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Measures of Spread - Accuplacer Qas

  1. A set consists of four distinct positive integers. The median is 10, the mean is 11, and the range is 12. What is the largest integer in the set?
    Answer: 18 Let the four integers in increasing order be a,b,c,da, b, c, d. Mean is 11, so the sum is a+b+c+d=4×11=44a+b+c+d = 4 \times 11 = 44. Median is 10, so the average of the middle two numbers is 10: (b+c)/2=10(b+c)/2 = 10, which means b+c=20b+c = 20. Range is 12, so d−a=12d-a = 12. We can substitute b+c=20b+c=20 into the sum equation: a+20+d=44a + 20 + d = 44, which simplifies to a+d=24a+d = 24. Now we have a system of two equations for aa and dd: d−a=12d-a=12 and d+a=24d+a=24. Adding the two equations gives 2d=362d = 36, so d=18d = 18. The largest integer is dd, which is 18.
  2. For any numerical dataset with at least two different values, which of the following statements about the range is always true?
    Answer: The range is heavily influenced by the presence of outliers. 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).
  3. The daily low temperatures, in degrees Celsius, for a week were recorded as: −5,2,−8,0,1,3,−4-5, 2, -8, 0, 1, 3, -4. What is the range of these temperatures?
    Answer: 11 The range is the difference between the maximum and minimum values in a dataset. The maximum temperature is 3°C and the minimum temperature is -8°C. The range is calculated as: Maximum - Minimum = 3−(−8)=3+8=113 - (-8) = 3 + 8 = 11.
  4. A dataset contains 5 numbers. If the smallest number is doubled, the range of the dataset decreases by 3. Which of the following statements must be true?
    Answer: The original smallest number was 3. Let the original minimum be mm and the original maximum be MM. The original range is Rold=M−mR_{old} = M - m. When the smallest number is doubled, the new minimum is (2m). We assume that (2m) is still the smallest number in the set, or at least smaller than M. The new range is Rnew=M−2mR_{new} = M - 2m. We are told that the range decreases by 3, so Rnew=Rold−3R_{new} = R_{old} - 3. Substituting the expressions for the ranges gives M−2m=(M−m)−3M - 2m = (M - m) - 3. This simplifies to M−2m=M−m−3M - 2m = M - m - 3. Subtracting MM from both sides gives −2m=−m−3-2m = -m - 3. Adding (2m) to both sides gives 0=m−30 = m - 3, so m=3m = 3.
  5. 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?
    Answer: The new range will be the same. 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.
  6. 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?
    Answer: Range only considers extreme values and may miss important variability patterns in the middle data points 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.
  7. The set of numbers 12,25,9,18,x{12, 25, 9, 18, x} has a range of 20. If xx is the maximum value in the set, what is the value of xx?
    Answer: 29 The range is the difference between the maximum and minimum values. The minimum value in the given set 12,25,9,18{12, 25, 9, 18} is 9. The problem states that xx is the maximum value of the entire set, so the minimum value remains 9. We can set up the equation: Range = Maximum - Minimum, which becomes 20=x−920 = x - 9. Solving for xx gives x=20+9=29x = 20 + 9 = 29.
  8. 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?
    Answer: 7 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.
  9. 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?
    Answer: The assumption that range accurately represents overall variability when it only reflects extreme values 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.
  10. For any number k>10k > 10, what is the range of the set {\(k-5, k+2k+2, kk, k−1k-1}\)?
    Answer: 7 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.
  11. 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?
    Answer: The highest temperature recorded was 15 degrees warmer than the lowest temperature recorded. 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.
  12. The range of a set of numbers {\(x, y, z}\) is RR. Assuming a>0a > 0, what is the range of the new set {\(ax-b, ay-b, az-b}\)?
    Answer: aRaR Let the minimum and maximum of the original set be minold\text{min}_{old} and maxold\text{max}_{old}. Then R=maxold−minoldR = \text{max}_{old} - \text{min}_{old}. When each number is transformed by multiplying by aa (where a>0a>0) and subtracting bb, the new minimum is a⋅minold−ba \cdot \text{min}_{old} - b and the new maximum is a⋅maxold−ba \cdot \text{max}_{old} - b. The new range is (a⋅maxold−b)−(a⋅minold−b)(a \cdot \text{max}_{old} - b) - (a \cdot \text{min}_{old} - b). This simplifies to a⋅maxold−b−a⋅minold+b=a⋅(maxold−minold)=aRa \cdot \text{max}_{old} - b - a \cdot \text{min}_{old} + b = a \cdot (\text{max}_{old} - \text{min}_{old}) = aR.
  13. In a class of 20 students, 5 students received a score of 75, 11 students received a score of 85, and 4 students received a score of 95 on a test. What is the range of the scores?
    Answer: 20 The range is the difference between the highest and lowest values in the data. The data values are the scores students received, which are 75, 85, and 95. The number of students who received each score are the frequencies. The range depends only on the scores themselves, not the frequencies. The highest score is 95 and the lowest score is 75. The range is 95−75=2095 - 75 = 20.
  14. 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?
    Answer: The machine's performance is not consistent because the range is 30. 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.
  15. 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?
    Answer: The number 5 must be changed to 7. 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.
  16. Two stores sell the same five models of a smartphone. At Store A, the prices are $500,$650,$700,$800,$1000{\$500, \$650, \$700, \$800, \$1000}. At Store B, the prices are $680,$700,$720,$750,$780{\$680, \$700, \$720, \$750, \$780}. Based only on the range of the prices, which statement is the most accurate conclusion?
    Answer: Store B offers more consistent pricing across its models than Store A. The range is a measure of variability or spread. The range for Store A is $1000−$500=$500\$1000 - \$500 = \$500. The range for Store B is $780−$680=$100\$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.
  17. 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?
    Answer: 16 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.
  18. 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?
    Answer: Interface X shows greater variability because its range of 2.4 seconds exceeds Interface Y's range of 1.4 seconds 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.
  19. 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?
    Answer: The standard deviation of each process to assess true variability beyond just range 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.
  20. 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?
    Answer: 2 data points must be removed to achieve a range of 14.5 or less 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.

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