Quartiles, IQR & Percentiles — Practice Set

Bank code: STA-IQR · Section: statistics · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

Percentile is a rank, not a score: quartiles split the sorted data into four equal counts, and IQR = Q3 - Q1.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · STA-IQR-001 · MCQ · 50s

The data set 3, 7, 9, 12, 15, 18, 21 is listed in increasing order. Using the standard method, in which the median is excluded from each half, what is the interquartile range?

(A) 8.5
(B) 11
(C) 12
(D) 18
(E) 25

Q2 · STA-IQR-002 · MCQ · 45s

A data set listed in increasing order is 2, 5, 8, 11, 14. Using the standard method, in which the median is excluded from each half, what is Q1?

(A) 2
(B) 3.5
(C) 5
(D) 8
(E) 12.5

Q3 · STA-IQR-003 · Numeric Entry · 50s

A data set listed in increasing order is 4, 8, 12, 16, 20, 24. What is its interquartile range? Enter your answer as a number.

Numeric entry — write the number.

Q4 · STA-IQR-009 · MCQ · 45s

On a class test, a score of 60 is at the 40th percentile, where the pth percentile is the score at or below which exactly p percent of the scores fall. Which of the following must be true?

(A) The student who scored 60 answered 40% of the questions correctly
(B) 40% of the students scored above 60
(C) 40% of the students scored 60 or below
(D) 60% of the students scored 60 or below
(E) 60 is the median score for the class

Q5 · STA-IQR-010 · QC · 55s

Data set A: 2, 4, 6, 8, 10

Quartiles are found by the standard method, with the median excluded from each half.

Column A: The interquartile range of Data set A Column B: The range of Data set A

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · STA-IQR-012 · Numeric Entry · 85s

A data set is listed in no particular order: 14, 22, 9, 37, 5, 28, 16, 41, 3, 19, 33, 7. What is its interquartile range? Enter your answer as a number.

Numeric entry — write the number.

Q7 · STA-IQR-013 · QC · 65s

Data set X: 10, 20, 30, 40, 50

Data set Y is formed by adding 100 to every value in Data set X.

Column A: The interquartile range of Data set Y Column B: The interquartile range of Data set X

Q8 · STA-IQR-015 · Select all that apply · 80s

A data set has Q1 = 15 and Q3 = 45. By the 1.5 x IQR rule, a value is an outlier only if it is strictly less than Q1 - 1.5 x IQR or strictly greater than Q3 + 1.5 x IQR. Which of the following values are outliers? Select all that apply.

(A) -35
(B) -10
(C) 0
(D) 92
(E) 100

Q9 · STA-IQR-017 · MCQ · 80s

A data set listed in increasing order is 2, 4, 6, 8, 10, 12, 14. The value 14 is deleted. Using the standard method, with the median excluded from each half, how does the interquartile range change?

(A) It increases by 2
(B) It increases by 4
(C) It stays the same
(D) It decreases by 2
(E) It decreases by 4

Q10 · STA-IQR-021 · Numeric Entry · 75s

Eight exam scores, listed in increasing order, are 55, 62, 68, 74, 79, 85, 91, 97. Using the standard method, what score is Q3, the 75th percentile? Enter your answer as a number.

Numeric entry — write the number.

Q11 · STA-IQR-022 · Select all that apply · 85s

A large data set has this five-number summary: minimum = 8, Q1 = 26, median = 40, Q3 = 62, maximum = 90. Which of the following statements are true? Select all that apply.

(A) The range is 82
(B) The interquartile range is 41
(C) About 25% of the values are greater than 62
(D) About half of the values lie between 26 and 62
(E) The mean of the data set is 40

Q12 · STA-IQR-024 · MCQ · 65s

In a box plot of a large data set, the left edge of the box is at Q1 = 30 and the right edge is at Q3 = 54. The whiskers reach the minimum, 5, and the maximum, 70. Approximately what fraction of the data values lie between 30 and 54?

(A) 1/4
(B) 3/8
(C) 1/2
(D) 3/4
(E) It cannot be determined without the individual data values

Q13 · STA-IQR-026 · Select all that apply · 90s

Which of the following statements about the interquartile range are true for EVERY data set? Select all that apply.

(A) IQR >= 0
(B) IQR <= the range of the data set
(C) IQR = 0 whenever every value in the data set is equal
(D) IQR = the range only when every value in the data set is equal
(E) Multiplying every value by 2 doubles the IQR

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · STA-IQR-027 · MCQ · 100s

A data set consists of 9 positive integers. Its first quartile is 20 and its third quartile is 50, found by the standard method with the median excluded from each half. What is the least possible value of the range of this data set?

(A) 0
(B) 30
(C) 31
(D) 49
(E) 60

Q15 · STA-IQR-028 · Numeric Entry · 105s

Seven numbers listed in increasing order are 3, k, 15, 20, 28, m, 42, where 3 < k < 15 and 28 < m < 42. Quartiles are found by the standard method, with the median excluded from each half. The interquartile range of the seven numbers is 20 and their mean is 22. What is the value of m? Enter your answer as a number.

Numeric entry — write the number.

Q16 · STA-IQR-029 · QC · 95s

Data set A consists of 9 numbers. Data set B is formed from A by replacing the largest number in A with a number that is 100 greater. Quartiles are found by the standard method, with the median excluded from each half.

