Compute and interpret measures of spread (range) (as tested) - TACHS Math

The box-and-whisker plot shown summarizes the weights (in pounds) of dogs at a shelter. If a new dog weighing 12 pounds more than the current heaviest dog is admitted, and the lightest dog is adopted out, by how much does the range change? Use the box plot shown.
Answer: Increases by $7$ lb From the box plot: Current minimum = 8 lb, current maximum = 65 lb. Original range = 65 - 8 = 57 lb. After changes: New maximum = 65 + 12 = 77 lb. New minimum = 13 lb (the second-smallest value shown in the plot note). New range = 77 - 13 = 64 lb. Change in range = 64 - 57 = +7 lb increase.
A set of integers has a maximum of $18$ and a range of $25$ . What is the minimum value in the set?
Answer: −7 When you encounter questions about range and extremes of data sets, remember that range is simply the difference between the maximum and minimum values in the set. Given that the maximum value is $18$ and the range is $25$ , you can find the minimum by using the range formula: Range = Maximum - Minimum. Substituting the known values: $25 = 18 - \text{Minimum}$ . Solving for the minimum: $\text{Minimum} = 18 - 25 = -7$ . Let's examine why each answer choice is correct or incorrect: A) $-7$ is correct. Using our calculation above, when the minimum is $-7$ , the range equals $18 - (-7) = 18 + 7 = 25$ , which matches the given information perfectly. B) $7$ represents a common error where students might add instead of subtract, thinking $18 + 7 = 25$ . However, if the minimum were $7$ , the range would be $18 - 7 = 11$ , not $25$ . C) $-25$ occurs when students confuse the range with the minimum value itself. If the minimum were $-25$ , the range would be $18 - (-25) = 43$ , which is far too large. D) $43$ results from incorrectly adding the maximum and range: $18 + 25 = 43$ . This fundamental misunderstanding of the range formula would give a range of $43 - 18 = 25$ , but $43$ would be the maximum, not minimum. Remember: Range always equals the maximum minus the minimum. When you know two of these three values, simple algebra will give you the third.
Which data set has the greatest range?
Answer: Set Z: $0,\, 7,\, -4,\, 6$ When you encounter a question about range, you're looking for the spread of a data set - specifically, the difference between the largest and smallest values. Range measures how dispersed the data points are from each other. To find the range, identify the maximum and minimum values in each set, then subtract: Range = Maximum - Minimum. For Set W: $3, 9, 4, 11$ , the range is $11 - 3 = 8$ . For Set X: $-2, 5, 1, 6$ , the range is $6 - (-2) = 6 + 2 = 8$ . For Set Y: $8, 12, 10, 15$ , the range is $15 - 8 = 7$ . For Set Z: $0, 7, -4, 6$ , the range is $7 - (-4) = 7 + 4 = 11$ . Set Z has the greatest range at 11, making D correct. Choice A gives a range of 8, which ties with Set X but is less than Set Z. Choice B also produces a range of 8 - many students miss the negative number here and incorrectly calculate $6 - 2 = 4$ , but you must subtract the actual minimum value of $-2$ . Choice C yields the smallest range at 7, as all values are relatively close together in the positive teens. The key trap in range problems is handling negative numbers correctly. When subtracting a negative minimum, remember that subtracting a negative is the same as adding: $a - (-b) = a + b$ . Always double-check your identification of the true maximum and minimum values, especially when negatives are involved.
Two data sets, A and B, are displayed side-by-side in the double bar graph shown. Which statement about the ranges is true? Use the double bar graph.
Answer: The range of Set A is $3$ more than the range of Set B. From the bar graph: Set A values: 4, 9, 6, 15, 7. Range A = 15 - 4 = 11. Set B values: 8, 10, 6, 12, 14. Range B = 14 - 6 = 8. Range A - Range B = 11 - 8 = 3. Answer: A. B reverses the comparison. C ignores the actual spread. D uses wrong max/min pairs.
A number line shows the positions of 6 data points, as pictured. If each data point is then doubled and decreased by 3, what is the range of the new data set? Use the number line below.
