7th Grade Math Quiz: Compare Two Populations Using Data

What this quiz covers

This quiz focuses on Compare Two Populations Using Data, giving you a quick way to practice the rules, question types, and explanations that matter most for 7th Grade Math.

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. League A has more consistent scoring because the right-skewed distribution indicates most players score close to the mean value.
2. Both leagues have identical scoring patterns since they have the same mean, regardless of the shape of their distributions.
3. League B likely has more consistent scoring patterns because symmetric distributions typically indicate less extreme variation than skewed distributions. (correct answer)
4. League A has better overall performance because right-skewed distributions always indicate higher maximum values than symmetric distributions.

Explanation: When you encounter questions about data distributions, focus on how the shape of the distribution affects the spread and consistency of the data, not just the central tendency. In this problem, both leagues have the same mean (2.4 goals), but their distribution shapes tell very different stories about scoring consistency. League A's right-skewed distribution means most players score fewer goals (clustered on the left), while a few high-scoring players pull the mean above the median. This creates more variability. League B's symmetric distribution means players' scores are evenly spread around the mean, with the mean and median being approximately equal, indicating more predictable, consistent scoring patterns. Choice C is correct because symmetric distributions typically have less extreme variation than skewed distributions, making League B's scoring more consistent and predictable. Choice A is wrong because right-skewed distributions actually indicate inconsistent scoring—most players score below average while a few score much higher. Choice B misses the key point that identical means don't guarantee identical patterns; the distribution shape matters significantly for understanding variability. Choice D makes an unfounded claim that right-skewed distributions "always" indicate better performance, when skewness actually suggests inconsistency rather than superior overall ability. Study tip: Remember that mean alone doesn't tell the whole story about data. Always consider the distribution shape—symmetric distributions generally indicate more consistent, predictable patterns than skewed ones, even when means are identical.

Question 2

1. The claim is correct because Team Blue's lower standard deviation clearly indicates superior and more reliable performance overall.
2. The claim is incorrect because Team Red has both faster times and their higher standard deviation shows more competitive depth.
3. The claim is partially correct about consistency, but Team Red actually performs better with faster average times despite more variability. (correct answer)
4. The claim cannot be properly evaluated without knowing the median times and interquartile ranges for both teams.

Explanation: When analyzing statistical claims about team performance, you need to distinguish between different aspects of what makes a team "better" - consistency versus actual performance level. Let's examine what the data tells us. Team Red averages 45.2 seconds with a standard deviation of 3.1 seconds, while Team Blue averages 46.8 seconds with a standard deviation of 1.9 seconds. The coach is correct that Team Blue is more consistent (lower standard deviation means times cluster closer to their average), but Team Red is actually faster on average by 1.6 seconds - a significant difference in competitive swimming. Answer C correctly identifies this nuanced situation: the coach is partially right about consistency, but Team Red performs better overall due to faster times, despite having more variable performance. Answer A incorrectly equates consistency with superior performance overall, ignoring that Team Blue's swimmers are actually slower on average. Answer B misses the point about consistency entirely and incorrectly suggests that higher standard deviation indicates "competitive depth" - variability doesn't necessarily mean depth of talent. Answer D unnecessarily complicates the analysis by requesting additional statistics when the mean and standard deviation provide sufficient information to evaluate both claims about speed and consistency. Study tip: In statistics problems involving performance comparisons, always separate the different claims being made. Consistency (measured by standard deviation) and performance level (measured by mean) are distinct concepts. A team can be consistent but slow, or fast but inconsistent - analyze each aspect independently before drawing conclusions.

