ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Measures Of Spread
20 questions · exam conditions
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Measures Of SpreadQuestion 1 of 20

The side lengths of a certain pentagon are 5 cm, 8 cm, 5 cm, 12 cm, and 7 cm. What is the range of the side lengths?

3
5
7
12
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Measures Of Spread

Practice Measures Of Spread in ACCUPLACER Quantitative Reasoning, Algebra & Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Measures Of Spread, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The side lengths of a certain pentagon are 5 cm, 8 cm, 5 cm, 12 cm, and 7 cm. What is the range of the side lengths?

  1. 3
  2. 5
  3. 7 (correct answer)
  4. 12
Explanation: The question asks for the range of the given side lengths. The set of side lengths is 5,8,5,12,7{5, 8, 5, 12, 7}. The range is the difference between the maximum and minimum values in this set. The maximum length is 12 cm and the minimum length is 5 cm. The range is 12−5=712 - 5 = 7 cm.

Question 2

A student's scores on five math quizzes were 88,92,81,85,94{88, 92, 81, 85, 94}. The same student's scores on five science quizzes were 78,95,83,89,80{78, 95, 83, 89, 80}. How much greater is the range of the science scores than the range of the math scores?

  1. 1
  2. 3
  3. 4 (correct answer)
  4. 30
Explanation: First, calculate the range for each set of scores. For the math scores, the maximum is 94 and the minimum is 81, so the range is 94−81=1394 - 81 = 13. For the science scores, the maximum is 95 and the minimum is 78, so the range is 95−78=1795 - 78 = 17. The question asks for the difference between the two ranges, which is 17−13=417 - 13 = 4.

Question 3

A list of 10 numbers has a range of 8. A new list is created by adding two additional numbers to the original list. Both of the new numbers are equal to the median of the original list. How does the range of the new list of 12 numbers compare to the range of the original list?

  1. The new range will be smaller.
  2. The new range will be the same. (correct answer)
  3. The new range will be larger.
  4. The relationship cannot be determined without knowing the numbers.
Explanation: The range is determined solely by the minimum and maximum values in a dataset. The median of a dataset must lie between (or be equal to) the minimum and maximum values. Therefore, adding numbers equal to the median will not change the original minimum or maximum. Since the minimum and maximum values of the list are unchanged, the range will also be the same.

Question 4

A set of 5 positive integers has a mean of 10. What is the smallest possible range for this set?

  1. 0 (correct answer)
  2. 1
  3. 4
  4. 5
Explanation: The range measures the spread of the data. To minimize the spread, all the numbers should be as close to each other as possible. If the mean of 5 integers is 10, their sum must be 5×10=505 \times 10 = 50. The smallest possible range is 0, which occurs if all numbers are the same. The set 10,10,10,10,10{10, 10, 10, 10, 10} consists of 5 positive integers, has a mean of 10, and its range is 10−10=010 - 10 = 0.

Question 5

The number of items produced by a factory machine on six consecutive days was 152, 161, 158, 155, 182, and 156. The manager considers the machine's performance "consistent" if the range of items produced is 10 or less. Based on this data, which conclusion is valid?

  1. The machine's performance is consistent because the range is 9.
  2. The machine's performance is consistent because most values are close together.
  3. The machine's performance is not consistent because the mean output is too low.
  4. The machine's performance is not consistent because the range is 30. (correct answer)
Explanation: First, calculate the range of the given data. The maximum value is 182 and the minimum value is 152. The range is 182−152=30182 - 152 = 30. The manager's criterion for consistency is a range of 10 or less. Since the calculated range of 30 is greater than 10, the machine's performance is not consistent according to the given rule.

Question 6

Two stores sell the same five models of a smartphone. At Store A, the prices are {\500, $650, $700, $800, $1000}.AtStoreB,thepricesare. At Store B, the prices are {$680, $700, $720, $750, $780}$. Based only on the range of the prices, which statement is the most accurate conclusion?

  1. The average price of a phone at Store A is higher than at Store B.
  2. Store B offers more consistent pricing across its models than Store A. (correct answer)
  3. The most expensive phone model is sold at Store A.
  4. Both stores sell a phone model for exactly $700.
Explanation: The range is a measure of variability or spread. The range for Store A is \1000 - $500 = $500.TherangeforStoreBis. The range for Store B is $780 - $680 = $100$. A smaller range indicates less spread and therefore more consistent pricing. Since Store B's range is much smaller, its pricing is more consistent. The other statements, while potentially true, are not conclusions drawn from the range itself.

