer: Median $=70.5$; Mean $=72$; Range $=17$; Pattern: most temperatures are around 70–72; Striking deviations: 85 is much higher than the rest (outlier) This question tests calculating measures of center like median and mean, variability like range, describing the overall pattern in words, and noting striking deviations such as outliers, gaps, or clusters. Measures: median ((70+71)/2=70.5), mean (576/8=72), range (85-68=17). Pattern: 'most temperatures around 70–72, tightly clustered.' Deviation: 85 outlier. For data 68,69,70,70,71,71,72,85, correct: median=70.5, mean=72, range=17, pattern: around 70-72, deviation: 85 high outlier. Common errors: median not averaging, wrong range, missing outlier, poor pattern description. To calculate: order, average middles for median, sum/divide mean, max-min range; narrate pattern, spot deviations.

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