Question 3

1. The argument is correct because Class A's mean is higher, indicating better overall performance regardless of other factors.
2. The argument cannot be evaluated because we need the median scores to make a proper comparison between classes.
3. The argument is incorrect because Class B's lower range means their mean is more reliable than Class A's mean.
4. The argument is incomplete because while Class A has a slightly higher mean, Class B shows more consistent performance. (correct answer)

Explanation: When comparing group performance using statistics, you need to consider both central tendency (like the mean) and variability (like the range) to get the complete picture. The student's argument focuses only on Class A's higher mean (78 vs. 76), but this tells just part of the story. While Class A did score slightly higher on average, Class B's much smaller range (16 vs. 24) reveals that their scores were clustered more tightly around their mean. This consistency suggests more reliable, predictable performance across the class, even though their average was 2 points lower. Choice A is wrong because it ignores the range entirely. A higher mean doesn't automatically indicate better performance when one group shows much more variability in scores. Choice B is incorrect because you don't need the median to evaluate this argument. The mean and range provide sufficient information to assess both average performance and consistency. Choice C goes too far by calling the argument completely incorrect. While Class B does show more consistency, Class A's higher mean is still a valid point worth considering. Choice D correctly identifies that the argument is incomplete rather than wrong. It acknowledges Class A's higher average while recognizing that Class B's greater consistency is an important factor that wasn't considered. Study tip: When comparing groups statistically, always examine both measures of center (mean, median) and measures of spread (range, standard deviation). Consistency can be just as important as average performance, especially in educational settings.

Question 4

1. High school students have more screen time and more varied usage patterns compared to middle school students overall. (correct answer)
2. Middle school students have more consistent screen time because their median is lower, making their IQR proportionally smaller.
3. High school students are less consistent in their screen time habits because their IQR is 50% larger than middle schoolers.
4. Both groups show similar consistency in screen time since the ratio of IQR to median is approximately the same for each group.

Explanation: High school students have both higher median screen time (5 vs 3.5 hours) and greater variability (IQR of 3 vs 2 hours), indicating more screen time and more varied usage patterns.

Question 5

1. 8th graders are generally taller and have more consistent heights because their median is higher and IQR is smaller. (correct answer)
2. 6th graders have more consistent heights because their median height is lower, making the IQR relatively smaller.
3. 8th graders are generally taller, but 6th graders have more consistent heights because their IQR is larger.
4. Both populations have similar height consistency because the difference between their IQRs is only 2 inches.

Explanation: 8th graders are generally taller (median of 64 vs 58 inches) and more consistent in height (IQR of 6 vs 8 inches). A smaller IQR indicates less variability and more consistency.

Question 6

1. Cashier A has a higher mean and a larger range than Cashier B.
2. Cashier B has a lower mean and a smaller range than Cashier A.
3. Cashier A has a lower mean and a smaller range than Cashier B. (correct answer)
4. Both cashiers have the same mean and the same range.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: 7th grade words mean 5.4 letters vs 4th grade mean 3.7 letters (difference 1.7 letters, 7th grade words generally longer—higher center). For these checkout times, Cashier A has a mean of 5 and range of 2, while Cashier B has a mean of 5.5 and range of 5, so Cashier A has a lower mean and a smaller range, inferring Cashier A generally faster with more consistency. Common errors include claiming same means when 5<5.5, miscalculating range (e.g., for A as larger), or confusing mean with median. Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]); uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics); mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 7

1. Team 2 has a higher median and is less variable than Team 1.
2. Team 1 has a lower median and is more variable than Team 2.
3. Team 1 has a higher median and is less variable than Team 2. (correct answer)
4. Both teams have the same median and the same range.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: 7th grade words mean 5.4 letters vs 4th grade mean 3.7 letters (difference 1.7 letters, 7th grade words generally longer—higher center). For these practice times, Team 1 has a median of 24.5 and range of 4, while Team 2 has a median of 23.5 and range of 15, so Team 1 has a higher median and is less variable, inferring Team 1 players generally practiced more with more consistency. Common errors include confusing median with mean (e.g., calculating averages instead), miscalculating range (e.g., using max-min incorrectly for Team 2 as smaller), or inferring the opposite variability like claiming Team 1 is more variable when its range is smaller. Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]); uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics); mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 8

