Data Distribution — Practice Set

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

The tail pulls the mean, and a normal curve splits 68-95-99.7 symmetrically about its centre.

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-DDI-001 · MCQ · 40s

In a normal distribution, which of the following must be true?

(A) The mean is greater than the median.
(B) The mode is less than the mean.
(C) The mean, the median, and the mode are all equal.
(D) The distribution is skewed to the right.
(E) The median is greater than the mode.

Q2 · STA-DDI-006 · Numeric Entry · 50s

Scores on a test are normally distributed with mean 100 and standard deviation 15. Approximately what percent of the scores are between 70 and 130? Enter your answer as a number.

Numeric entry — write the number.

Q3 · STA-DDI-007 · MCQ · 45s

A histogram shows most of its values clustered near the right-hand end of the range, with a long thin tail stretching toward the low values on the left. How is this distribution described?

(A) Right-skewed (positively skewed)
(B) Left-skewed (negatively skewed)
(C) Normal
(D) Uniform
(E) Symmetric with two peaks

Q4 · STA-DDI-008 · QC · 40s

A variable is normally distributed with mean m and standard deviation s, where s > 0.

Column A: The probability that a randomly selected value is greater than m Column B: The probability that a randomly selected value is less than m

Q5 · STA-DDI-009 · MCQ · 55s

The list 3, 5, 7, 9, 11, 13 has one further value, 100, appended to it. Compared with the original six values, the new list of seven values has:

(A) a mean that rises sharply and a median that rises slightly
(B) a median that rises sharply and a mean that rises slightly
(C) a mean and a median that rise by the same amount
(D) a mean and a median that are both unchanged
(E) a mean that falls and a median that rises

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · STA-DDI-011 · MCQ · 65s

A large data set has mean 40, median 50, and mode 55, and its histogram has a single peak. Which of the following best describes the distribution?

(A) Symmetric and bell-shaped
(B) Right-skewed, with a long tail toward higher values
(C) Left-skewed, with a long tail toward lower values
(D) Uniform, with all values equally likely
(E) Normal, with mean equal to median

Q7 · STA-DDI-013 · Numeric Entry · 80s

The seven values 12, 15, 18, 21, 24, 27, 30 have mean 21. One additional value is appended to the list, and the mean of the eight values is 10.5 greater than the mean of the seven. What is the additional value? Enter your answer as a number.

Numeric entry — write the number.

Q8 · STA-DDI-017 · MCQ · 85s

The 50 scores on an exam are grouped as follows:

60 to 70: 5 scores

70 to 80: 10 scores

80 to 90: 20 scores

90 to 100: 15 scores

Estimating every score in an interval by the midpoint of that interval, what is the estimated mean score?

(A) 79
(B) 80
(C) 84
(D) 85
(E) 89

Q9 · STA-DDI-019 · Select all that apply · 90s

A data set is skewed to the right: its histogram has a single peak and a long tail stretching toward the higher values. Its mean is 50 and its median is 44. Which of the following must be true? Select all that apply.

(A) The tail of the distribution stretches toward values above the peak.
(B) At least half of the values are 44 or less.
(C) Removing the single largest value would lower the mean.
(D) The mode of the data set is less than 44.
(E) The number 50 is one of the values in the data set.

Q10 · STA-DDI-022 · Numeric Entry · 70s

Scores on a test are normally distributed with mean 500 and standard deviation 100. Using the 68-95-99.7 rule, what score is approximately at the 16th percentile? Enter your answer as a number.

Numeric entry — write the number.

Q11 · STA-DDI-023 · MCQ · 65s

Two classes took the same exam. Class A has mean 70 and standard deviation 5. Class B has mean 70 and standard deviation 15. Which of the following must be true?

(A) Class A's scores are more tightly clustered around 70 than Class B's.
(B) Class B's scores are more tightly clustered around 70 than Class A's.
(C) The two classes have the same spread of scores.
(D) Class A's average score is higher than Class B's.
(E) Class B's highest score is higher than Class A's highest score.

Q12 · STA-DDI-024 · QC · 85s

A data set of 11 values has median 45. A twelfth value, 200, is added to the data set.

Column A: The median of the 12-value data set Column B: 45

Q13 · STA-DDI-025 · Select all that apply · 85s

A data set has Q1 = 40 and Q3 = 60. A value is called an outlier if it is below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR. Which of the following values are outliers? Select all that apply.

(A) 8
(B) 35
(C) 55
(D) 85
(E) 92

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · STA-DDI-027 · QC · 90s

A variable is normally distributed with mean m and standard deviation s > 0. Using the 68-95-99.7 rule, and treating those three percentages as exact, let p be the probability that a randomly selected value lies between m - 2s and m + s.

Column A: p Column B: 0.815

Q15 · STA-DDI-028 · MCQ · 95s

A data set of 100 values has mean 60 and median 60. Its smallest value, 5, is replaced by 500; no other value changes. Which of the following must be true of the new data set?

(A) The mean is 64.95 and the median is 60 or greater.
(B) The mean is 555 and the median is 60.
(C) The mean is 64.95 and the median is 64.95.
(D) The mean is 60 and the median is greater than 60.
(E) The mean is 64.95 and the median is less than 60.

Q16 · STA-DDI-029 · Numeric Entry · 95s

The scores of 4,000 test takers are normally distributed with mean 500 and standard deviation 100. Approximately how many of them scored between 600 and 700? Enter your answer as a number.

