ISEE Upper Level Mathematics Achievement Quiz: Range Median And Mode

What this quiz covers

This quiz focuses on Range Median And Mode, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Mathematics Achievement.

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. 3 (correct answer)
2. 4
3. 3.5
4. 2.5

Explanation: First, find the total number of students: 2 + 5 + 8 + 4 + 1 = 20 students. The median is the average of the 10th and 11th values. Counting from the lowest: positions 1-2 have 1 book, positions 3-7 have 2 books, positions 8-15 have 3 books, positions 16-19 have 4 books, position 20 has 5 books. The 10th and 11th students both read 3 books, so the median is 3.

Question 2

1. 7 (correct answer)
2. 8
3. 9
4. 10

Explanation: From the dot plot, the minimum number of books is 2 and the maximum number of books is 9. The range is 9 - 2 = 7.

Question 3

1. 1 (correct answer)
2. 2
3. 3
4. 4

Explanation: From the line plot, the data values are: 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5. The range is 5 - 1 = 4. For the median of 12 values, we take the average of the 6th and 7th values: (3 + 3)/2 = 3. The difference between range and median is 4 - 3 = 1.

Question 4

1. 16 (correct answer)
2. 18
3. 20
4. 22

Explanation: When you encounter range problems, remember that range measures the spread of data by finding the difference between the largest and smallest values. Understanding how changes to these extreme values affect the range is crucial for these types of questions. Let's work through this step by step. Initially, you have a set of positive integers with a range of 24, meaning the difference between the largest and smallest values is 24. If we call the smallest value $s$ and the largest value $L$, then $L - s = 24$. Now, the smallest value increases by 5 (becomes $s + 5$) and the largest value decreases by 3 (becomes $L - 3$). The new range is: $(L - 3) - (s + 5) = L - 3 - s - 5 = L - s - 8 = 24 - 8 = 16$. Looking at the wrong answers: Choice B (18) likely comes from subtracting only 5 instead of 8, perhaps forgetting that both endpoints change. Choice C (20) results from subtracting only the decrease in the maximum value (3) from the original range. Choice D (22) occurs if you subtract only the increase in the minimum value (5) but ignore the decrease in the maximum. The correct answer is A) 16. Study tip: When the range changes due to modifications at both endpoints, always track how each change affects the difference. Increases to the minimum value decrease the range, while decreases to the maximum value also decrease the range. The total change is the sum of both individual effects.

Question 5

1. 8
2. 12
3. 16 (correct answer)
4. 24

Explanation: When you encounter problems involving consecutive integers and their median, remember that the median of an odd number of values is always the middle value when arranged in order. Since we have 9 consecutive even integers with a median of 16, we know 16 is the 5th value (middle position). The consecutive even integers are: 8, 10, 12, 14, 16, 18, 20, 22, 24. The range is the difference between the largest and smallest values: $24 - 8 = 16$. So the correct answer is C) 16. Let's examine why the other options are incorrect. Answer A) 8 represents the difference between consecutive terms (like $16 - 8 = 8$), but this isn't what range measures. Answer B) 12 might come from incorrectly calculating the difference between the median and one of the endpoints ($16 - 4 = 12$ if you mistakenly thought the sequence started at 4), but this doesn't represent the full spread of data. Answer D) 24 is actually the largest value in our data set, not the range—this is a common trap where students confuse a data point with the measure of spread. Strategy tip: For consecutive integer problems, always write out the actual sequence when possible. This prevents calculation errors and helps you clearly identify the minimum and maximum values needed for range. Remember that range = maximum - minimum, and with consecutive integers, you're looking at the difference between the first and last terms in your sequence.

Question 6

1. 103
2. 108 (correct answer)
3. 113
4. 118

Explanation: When you encounter questions about transforming data sets, focus on how different operations affect measures of spread like range. Range measures the spread between the highest and lowest values, so you need to understand how mathematical transformations impact this spread. Let's work through this step-by-step. The original data set has a range of 36, meaning the difference between the maximum and minimum values is 36. When you multiply every value by 3, you're scaling the entire data set. This scaling affects the range by the same factor - the new range becomes $36 \times 3 = 108$. Here's the key insight: subtracting 5 from each value shifts every data point by the same amount. Since range only depends on the difference between maximum and minimum values, adding or subtracting the same constant to all values doesn't change the range at all. The range remains 108. Looking at the wrong answers: Choice A (103) might result from incorrectly thinking you subtract 5 from the range after multiplying by 3: $108 - 5 = 103$. Choice C (113) could come from adding 5 instead of recognizing that constant shifts don't affect range: $108 + 5 = 113$. Choice D (118) might result from incorrectly applying both operations to the range: $36 \times 3 + 5 + 5 = 118$. The correct answer is B (108). Strategy tip: Remember that multiplication/division changes the range by the same factor, while addition/subtraction leaves the range unchanged. Focus on how each transformation affects spread, not individual values.