1. More Sunlight has a higher mean growth and a smaller range, so it is more consistent. (correct answer)
2. More Sunlight has a lower mean growth and a larger range, so it is less consistent.
3. Less Sunlight has a higher mean growth and a smaller range, so it is more consistent.
4. The means are the same, but More Sunlight has a larger range.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: 7th grade words mean 5.4 letters vs 4th grade mean 3.7 letters (difference 1.7 letters, 7th grade words generally longer—higher center). For these plant growths, More Sunlight has a mean of 10 and range of 2, while Less Sunlight has a mean of 8 and range of 7, so More Sunlight has higher mean growth and smaller range, inferring more sunlight leads to greater and more consistent growth. Common errors include claiming Less Sunlight higher mean when 8<10, miscalculating range (e.g., for More as larger), or overstating similarity when centers differ substantially. Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]); uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics); mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 9

1. The groups have about the same mean, so no comparison of variability can be made.
2. Group A generally sends more texts, and Group A is more variable.
3. Group A generally sends more texts, and Group B is less variable.
4. Group B generally sends more texts, and Group B is more variable. (correct answer)

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: 7th grade words mean 5.4 letters vs 4th grade mean 3.7 letters (difference 1.7 letters, 7th grade words generally longer—higher center). For these text messages, Group A has a mean of 24 and range of 4, while Group B has a mean of 26.25 and range of 17, so Group B generally sends more texts and is more variable, inferring Group B students text more on average but with less consistency. Common errors include claiming similar means when 26.25>24, miscalculating range (e.g., for Group B as smaller), or inferring no comparison possible when centers differ. Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]); uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics); mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 10

1. Class 2 has a lower median and a smaller IQR than Class 1.
2. Class 1 has about the same median as Class 2, but a smaller IQR. (correct answer)
3. Class 1 has a higher median and a larger IQR than Class 2.
4. Class 2 has a higher median and a smaller IQR than Class 1.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: 7th grade words mean 5.4 letters vs 4th grade mean 3.7 letters (difference 1.7 letters, 7th grade words generally longer—higher center). For these airplane lengths, both classes have a median of 25, Class 1 has IQR of 2 and Class 2 has IQR of 4, so medians are the same but Class 1 has a smaller IQR, inferring similar typical lengths but Class 1 airplanes more consistent in length. Common errors include miscalculating IQR (e.g., wrong quartiles for Class 2), claiming different medians when both are 25, or confusing IQR with range. Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]); uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics); mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 11

1. Group B has a higher mean and is more variable (higher MAD) than Group A. (correct answer)
2. Group A has a higher mean and is more variable (higher MAD) than Group B.
3. Group B has a lower mean and is less variable (lower MAD) than Group A.
4. Group A has a higher mean and is less variable (lower MAD) than Group B.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: 7th grade words mean 5.4 letters vs 4th grade mean 3.7 letters (difference 1.7 letters, 7th grade words generally longer—higher center). For these jump distances, Group A has a mean of about 147.14 and MAD of about 2.98, while Group B has a mean of about 147.86 and MAD of about 8.41, so Group B has a higher mean and is more variable, inferring Group B generally jumps farther but with less consistency. Common errors include reversing means (claiming A higher when 147.14<147.86), miscalculating MAD (e.g., not averaging deviations correctly), or understating the variability difference. Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]); uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics); mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 12

1. Class A has a higher mean, and Class B has a larger range. (correct answer)
2. Class B has a higher mean, and both classes have the same range.
3. Class B has a higher mean, and Class B has a smaller range.
4. Class A has a higher mean, and Class A has a larger range.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: Class A push-ups mean 21 vs Class B mean 20 (difference 1, Class A generally more push-ups—higher center). Example: two samples: Class A 18,20,21,22,24 (mean:105/5=21, range:24-18=6), Class B 12,16,20,24,28 (mean:100/5=20, range:28-12=16), compare:21>20 (Class A higher by 1), Class B range larger (16>6, more variable), inference: Class A generally more push-ups than Class B, but Class B more variable. Correct comparison and inference: Class A has a higher mean, and Class B has a larger range, so Class A appears stronger overall but Class B has more spread in performance. Error like comparison reversed (lower mean claimed higher:20>21), variability wrong (range 6 claimed larger than 16 when opposite), calculations arithmetic errors (mean sum/count wrong), inference contradicting data (claims classes similar when centers differ), or measure confusion (uses median as mean). Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]). Uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics). Mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 13