Numeric entry — write the number.

Q17 · STA-DDI-030 · Select all that apply · 105s

A data set has mean 80, median 70, and standard deviation 20. Which of the following must be true? Select all that apply.

(A) The data set is not normally distributed.
(B) At least one of the values is greater than 80.
(C) Approximately 68 percent of the values lie between 60 and 100.
(D) The sum of the values equals 80 times the number of values.
(E) The distribution has a longer tail on the high side than on the low side.

Q18 · STA-DDI-031 · MCQ · 100s

A histogram with unequal class widths plots frequency density (frequency divided by class width) as the height of each bar:

0 to 10: height 4

10 to 20: height 6

20 to 40: height 3

40 to 60: height 1

How many data values does the histogram represent in total?

(A) 14
(B) 140
(C) 160
(D) 180
(E) 280

Q19 · STA-DDI-032 · QC · 105s

Data set A consists of 10 values with mean 50 and standard deviation 10. Data set B consists of those same 10 values together with one additional value, 50. (Standard deviation here is the population standard deviation, as on the GRE.)

Column A: The standard deviation of data set A Column B: The standard deviation of data set B

Q20 · STA-DDI-033 · Numeric Entry · 90s

The 200 salaries at a company form a right-skewed distribution with mean $75,000. The 50 highest-paid employees have a mean salary of $150,000. What is the mean salary, in dollars, of the other 150 employees? Enter your answer as a number.

Numeric entry — write the number.

Q21 · STA-DDI-034 · Select all that apply · 110s

A data set consists of the 10 values 2, 4, 4, 6, 8, 8, 8, 10, 12, 14. Take Q1 to be the median of the lower five values and Q3 the median of the upper five values, and take the standard deviation to be the population standard deviation. Which of the following are true? Select all that apply.

(A) The mean is less than the median.
(B) The mode is 8.
(C) The range is 3 times the interquartile range.
(D) Removing the value 14 would raise the mean.
(E) The standard deviation is greater than 3.

Q22 · STA-DDI-035 · MCQ · 100s

Scores on a test are normally distributed with mean 500 and standard deviation 100. Exactly 1,600 test takers scored above 600. Approximately how many people took the test?

(A) 3,200
(B) 4,700
(C) 5,000
(D) 10,000
(E) 64,000

Q23 · STA-DDI-036 · QC · 85s

For a certain data set, Q1 = 20 and Q3 = 50. The lower outlier fence is Q1 - 1.5 x IQR and the upper outlier fence is Q3 + 1.5 x IQR.

Column A: The distance from the lower fence to the upper fence Column B: 120

Q24 · STA-DDI-041 · MCQ · 110s

The table below records the number of cars owned by each of 60 households, where a and b are the two missing frequencies.

Cars owned 0: 6 households

Cars owned 1: a households

Cars owned 2: b households

Cars owned 3: 9 households

Cars owned 4: 4 households

The mean number of cars per household is exactly 1.7. What is the median number of cars per household?

(A) 1
(B) 1.5
(C) 1.7
(D) 2
(E) 3

Q25 · STA-DDI-042 · Numeric Entry · 105s

A list of 5 positive integers, not necessarily distinct, has mean 20 and median 12. What is the greatest possible value of the largest integer in the list? Enter your answer as a number.

Numeric entry — write the number.

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Level 4 · Extreme — 5 questions · ~10 min

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

Q26 · STA-DDI-037 · MCQ · 120s

A variable is normally distributed with mean m and standard deviation s. Define three regions:

Region I: values more than 2 SDs below the mean

Region II: values between 1 and 2 SDs above the mean

Region III: values more than 3 SDs above the mean

Using the 68-95-99.7 rule, approximately what percent of the values lie in NONE of these three regions?

(A) 16.15%
(B) 81.35%
(C) 83.7%
(D) 83.85%
(E) 84%

Q27 · STA-DDI-038 · QC · 125s

Data set P is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Data set Q is 1, 1, 1, 5, 5, 6, 6, 10, 10, 10. Both have mean 5.5. (Standard deviation here is the population standard deviation, as on the GRE.)

Column A: The standard deviation of data set P Column B: The standard deviation of data set Q

Q28 · STA-DDI-039 · Numeric Entry · 120s

The 50 exam scores in a class have mean 72 and standard deviation 8. The teacher rescales every score x to a new score a(x + 5), where a is a positive constant, and the mean of the new scores is 115.5. What is the standard deviation of the new scores? Enter your answer as a number.

Numeric entry — write the number.

Q29 · STA-DDI-040 · Select all that apply · 125s

A data set of 100 values has mean 60, median 55, standard deviation 12, Q1 = 45, and Q3 = 70. Which of the following must be true? Select all that apply.

(A) The sum of the 100 values is 6,000.
(B) The interquartile range is 25.
(C) If every value is multiplied by 3, the new standard deviation is 36 and the new median is 165.
(D) Exactly 25 of the values are less than 45.
(E) If 8 is added to every value, the new standard deviation is 20.

Q30 · STA-DDI-043 · MCQ · 125s

The scores on a test are normally distributed. A score of 88 is at the 97.5th percentile and a score of 76 is at the 16th percentile. Using the 68-95-99.7 rule, what score is at the 84th percentile?

(A) 80
(B) 82
(C) 84
(D) 85
(E) 88

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