Question 7

1. 10
2. 11 (correct answer)
3. 12
4. Cannot be determined

Explanation: When you encounter statistics problems involving transformations of data, remember that adding or subtracting the same value to every data point shifts the entire distribution without changing its shape or spread. First, let's identify the mode in the original data set: 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10. The mode is the value that appears most frequently. Counting the frequencies: 6 appears once, 7 appears twice, 8 appears four times, 9 appears five times, and 10 appears three times. Since 9 appears most often (5 times), the original mode is 9. When the teacher adds 2 bonus points to every student's score, each value increases by 2: the new scores become 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12. The value that was most frequent before (9) becomes 11, and it still appears most frequently (5 times). Therefore, the new mode is 11, making choice B correct. Choice A (10) represents the original maximum score, not the new mode. Choice C (12) is the new maximum score after adding bonus points, but it only appears 3 times. Choice D (Cannot be determined) is incorrect because adding a constant to all values preserves the frequency relationships. Remember: when you add or subtract the same amount from every data point, the mode shifts by that same amount. The relative frequencies don't change, just the actual values.

Question 8

1. 1 (correct answer)
2. 1.5
3. 2
4. 0

Explanation: When you encounter a question about finding the median, you need to arrange all data values in order and find the middle value (or average of the two middle values if there's an even number of data points). First, let's determine the total number of students: 8 + 6 + 4 + 3 + 2 + 1 = 24 students. Since we have 24 students (an even number), the median will be the average of the 12th and 13th values when all absence counts are listed in order. Now, let's list out the data in order: The first 8 students have 0 absences, the next 6 students (positions 9-14) have 1 absence, the next 4 students (positions 15-18) have 2 absences, and so on. This means the 12th student has 1 absence (since positions 9-14 all have 1 absence), and the 13th student also has 1 absence. Therefore, the median is $\frac{1 + 1}{2} = 1$. Looking at the wrong answers: B) 1.5 might tempt you if you incorrectly identified which positions represent the middle values or made an arithmetic error. C) 2 could result from confusing the median with another measure like the mode of the frequencies, or miscounting positions. D) 0 might seem appealing since it's the most frequent value (the mode), but that's not what median measures. Remember: for median problems, always count the total data points first, then systematically locate the middle position(s). Don't confuse median with mode (most frequent value) or mean (average).

Question 9

1. 5.1
2. 6.0
3. 6.12 (correct answer)
4. 6.84

Explanation: When you encounter problems involving percentage changes to data sets, remember that the median's position stays the same, but its value changes proportionally with the data transformation. First, let's find the median of the original data set: 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0. With 7 values, the median is the 4th value when arranged in order, which is 5.1. Now, when each measurement increases by 20%, we multiply each value by 1.20. The key insight is that you don't need to calculate all seven new values—since every number increases by the same percentage, the median will also increase by exactly 20%. So the new median is $5.1 \times 1.20 = 6.12$. Looking at the wrong answers: Choice (A) gives 5.1, which is the original median before the 20% increase—this ignores the percentage change entirely. Choice (B) shows 6.0, which was the largest value in the original data set, suggesting confusion about which position represents the median. Choice (D) gives 6.84, which would result from incorrectly taking 20% of the largest original value (6.0) and adding it to the original median: $5.1 + (0.20 \times 6.0) = 6.3$—though even this calculation doesn't quite match 6.84. The correct answer is (C) 6.12. Study tip: When a data set undergoes a uniform transformation (like adding a constant or multiplying by a percentage), all measures of central tendency change by that same transformation. This saves you time—transform the median directly rather than recalculating the entire data set.

Question 10

1. 6 hours
2. 7 hours (correct answer)
3. 8 hours
4. There is no mode

Explanation: When you encounter questions about measures of central tendency, remember that removing data points can change which measure (mean, median, or mode) best describes your dataset. The mode is the value that appears most frequently in a dataset. Let's first identify the original data distribution: 5 hours (3 students), 6 hours (5 students), 7 hours (8 students), 8 hours (6 students), and 9 hours (3 students). The original mode is 7 hours since it appears 8 times, more than any other value. Now we need to remove the two students who spent the least time. The least time spent was 5 hours by 3 students, so removing 2 of these students leaves us with: 5 hours (1 student), 6 hours (5 students), 7 hours (8 students), 8 hours (6 students), and 9 hours (3 students). Looking at our new frequencies, 7 hours still appears most often with 8 students, making it the new mode. Let's examine why the other choices are incorrect: Choice A (6 hours) appears 5 times, which is less than 7 hours' frequency of 8. Choice C (8 hours) appears 6 times, also less than 7 hours. Choice D suggests there's no mode, but this would only be true if no value appeared more frequently than others or if multiple values tied for the highest frequency. Remember that when data is removed from a dataset, always recalculate the frequencies completely rather than assuming the mode stays the same. The mode depends entirely on which value appears most often in your current dataset.