Question 7

Consider the set of integers S = 4,7,9,11,15,18,23{4, 7, 9, 11, 15, 18, 23}. What is the range of the prime numbers within set S?

  1. 16 (correct answer)
  2. 19
  3. 21
  4. 23
Explanation: This is a two-step problem. First, identify the prime numbers in the set S. A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. The prime numbers in S are 7, 11, and 23. Second, calculate the range of this new subset of numbers 7,11,23{7, 11, 23}. The maximum value is 23 and the minimum value is 7. The range is 23−7=1623 - 7 = 16.

Question 8

A researcher wants to compare the spread of response times between two different user interfaces. Interface X has response times ranging from 0.8 to 3.2 seconds, while Interface Y has response times ranging from 1.5 to 2.9 seconds. However, Interface Y has twice as many data points as Interface X. Which statement about comparing their variability is most accurate?

  1. Interface X shows greater variability because its range of 2.4 seconds exceeds Interface Y's range of 1.4 seconds (correct answer)
  2. Interface Y shows greater variability because having more data points increases the likelihood of extreme values
  3. The comparison is inconclusive because range alone cannot account for the different sample sizes
  4. Interface Y shows greater variability per data point when adjusting range for sample size differences
Explanation: Range is calculated as maximum minus minimum value and is independent of sample size. Interface X has a range of 3.2 - 0.8 = 2.4 seconds, while Interface Y has a range of 2.9 - 1.5 = 1.4 seconds. Interface X shows greater spread regardless of sample sizes. Choice B incorrectly assumes more data points automatically mean greater variability. Choice C is wrong because range comparison is valid regardless of sample size. Choice D incorrectly suggests adjusting range for sample size, which is not a standard practice.

Question 9

A dataset contains the following values: 12, 15, 18, 23, 28, 35, 41. A researcher wants to reduce the range by exactly 50% through removing data points. What is the minimum number of data points that must be removed to achieve this goal?

  1. 1 data point must be removed to reduce the range from 29 to 14.5
  2. 4 data points must be removed to create sufficient reduction in extreme values
  3. 3 data points must be removed to ensure the range is exactly half the original
  4. 2 data points must be removed to achieve a range of 14.5 or less (correct answer)
Explanation: When you encounter questions about reducing range through data removal, you need to understand that range equals the difference between maximum and minimum values, and removing extreme values has the greatest impact. First, calculate the original range: 41−12=2941 - 12 = 29. To reduce this by exactly 50%, you need a new range of 29×0.5=14.529 \times 0.5 = 14.5. To minimize data point removal, you should remove the most extreme values first. If you remove just the highest value (41) and lowest value (12), your new dataset becomes: 15, 18, 23, 28, 35. The new range is 35−15=2035 - 15 = 20, which is still too large. However, if you remove 41 and 23 (keeping 12), you get a range of 35−12=2335 - 12 = 23, still too large. The key insight is finding any combination where exactly 2 removals achieve the target. Removing 41 and 35 gives you 12, 15, 18, 23, 28 with range 28−12=1628 - 12 = 16, and removing 41 and 28 gives range 23−12=1123 - 12 = 11. Since 11 < 14.5, this works. Choice A incorrectly states the original range as 29 when targeting 14.5, but claims only 1 removal is sufficient—this is mathematically impossible. Choice B suggests 4 removals, which is excessive since 2 strategic removals can achieve the goal. Choice C claims 3 removals are needed for exactly half the range, but this overshoots the minimum requirement. Remember: when minimizing data removal to achieve a specific range reduction, always consider removing extreme values strategically rather than assuming you need to remove the absolute maximum and minimum.

Question 10

A data analyst is comparing the variability of two manufacturing processes. Process A has measurements with a range of 24 units and a total of 8 data points. Process B has measurements with a range of 18 units and a total of 12 data points. If the analyst wants to determine which process shows more consistent performance per data point, what additional information is most critical?

  1. The standard deviation of each process to assess true variability beyond just range (correct answer)
  2. The median values of both processes to determine central tendency differences
  3. The total production time for each process to calculate efficiency ratios
  4. The average range per data point by dividing range by sample size
Explanation: Range only considers the difference between maximum and minimum values, making it sensitive to outliers and not accounting for how the other data points are distributed. Standard deviation considers all data points and their distances from the mean, providing a more comprehensive measure of variability. Choice B focuses on central tendency rather than spread. Choice C introduces irrelevant information about production time. Choice D incorrectly suggests dividing range by sample size, which is not a meaningful measure of spread.