1. Class 2 has a higher mean and a lower MAD, so it did better and was more consistent.
2. Class 1 has a higher mean and a lower MAD, so it did better and was more consistent. (correct answer)
3. Class 1 has a lower mean and a higher MAD, so it did worse and was less consistent.
4. Both classes have the same mean and the same MAD.

Explanation: Tests comparing two populations using random sample data, calculating measures of center (mean, median) and variability (range, MAD), drawing informal inferences about population differences. Comparing populations from samples: calculate center (mean or median for each), calculate variability (range=max-min, or MAD=average distance from mean), compare centers (which higher? mean₁ vs mean₂), compare variability (which more spread? range₁ vs range₂), draw inference (population with higher center generally higher values, population with larger variability more spread/less consistent). Example: Class 1 scores mean 8.5 vs Class 2 mean 7.5 (difference 1, Class 1 generally higher scores—higher center). Example: two samples: Class 1 7,8,8,9,9,10 (mean:51/6=8.5, MAD≈0.833), Class 2 5,6,7,8,9,10 (mean:45/6=7.5, MAD=1.5), compare:8.5>7.5 (Class 1 higher by 1), Class 1 MAD smaller (0.833<1.5, less variable), inference: Class 1 generally higher quiz scores than Class 2 and more consistent. Correct comparison and inference: Class 1 has a higher mean and a lower MAD, so it did better and was more consistent. Error like comparison reversed (lower mean claimed higher:7.5>8.5), variability wrong (MAD 0.833 claimed larger than 1.5 when opposite), calculations arithmetic errors (mean sum/count wrong), inference contradicting data (claims classes similar when centers differ), or measure confusion (uses range as MAD). Steps: (1) calculate centers (mean=sum/count, or median=middle value when ordered), (2) calculate variability (range=max-min, or MAD=average of |x-mean|), (3) compare centers (mean₁ vs mean₂: which larger?), (4) compare variability (range₁ vs range₂: which larger?), (5) infer (if mean₁>mean₂ substantially: population 1 generally higher values; if range₁>range₂: population 1 more variable/spread out/less consistent). Informal inference: not formal statistical test (no p-values), just observation (centers differ by X, which is [small/large] relative to variabilities, so populations [appear similar/different]). Uses: comparing grade levels (vocabulary complexity), teams (performance), classes (achievement), groups (characteristics). Mistakes: calculating centers/variability wrong, comparing without both measures (center only or variability only insufficient), reversing comparisons, overstating small differences or understating large ones.

Question 14

1. School 2 has a higher median score and lower standard deviation, showing both better performance and more consistency. (correct answer)
2. School 2 has a higher maximum score and smaller range, demonstrating both peak performance and better overall consistency.
3. School 2 has a higher minimum score and lower mean, indicating that even their lowest performers did better than School 1's lowest.
4. School 2 has a higher mode and lower median, showing that more students achieved the most common score successfully.

Explanation: A higher median indicates better overall performance (center), while a lower standard deviation indicates more consistency (less variability). This combination provides the strongest evidence for better overall performance.

Question 15

1. Dataset X has a higher median and greater variability than Dataset Y, suggesting X represents a more diverse population.
2. Dataset Y has a lower median but similar variability to Dataset X, indicating both populations are equally spread out.
3. Dataset X has both a higher median and lower variability than Dataset Y, suggesting X represents higher and more consistent values. (correct answer)
4. Dataset Y has a higher median but Dataset X has lower variability, making it difficult to determine which population performs better.

Explanation: From the box plots, Dataset X shows a higher median (around 75 vs 65) and smaller IQR (indicating lower variability) compared to Dataset Y, suggesting Dataset X has both higher and more consistent values.