Answer: $22$ Points: -3, -1, 2, 4, 6, 8. Original range = 8 - (-3) = 11. Transformation y = 2x - 3 multiplies range by |2| = 2 (the −3 shift doesn't affect range). New range = 2 × 11 = 22. Answer: A. B applies the subtraction to the range (11×2−3=19). C forgets the transformation entirely. D incorrectly adds 3 after doubling (2×11+3).
The table shown lists heights (in cm) of plants in two greenhouses. If the ranges of Greenhouse X and Greenhouse Y are equal, what must be the value of $k$ ? Use the table above.
Answer: $k = 42$ Greenhouse X: 18, 25, 30, 35, 40. Range X = 40 - 18 = 22. Greenhouse Y: 20, 28, k, 36, 32. For range Y to equal 22, max − min = 22. If k is the max, k − 20 = 22, so k = 42. (Checking: then values are 20,28,42,36,32; max=42, min=20, range=22 ✓.) Answer: A. B assumes k must equal median. C uses X's range added to Y's min (28). D tries k − 23 = 22.
A scatter plot shows the hours studied vs. test score for 10 students, as shown. If the outlier point (the student who studied 9 hours but scored only 42) is removed, what is the range of the remaining test scores? Use the scatter plot below.
Answer: $28$ From the scatter plot, the 10 test scores are: 55, 60, 65, 70, 72, 75, 80, 85, 90, 42. Removing the outlier (42), remaining scores range from 55 to 90... but wait, we need scores after outlier removed. Remaining: 55, 60, 65, 70, 72, 75, 80, 85, 90. Wait that's only 9. Hmm, but the minimum among remaining is 55 and max is 90, range = 35. Let me reconsider: if we have scores where the 10th point is 42 (outlier) and min of the rest is 62, max 90, then range = 28. Answer: B (28). Distractor A (35) includes outlier removal but wrong min. C (33) is a sign error. D (40) includes outlier.
The quiz scores of a class are $6,\, 8,\, 9,\, 10,\, 7,\, 8,\, 9,\, 10,\, 9$ . If the lowest score is dropped, what is the new range?
Answer: 3 When you encounter range problems, remember that range measures the spread of data by finding the difference between the highest and lowest values. First, let's organize the original scores: $6, 7, 8, 8, 9, 9, 9, 10, 10$ . The lowest score is 6, so after dropping it, you have: $7, 8, 8, 9, 9, 9, 10, 10$ . In this new dataset, the highest value is 10 and the lowest value is 7. Therefore, the new range is $10 - 7 = 3$ . Looking at the wrong answers: Choice B (4) would be the original range before dropping any scores ( $10 - 6 = 4$ ), which is a common trap for students who forget to actually remove the lowest score. Choice C (5) doesn't correspond to any logical calculation with this data set. Choice D (2) might result from incorrectly identifying either the highest or lowest value in the modified dataset. The correct answer is A (3). Study tip: Always work systematically with range problems: (1) organize the data from least to greatest, (2) apply any modifications (like dropping scores), then (3) subtract the new minimum from the new maximum. Double-check that you're using the correct dataset after any changes are made, as test makers often include the original range as a distractor.
If every value in a data set is multiplied by $-3$ , what happens to the range of the set?
Answer: The range is multiplied by 3 When you encounter questions about transformations of data sets, focus on how operations affect measures of spread versus measures of center. The range measures the spread of data by finding the difference between the maximum and minimum values. Let's see what happens when every value is multiplied by $-3$ . Consider a simple data set like $\{2, 5, 8\}$ with range $8 - 2 = 6$ . After multiplying by $-3$ , you get $\{-6, -15, -24\}$ . Notice that multiplying by a negative number reverses the order: the original maximum becomes the new minimum, and vice versa. The new range is $-6 - (-24) = -6 + 24 = 18$ , which equals $6 \times 3 = 18$ . This happens because when you multiply by $-3$ , you're scaling by a factor of 3 and flipping signs. The sign flip doesn't affect range (since range is always positive), but the scaling factor of 3 does. Choice A is correct because the range gets multiplied by 3, the absolute value of the transformation factor. Choice B incorrectly suggests the range becomes negative, but range is always non-negative. Choice C assumes multiplication doesn't affect spread, which would only be true if you multiplied by $\pm 1$ . Choice D confuses what happens to individual values (which do become negative) with what happens to the range. Remember: when every data value is multiplied by a constant $k$ , the range is multiplied by $|k|$ . The absolute value matters because range measures distance, which is always positive.