Question 11

A quality control manager examines defect rates across four production shifts. She notices that the morning shift has the smallest range of defect rates but wants to determine if this necessarily indicates the most consistent performance. What additional consideration about range as a measure of spread should influence her analysis?

  1. Range may not reflect consistency if the morning shift has outliers that don't affect the maximum and minimum values
  2. Range only considers extreme values and may miss important variability patterns in the middle data points (correct answer)
  3. Range calculations require equal sample sizes across shifts to make meaningful comparisons between groups
  4. Range values need to be adjusted for the time duration of each shift to account for operational differences
Explanation: Range only uses the highest and lowest values, ignoring how the remaining data points are distributed. A shift could have a small range but still show inconsistent performance if the middle values vary significantly. For example, values of 2, 5, 8, 11, 14, 17, 20 and values of 2, 2, 2, 20, 20, 20, 20 both have the same range (18) but very different consistency patterns. Choice A incorrectly describes how outliers affect range. Choice C incorrectly states that equal sample sizes are required for range comparison. Choice D introduces irrelevant timing considerations.

Question 12

A meteorologist analyzes temperature data for two cities over the same week. City A had temperatures ranging from 45°F to 78°F, while City B had temperatures ranging from 52°F to 71°F. She wants to determine which city experienced more variable weather conditions. What important limitation should she consider when using range for this comparison?

  1. Range measurements need normalization based on historical temperature patterns for each city
  2. Range calculations require adjustment for different geographical elevations between the two cities
  3. Range comparisons are invalid when the average temperatures of the cities differ significantly
  4. Range may not reflect daily temperature patterns since it only captures the week's extreme measurements (correct answer)
Explanation: When analyzing variability in data sets, you need to understand what different statistical measures actually capture and their inherent limitations. Range measures the spread between the highest and lowest values in a dataset, which seems useful for comparing variability. However, range has a critical weakness: it only considers two data points—the extremes—while completely ignoring everything that happens in between. In this temperature scenario, City A's range is 33°F (78° - 45°) versus City B's range of 19°F (71° - 52°), suggesting City A had more variable weather. But what if City A had steady 60°F temperatures all week except for one unusually hot and one unusually cold day? City B might have actually experienced more day-to-day fluctuation despite its smaller range. Option A is incorrect because range doesn't require historical normalization—it's simply the difference between extremes. Option B is wrong because elevation adjustments aren't relevant to basic range calculations. Option C incorrectly suggests that different average temperatures invalidate range comparisons, but ranges can be meaningfully compared regardless of the datasets' central tendencies. Option D correctly identifies that range may not reflect the actual daily temperature patterns since it only captures the week's extreme measurements, missing all the variation that occurred between those extremes. Study tip: Remember that range is the most limited measure of variability because it ignores all middle values. When you see questions about data variability, consider whether the proposed measure actually captures what's being investigated.

Question 13

A sports analyst is comparing the consistency of two basketball players' scoring. Player A's game scores have a range of 18 points, while Player B's scores have a range of 12 points. The analyst concludes that Player B is more consistent. Which assumption makes this conclusion potentially flawed?

  1. The assumption that both players have played the same number of games this season
  2. The assumption that range accurately represents overall variability when it only reflects extreme values (correct answer)
  3. The assumption that both players face opponents of similar defensive strength in their games
  4. The assumption that scoring consistency should be measured in absolute points rather than percentages
Explanation: Range only considers the difference between the highest and lowest values, ignoring the distribution of all other data points. Player A might have one very high game and one very low game (creating a large range) but otherwise consistent scores, while Player B might have moderate extremes but highly variable middle scores. Range can be misleading about overall consistency. Choice A incorrectly suggests sample size affects range comparison validity. Choice C introduces external factors irrelevant to the statistical measure. Choice D incorrectly focuses on measurement units rather than the limitation of range as a measure.

Question 14

Two manufacturing processes produce components with the following specifications: Process 1 creates components measuring 50±15 mm, while Process 2 creates components measuring 200±30 mm. An engineer claims that both processes have equal variability because their ranges are proportional to their target values. What is the most significant limitation of using range to support this claim?

  1. Range calculations assume normal distribution patterns which may not apply to manufacturing processes
  2. Range represents maximum possible spread rather than actual observed spread in the sample data
  3. Range ignores the concentration of data points and may not reflect true process consistency patterns (correct answer)
  4. Range comparisons require identical sample sizes to make valid statistical inferences about processes
Explanation: While both processes have ranges that are 30% of their target values (30/100 for Process 1, 60/200 for Process 2), range only shows the spread between extreme values. It doesn't reveal whether most components cluster near the target value or are spread throughout the range. Process consistency depends on how tightly controlled the entire production is, not just the extreme cases. Choice A incorrectly assumes range requires normal distribution. Choice B misunderstands what range measures (actual, not theoretical spread). Choice D incorrectly states sample size requirements for range comparison.