Column A: The interquartile range of Data set A Column B: The interquartile range of Data set B

Q17 · STA-IQR-030 · MCQ · 85s

A data set of 12 values has mean 50, median 48, Q1 = 35 and Q3 = 63. Every value in the data set is multiplied by 1.5 and then decreased by 6. What are the new interquartile range and the new median?

(A) IQR = 42, median = 66
(B) IQR = 36, median = 66
(C) IQR = 42, median = 72
(D) IQR = 28, median = 66
(E) IQR = 36, median = 72

Q18 · STA-IQR-031 · Numeric Entry · 95s

In a certain data set Q1 = 24 and Q3 = 60. By the 1.5 x IQR rule, a value is an outlier only if it is strictly beyond a fence. Let w be the least integer that is an upper outlier and let u be the greatest integer that is a lower outlier. What is w - u? Enter your answer as a number.

Numeric entry — write the number.

Q19 · STA-IQR-032 · QC · 90s

Data set A: 10, 20, 30, 40, 50

Data set B is Data set A with the value 50 replaced by 500.

Quartiles are found by the standard method, with the median excluded from each half.

Column A: The interquartile range of Data set A Column B: The interquartile range of Data set B

Q20 · STA-IQR-033 · Select all that apply · 110s

A data set of 11 values, listed in increasing order, is 2, 5, 6, 8, 11, 13, 15, 16, 19, 22, 48. Quartiles are found by the standard method, with the median excluded from each half, and the 1.5 x IQR rule marks a value as an outlier only if it is strictly beyond a fence. Which of the following statements are true? Select all that apply.

(A) The interquartile range is 13
(B) The data set contains exactly one outlier
(C) The range is more than three times the interquartile range
(D) Deleting the value 48 leaves the interquartile range unchanged
(E) The median is greater than the mean

Q21 · STA-IQR-034 · MCQ · 95s

In a group of 500 students who each received a different score on an exam, Arun's score is at the 92nd percentile and Bela's score is at the 68th percentile. Here the pth percentile is the score at or below which exactly p percent of the 500 scores fall. How many students scored higher than Bela and lower than Arun?

(A) 24
(B) 118
(C) 119
(D) 120
(E) 121

Q22 · STA-IQR-035 · Numeric Entry · 100s

A data set consists of exactly 5 numbers. Its first quartile is 7, its median is 12, and its third quartile is 20, where Q1 is the median of the two values below the median and Q3 is the median of the two values above it. What is the mean of the 5 numbers? Enter your answer as a number.

Numeric entry — write the number.

Q23 · STA-IQR-036 · QC · 90s

Data set S consists of 10 values, all equal to 50.

Data set T consists of 10 values: one 0, eight 50s, and one 100.

Quartiles are found by the standard method.

Column A: The interquartile range of Data set S Column B: The interquartile range of Data set T

Q24 · STA-IQR-041 · MCQ · 105s

In a class of 20 students, the number of books each student read last month was recorded: 5 students read 1 book, 6 students read 2 books, 4 students read 3 books, 3 students read 5 books, and 2 students read 9 books. What is the interquartile range of the 20 values?

(A) 2
(B) 2.5
(C) 4
(D) 5.5
(E) 8

Q25 · STA-IQR-042 · MCQ · 100s

List L consists of 15 numbers. Each of the 3 largest numbers in L is increased by 20, and no other number is changed. Quartiles are found by the standard method, with the median excluded from each half. Which of the following statistics of the list must be unchanged?

(A) The mean only
(B) The range only
(C) The median and the interquartile range only
(D) The mean, the median and the interquartile range
(E) The median, the interquartile range and the range

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · STA-IQR-037 · MCQ · 120s

A data set of n values, where n > 4, has an interquartile range of 0. Which of the following must be true?

I. Every value in the data set is equal.

II. At least 50% of the values are equal to one another.

III. The median equals Q1.

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II and III

Q27 · STA-IQR-038 · Numeric Entry · 130s

A data set consists of 8 positive integers. Its first quartile is 12, its median is 20, its third quartile is 31, and its largest value is 40. Quartiles are found by the standard method: with 8 values the lower half is the 4 smallest and the upper half is the 4 largest. What is the greatest possible value of the mean of the data set? Enter your answer as a number.

Numeric entry — write the number.

Q28 · STA-IQR-039 · QC · 125s

A data set of 12 integers has an interquartile range of 20, and every value in it is less than 1000. Three more values, each equal to 1000, are appended to form a new data set of 15 values. Quartiles are found by the standard method, with the median excluded from each half.

Column A: The interquartile range of the new 15-value data set Column B: 20

Q29 · STA-IQR-040 · Select all that apply · 120s

A data set of 20 values has Q1 = 30, Q3 = 70, mean 50 and median 48. Every value x in the data set is replaced by 2x - 10. Which of the following statements about the transformed data set are true? Select all that apply.

(A) The new interquartile range is 80
(B) The new mean is 100
(C) The new median is 86
(D) The new Q1 is 50
(E) The new range is 10 less than twice the original range

Q30 · STA-IQR-043 · MCQ · 130s

A data set consists of 20 positive integers, no two of which are equal. The least of them is 1 and the greatest is 100. Quartiles are found by the standard method: with 20 values the lower half is the 10 smallest and the upper half is the 10 largest. What is the greatest possible value of Q1?

(A) 84.5
(B) 85.5
(C) 86
(D) 89.5
(E) 95.5

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