Question 11

1. 0 (correct answer)
2. 0.5
3. 1
4. 1.5

Explanation: When you encounter a statistics problem asking about mode and median, you need to understand what each measure represents and how to calculate them from frequency data. First, let's organize the data: 5 students own 0 pets, 8 students own 1 pet, 12 students own 2 pets, 8 students own 3 pets, and 2 students own 4 pets. The total number of students is $5 + 8 + 12 + 8 + 2 = 35$. The mode is the value that appears most frequently. Looking at our data, 12 students own 2 pets, which is the highest frequency. Therefore, the mode is 2 pets. For the median, you need the middle value when all 35 students are arranged in order. The median position is $\frac{35 + 1}{2} = 18$. Counting through the ordered data: positions 1-5 have 0 pets, positions 6-13 have 1 pet, positions 14-25 have 2 pets. Since the 18th student falls in the group with 2 pets, the median is 2 pets. The difference between mode and median is $2 - 2 = 0$, making A correct. Choice B (0.5) might result from incorrectly calculating the median position or misunderstanding how to find the median with grouped data. Choice C (1) could come from miscounting frequencies or confusing which value has the highest frequency. Choice D (1.5) might stem from incorrectly averaging values instead of finding the true median position. Remember: always organize frequency data systematically and count positions carefully when finding the median in grouped data.

Question 12

1. 13
2. 16 (correct answer)
3. 19
4. 25

Explanation: When you encounter a problem about range after removing data points, you need to carefully identify which values remain and then find the difference between the highest and lowest of those remaining values. Starting with the original scores: 78, 82, 85, 88, 91, 94, 98, 103. The coach removes the highest score (103) and the lowest score (78), leaving you with: 82, 85, 88, 91, 94, 98. To find the range of these remaining 6 scores, you subtract the lowest remaining value from the highest remaining value: $98 - 82 = 16$. Let's examine why the other answers are incorrect. Choice A (13) might come from finding the difference between consecutive scores or making an arithmetic error in the subtraction. Choice C (19) could result from incorrectly including one of the removed scores in your calculation—for example, if you mistakenly used 85 as your lowest value instead of 82. Choice D (25) is the range of the original complete dataset ($103 - 78 = 25$), which represents the trap of forgetting to remove the highest and lowest scores entirely. The key strategy here is to work methodically: first identify and remove the specified values, then clearly identify the new highest and lowest values from what remains. Many students rush and either forget to remove scores or accidentally use values from the original dataset. Always double-check that you're working with the correct modified dataset before calculating your final answer.

Question 13

1. The mode is 65.
2. The mode is 62. (correct answer)
3. The mode is 68.
4. The mode is 63.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (62, 65, 60, 63, 66, 62, 64, 67, 61, 62, 65, 68, 62, 64), you can find the mode by counting frequencies: 60 appears once, 61 once, 62 appears four times, 63 once, 64 appears twice, 65 appears twice, 66 once, 67 once, and 68 once. Choice B is correct because 62 appears four times, more frequently than any other temperature value. Choice A is incorrect because 65 only appears twice, which is less frequent than 62 - this error occurs when students don't systematically count all occurrences. To help students: Use tally marks or a frequency table to track each value's occurrences systematically. Always recount to verify, especially with larger data sets.

Question 14

1. 40
2. 45
3. 50 (correct answer)
4. 55

Explanation: This question tests your understanding of statistical measures and how they relate to each other in a data set. When you see problems involving mode, median, and range together, you need to think systematically about what each measure tells you about the data's distribution. Let's work through this step by step. You know the data set has 11 values, so the median (middle value) is the 6th value when arranged in order, which is 28. The mode is 25, meaning 25 appears most frequently. The range is 40, and since the smallest value is 10, you can find the largest value using the range formula: Range = Largest value - Smallest value. Therefore: $40 = \text{Largest value} - 10$, so the largest value is $40 + 10 = 50$. Looking at the wrong answers: Choice A) 40 represents the range itself, not the largest value—this is a common trap where students confuse the range with an actual data point. Choice B) 45 might result from incorrectly calculating $10 + 40 - 5$, perhaps from some confusion about how range works. Choice D) 55 could come from adding the range to something other than the minimum value, like $28 + 40 - 13 = 55$, showing misunderstanding of the range formula. Remember that range is always the difference between the highest and lowest values in a data set. When you know the range and one extreme value, you can always find the other extreme by adding or subtracting the range accordingly.