Question 15

The dataset 10,12,15,18,20{10, 12, 15, 18, 20} has a range of 10. If the number 8 is added to the set, what is the new range?

  1. 2
  2. 10
  3. 12 (correct answer)
  4. 14
Explanation: The original set is 10,12,15,18,20{10, 12, 15, 18, 20}, with a minimum of 10 and a maximum of 20. When the number 8 is added, the new set becomes 8,10,12,15,18,20{8, 10, 12, 15, 18, 20}. The maximum value remains 20, but the minimum value is now 8. The new range is the difference between the new maximum and minimum: 20−8=1220 - 8 = 12.

Question 16

For any number k>10k > 10, what is the range of the set {\(k-5, k+2k+2, kk, k−1k-1})?

  1. 3
  2. 5
  3. 2k−32k - 3
  4. 7 (correct answer)
Explanation: To find the range, we must identify the maximum and minimum values in the set. For any k>10k > 10, the values in increasing order are k−5k-5, k−1k-1, kk, and k+2k+2. The minimum value is k−5k-5 and the maximum value is k+2k+2. The range is the difference between the maximum and minimum: (k+2)−(k−5)=k+2−k+5=7(k+2) - (k-5) = k + 2 - k + 5 = 7.

Question 17

The range of the daily high temperatures in a city during a 30-day period was 15 degrees Fahrenheit. Which of the following must be true?

  1. The highest temperature recorded was 15 degrees warmer than the lowest temperature recorded. (correct answer)
  2. The temperature was never higher than 15 degrees during the period.
  3. The average temperature for the period was approximately 15 degrees.
  4. The temperature changed by at most 15 degrees from one day to the next.
Explanation: The range of a dataset is defined as the difference between the maximum and minimum values. If the range of temperatures is 15 degrees, it means that Maximum Temperature−Minimum Temperature=15\text{Maximum Temperature} - \text{Minimum Temperature} = 15. This is exactly what statement A says. The other options are incorrect interpretations. The actual temperatures could be high (e.g., 80 and 65), the range gives no information about the average, and it describes the overall spread, not the day-to-day change.

Question 18

Dataset P is 5,8,11,15{5, 8, 11, 15}. Dataset Q is 2,9,12,19{2, 9, 12, 19}. If Dataset R is formed by combining all elements of P and Q, what is the range of R?

  1. 10
  2. 13
  3. 17 (correct answer)
  4. 27
Explanation: To find the range of the combined dataset R, we need to find the overall minimum and maximum values from both P and Q. The combined dataset is 5,8,11,15,2,9,12,19{5, 8, 11, 15, 2, 9, 12, 19}. The smallest value in this set is 2 (from dataset Q). The largest value is 19 (from dataset Q). The range of R is the difference between this overall maximum and minimum: 19−2=1719 - 2 = 17.

Question 19

For any numerical dataset with at least two different values, which of the following statements about the range is always true?

  1. The range is always greater than the mean of the dataset.
  2. The range is heavily influenced by the presence of outliers. (correct answer)
  3. If a constant is added to every value, the range increases by that constant.
  4. The range must be one of the values present in the dataset.
Explanation: The range is calculated as Maximum - Minimum. Because it depends only on these two extreme values, a single very high or very low number (an outlier) will directly affect either the maximum or minimum, and thus can dramatically change the range. Statement A is false (e.g., for {10, 11, 12}, mean is 11 and range is 2). Statement C is false; adding a constant to all values does not change the range. Statement D is false (e.g., for {3, 5, 9}, the range is 6, which is not in the set).

Question 20

The set of data is 5,9,12,14,20{5, 9, 12, 14, 20}. Its range is 15. If the maximum value, 20, is changed to 22, which other number must be changed, and to what value, for the range to remain 15?

  1. The number 5 must be changed to 3.
  2. The number 5 must be changed to 7. (correct answer)
  3. The number 14 must be changed to 16.
  4. The number 9 must be changed to 11.
Explanation: The original range is 20−5=1520 - 5 = 15. The maximum value is changed from 20 to 22. To keep the range at 15, the new minimum value must satisfy the equation: 22−new minimum=1522 - \text{new minimum} = 15. Solving for the new minimum gives 22−15=722 - 15 = 7. Therefore, the original minimum value, 5, must be changed to 7.