Question 16

1. Team A players are more consistent scorers because they have a higher mean score than Team B players.
2. Team B players are more consistent scorers because they have a lower MAD, meaning less variability in their scoring than Team A players. (correct answer)
3. Team A players are more consistent scorers because their MAD is twice as large as Team B's MAD.
4. Both teams have equally consistent scoring because the difference in their mean scores is less than 1 point.

Explanation: Consistency is measured by variability, not by the mean, and MAD tells us how far scores typically are from the mean. Team B's MAD of 2.1 is lower than Team A's MAD of 4.2, which means Team B's scores are more tightly clustered around their mean and therefore more consistent. Choice A is wrong because it confuses a higher mean with more consistency. Choice C is wrong because a larger MAD means more variability, not more consistency. Choice D is wrong because comparing means doesn't tell us anything about consistency; that requires comparing variability.

Question 17

1. About 3.7 letters
2. About 1.0 letter
3. About 1.7 letters (correct answer)
4. About 0.7 letters

Explanation: The mean word length for 7th grade is (5+6+4+7+5+6+5)/7 = 38/7, or about 5.4 letters. The mean word length for 4th grade is (3+4+3+5+4+3+4)/7 = 26/7, or about 3.7 letters. Subtracting these means gives about 5.4 - 3.7 = 1.7 letters. Choice A mistakes the 4th grade mean itself for the difference. Choice B and choice D come from arithmetic slips when calculating one or both of the means.

Question 18

1. Class B has a higher mean, but Class A has a larger range.
2. Class B has a higher mean and a smaller range than Class A.
3. Class A has a higher mean and a larger range than Class B.
4. Class A has a higher mean, and Class B has a larger range. (correct answer)

Explanation: Class A's mean is (78+82+80+76+84+79+81)/7 = 560/7 = 80, and its range is 84 - 76 = 8. Class B's mean is (70+88+75+92+68+85+72)/7 = 550/7, or about 78.6, and its range is 92 - 68 = 24. Comparing these, Class A has the higher mean, 80 versus about 78.6, and Class B has the much larger range, 24 versus 8. Choice A is wrong because it has the means reversed. Choice B is wrong for the same reason, and also gets the range comparison backward. Choice C is wrong because it correctly identifies the higher mean but incorrectly claims Class A also has the larger range.

Question 19

1. Period 1 performed better overall because they have more students scoring 9 and 10, indicating higher achievement levels.
2. Period 2 performed better because their scores are more spread out, showing a wider range of student abilities.
3. Period 1 shows more consistent performance with less variability, while Period 2 has more diverse performance levels overall. (correct answer)
4. Both periods performed equally well since they have the same number of total students and similar score ranges.

Explanation: According to the dot plots, Period 1's scores cluster closely around 8 to 10, showing less spread and more consistent performance, while Period 2's scores are distributed more widely across 6 to 10, showing more variability from student to student, which matches choice C. Choice A wrongly treats a higher cluster of scores as automatically 'better' performance rather than describing the spread and consistency the data actually shows. Choice B wrongly treats a wider spread of scores as an advantage, when a wider spread actually indicates more inconsistent performance across students. Choice D is incorrect because the two groups show clearly different patterns of spread, even if their overall score ranges overlap somewhat. Comparing two data distributions means describing both where the data is centered and how spread out or consistent it is, not just which group scored higher on average.

Question 20

1. Class X performed better because more students scored in the highest interval, even though the distributions have similar spreads overall.
2. Class Y performed better because their scores are more evenly distributed, indicating more consistent performance across all students.
3. Both classes performed equally well since they both have students scoring across the full range from 60 to 100 points.
4. Class X shows more variability with peaks at both ends, while Class Y shows more central clustering around average scores. (correct answer)

Explanation: Class X has a bimodal-like distribution with students clustered at both lower and higher scores (more variability), while Class Y shows more central clustering around middle scores, indicating different patterns of performance distribution.