Question 15

1. The range is 9. (correct answer)
2. The range is 23.
3. The range is 16.
4. The range is 8.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (7, 11, 9, 16, 11, 13, 10, 8, 14, 11), you can find the range by first identifying the highest value (16) and the lowest value (7), then calculating 16 - 7 = 9. Choice A is correct because it accurately reflects the calculation of the range as the difference between the maximum and minimum values. Choice B is incorrect because 23 represents adding the highest and lowest values (16 + 7) rather than subtracting them, a common conceptual error. To help students: Always arrange data from smallest to largest first, clearly identify the maximum and minimum values, then subtract minimum from maximum. Remind students that range measures spread, not sum.

Question 16

1. The range is 35. (correct answer)
2. The range is 28.
3. The range is 145.
4. The range is 90.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (55, 62, 70, 68, 74, 81, 77, 90, 85, 62, 79, 88, 73, 66, 62), you can find the range by identifying the highest value (90) and the lowest value (55), then calculating 90 - 55 = 35. Choice A is correct because it accurately reflects the calculation of the range as the difference between the maximum and minimum values. Choice C is incorrect because 145 represents adding the highest and lowest values (90 + 55) rather than subtracting them, demonstrating confusion about the range formula. To help students: Remember that range measures the spread of data, calculated as maximum minus minimum. Create a habit of circling or highlighting the highest and lowest values before calculating.

Question 17

1. The median is 79. (correct answer)
2. The median is 82.
3. The median is 71.
4. The median is 77.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (64, 71, 85, 90, 77, 82, 68, 75, 88, 92, 71, 79, 83, 71, 86), first order the 15 values: 64, 68, 71, 71, 71, 75, 77, 79, 82, 83, 85, 86, 88, 90, 92, then find the middle (8th) value: 79. Choice A is correct because with 15 values, the median is the 8th value in the ordered list, which is 79. Choice D is incorrect because 77 is the 7th value, not the 8th - this error occurs when students miscount the position or start counting from zero instead of one. To help students: For odd-numbered data sets, the median position is (n+1)/2 = (15+1)/2 = 8. Always verify by counting from both ends to ensure you've found the true middle.

Question 18

1. 35
2. 38 (correct answer)
3. 40
4. 42

Explanation: When you encounter range problems, remember that range measures the spread of data by finding the difference between the highest and lowest values. Adding new data points can potentially change both extremes. First, identify the current range boundaries. The original dataset has a minimum of 15 and maximum of 47. Now examine each new value: 12, 18, 35, 42, and 50. You need to determine if any of these extend beyond the current boundaries. Looking at the new values, 12 is smaller than the current minimum of 15, so it becomes the new minimum. The value 50 is larger than the current maximum of 47, so it becomes the new maximum. The other new values (18, 35, 42) fall within the original range and don't affect the extremes. The new range is $50 - 12 = 38$. Choice A (35) represents the original range of $47 - 15 = 32$, but this fails to account for the new extreme values. Choice C (40) might result from incorrectly using 50 as the new maximum but keeping 15 as the minimum ($50 - 15 = 35$), then making an arithmetic error. Choice D (42) could come from correctly identifying the new maximum of 50 but mistakenly using the original minimum of 15, then calculating $50 - 8 = 42$ through computational error. The correct answer is B (38). Strategy tip: Always check if new data points extend beyond the existing minimum or maximum values. The range can only stay the same or increase when adding data—it can never decrease.

Question 19

1. The mode is 3.
2. The mode is 5.
3. The mode is 2. (correct answer)
4. The mode is 6.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (2, 3, 4, 2, 5, 3, 2, 6, 4, 3, 2, 5), you can find the mode by counting frequencies: 2 appears four times, 3 appears three times, 4 appears twice, 5 appears twice, and 6 appears once. Choice C is correct because 2 appears four times, more frequently than any other value in the data set. Choice B is incorrect because 5 only appears twice, which is less frequent than both 2 and 3 - this error may occur when students confuse mode with other measures or don't count systematically. To help students: Create a frequency distribution table listing each unique value and its count. The mode is always the value with the highest frequency, not necessarily the highest value.

Question 20

1. The median is 70. (correct answer)
2. The median is 71.
3. The median is 72.
4. The median is 69.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (70, 72, 68, 71, 69, 70, 73, 74, 70, 67, 75), first order the 11 values: 67, 68, 69, 70, 70, 70, 71, 72, 73, 74, 75, then find the middle (6th) value: 70. Choice A is correct because with 11 values, the median is the 6th value in the ordered list, which is 70. Choice B is incorrect because 71 is the 7th value, not the 6th - this error occurs when students count the middle position incorrectly or fail to order the data properly first. To help students: For odd-numbered data sets, use (n+1)/2 to find the median position. Always order data from least to greatest before finding the median.