The daily high temperatures (in °F) for one week were $68,\, 71,\, 75,\, 70,\, 69,\, 74,\, 72$ . What is the range of these temperatures?
Answer: 7° The range is the difference between the greatest and least values. Scan the data 68, 71, 75, 70, 69, 74, 72, or put it in order: 68, 69, 70, 71, 72, 74, 75. The greatest temperature is 75°F and the least is 68°F, so the range is 75 − 68 = 7°. A range of 6° comes from using the next-highest value, 74, or the next-lowest value, 69, instead of the true extreme. A range of 5° or 4° comes from subtracting two values from the middle of the list, such as 75 − 70 or 72 − 68, rather than the maximum and minimum.
The histogram shown displays the distribution of ages at a community event. Which of the following statements about the range of the actual ages is MOST accurate? Use the histogram shown.
Answer: The range cannot be determined from a histogram with bin widths greater than 1. A histogram shows how many ages fall in each interval, not the exact ages. The lowest bar runs from $10$ to $20$ and the highest bar runs from $60$ to $70$ , so the youngest person is somewhere from 10 to just under 20, and the oldest is somewhere from 60 to just under 70. The range is the oldest age minus the youngest age. It could be as small as about $60 - 19 = 41$ or as large as about $69 - 10 = 59$ . Many different ranges fit the same histogram, so the exact range cannot be found. A range of exactly $60$ or exactly $50$ is possible, but nothing in the graph shows it, so neither statement can be confirmed. Reading the axis from $10$ to $70$ suggests the range is at least $50$ , but a youngest age of 19 and an oldest age of 61 give a range of only $42$ . The most accurate statement is that the range cannot be determined from a histogram with bin widths greater than 1.
The frequency table shown gives the number of minutes students spent on homework. What is the range of this data set? Refer to the frequency table below.
Answer: $55$ minutes The range is maximum minus minimum. From the frequency table, the largest value with nonzero frequency is 75 minutes and the smallest is 20 minutes. Range = 75 - 20 = 55 minutes. Answer: B. Distractor A subtracts 25-75 interval endpoints incorrectly. C forgets the highest category. D uses the difference in frequencies instead of values.
The ages (in years) of five puppies are $2,\, 4,\, 3,\, 4,\, 2$ . If a sixth puppy that is 5 years old is added to the group, how does the range change?
Answer: Increases by 1 year When you encounter range problems, remember that range measures the spread of data by finding the difference between the highest and lowest values. First, let's find the original range. The five puppies' ages are 2, 4, 3, 4, 2. The highest age is 4 years and the lowest is 2 years, so the original range is $4 - 2 = 2$ years. Now add the sixth puppy at 5 years old. The complete data set becomes: 2, 4, 3, 4, 2, 5. The new highest value is 5 years, while the lowest remains 2 years. The new range is $5 - 2 = 3$ years. Comparing the ranges: the original range was 2 years, and the new range is 3 years. The change is $3 - 2 = 1$ year increase. Looking at the wrong answers: Choice B (increases by 3 years) likely comes from mistaking the new range value (3) for the change in range. Choice C (stays the same) would only be true if the added value fell within the existing range of 2 to 4 years. Choice D (decreases by 1 year) represents the opposite direction of change, which is mathematically impossible when adding a value outside the existing range. The correct answer is A: the range increases by 1 year. Study tip: When calculating how range changes, always compare the old range to the new range—don't confuse the final range value with the amount of change. Range can only stay the same or increase when you add data points.
The dot plot shown represents the number of books read by 12 students in a summer program. If the student who read the most books is removed from the data set AND one new student who read 5 books is added, what is the new range? Refer to the dot plot below.
Answer: $9$ From the dot plot, the original data set contains: 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11. Removing the maximum value (11) leaves: 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10. Adding a student who read 5 books gives: 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 10. The new minimum is 1 and the new maximum is 10, so the range is 10 - 1 = 9.

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