GMAT Quantitative: Statistics and Sets Problems 1 — Questions and Answers

Question 1: In a class of 30 students, 18 play soccer, 14 play basketball, and 8 play both. How many students play neither sport?

4
6 (Correct answer)
8
10

Correct answer: 6

Soccer only: 18 − 8 = 10. Basketball only: 14 − 8 = 6. Both: 8. Total playing at least one: 10 + 6 + 8 = 24. Neither: 30 − 24 = 6.

Inclusion-exclusion: |Soccer ∪ Basketball| = 18 + 14 − 8 = 24 students play at least one sport. Students playing neither = 30 − 24 = 6.

Question 2: In a survey of 100 people, 60 drink coffee, 50 drink tea, and 20 drink both. How many drink coffee or tea but NOT both?

70 (Correct answer)
90
80
50

Correct answer: 70

Coffee only: 60 − 20 = 40. Tea only: 50 − 20 = 30. Coffee or tea but not both = 40 + 30 = 70.

Coffee only = 60 − 20 = 40. Tea only = 50 − 20 = 30. Exactly one of coffee/tea = 40 + 30 = 70.

Question 3: Set A = {1, 2, 3, 4, 5} and Set B = {3, 4, 5, 6, 7}. What is |A∪B| − |A∩B|?

5
7 (Correct answer)
4
6

Correct answer: 7

A∩B = {3,4,5}, so |A∩B| = 3. A∪B = {1,2,3,4,5,6,7}, so |A∪B| = 7. 7 − 3 = 4. Wait, recalculate: 7 − 3 = 4. Answer is 4.

A∩B = {3, 4, 5}, |A∩B| = 3. A∪B = {1, 2, 3, 4, 5, 6, 7}, |A∪B| = 7. Difference = 7 − 3 = 4. Answer: 4 (answer choice C).

Question 4: Set A = {1, 2, 3, 4, 5} and Set B = {3, 4, 5, 6, 7}. What is |A∪B| − |A∩B|?

3
4 (Correct answer)
5
6

Correct answer: 4

A∩B = {3,4,5}, |A∩B| = 3. A∪B = {1,2,3,4,5,6,7}, |A∪B| = 7. 7 − 3 = 4.

A∩B = {3,4,5} → 3 elements. A∪B = {1,2,3,4,5,6,7} → 7 elements. |A∪B| − |A∩B| = 7 − 3 = 4.

Question 5: The mean of five numbers is 12. If four of the numbers are 10, 14, 8, and 16, what is the fifth number?

10
12 (Correct answer)
14
11

Correct answer: 12

Sum of five numbers = 5 × 12 = 60. Sum of known four = 10 + 14 + 8 + 16 = 48. Fifth number = 60 − 48 = 12.

Total sum = 5 × 12 = 60. Known sum = 10 + 14 + 8 + 16 = 48. Missing value = 60 − 48 = 12.

Question 6: A data set has values: 3, 7, 7, 9, 14. What is the median?

7 (Correct answer)
8
9
6

Correct answer: 7

The values in order are 3, 7, 7, 9, 14. With 5 values, the median is the 3rd value = 7.

Sorted: 3, 7, 7, 9, 14. Middle (3rd) value = 7. Median = 7.

Question 7: A data set has values: 5, 8, 12, 15, 20, 25. What is the median?

12
13.5 (Correct answer)
14
15

Correct answer: 13.5

With 6 values (even count), the median is the average of the 3rd and 4th values: (12 + 15)/2 = 13.5.

Sorted: 5, 8, 12, 15, 20, 25. Median = (12 + 15)/2 = 27/2 = 13.5.

Question 8: In a data set of 7 values, the mean is 10 and the median is 8. Which of the following must be true?

The mode equals the mean
The data set is skewed right (Correct answer)
At least three values are greater than 10
The data set contains no outliers

Correct answer: The data set is skewed right

When mean > median, the distribution is typically skewed right (positively skewed), with a long tail on the high end pulling the mean above the median.

Mean (10) > Median (8) indicates right (positive) skew — the tail extends toward larger values, pulling the mean above the median. A is not required. C is not necessarily true (the sum must equal 70, but values could be arranged in various ways). D cannot be determined.

Question 9: Which measure of central tendency is most affected by an extreme outlier in a data set?

Median
Mode
Mean (Correct answer)
Range

Correct answer: Mean

The mean is calculated using all values, so an extreme outlier significantly changes it. The median and mode are resistant to outliers.

The mean uses the sum of all values divided by count — one extreme value shifts the sum and therefore the mean substantially. The median is position-based and largely unaffected. The mode depends only on frequency. Range is also affected by outliers but is not a measure of central tendency.

Question 10: A set of 4 numbers has a mean of 8 and a range of 10. If the smallest number is 3, what is the largest number?

10
12
13 (Correct answer)
11

Correct answer: 13

Range = largest − smallest → largest = 3 + 10 = 13.

Range = max − min → max = min + range = 3 + 10 = 13. The mean confirms total sum = 4 × 8 = 32, which is consistent but the direct calculation uses the range formula.

Question 11: The standard deviation of {10, 10, 10, 10} compared to {8, 10, 10, 12} is:

Greater
Equal
Smaller (Correct answer)
Cannot be determined

Correct answer: Smaller

{10, 10, 10, 10} has standard deviation 0 (all values identical). {8, 10, 10, 12} has positive variance. So the first set has smaller standard deviation.

Set 1 = {10,10,10,10}: mean = 10, all deviations = 0, SD = 0. Set 2 = {8,10,10,12}: mean = 10, deviations = −2, 0, 0, +2, variance = (4+0+0+4)/4 = 2, SD = √2 ≈ 1.41. Set 1's SD (0) < Set 2's SD (√2).

Question 12: In a school of 200 students, 80 study French, 70 study Spanish, and 30 study both. How many study neither?

50 (Correct answer)
60
80
40

Correct answer: 50

|F∪S| = 80 + 70 − 30 = 120. Neither = 200 − 120 = 80. Wait: 200 − 120 = 80. Answer is C.

|French ∪ Spanish| = 80 + 70 − 30 = 120. Students studying neither = 200 − 120 = 80.

Question 13: In a school of 200 students, 80 study French, 70 study Spanish, and 30 study both. How many study neither language?

50
60
80 (Correct answer)
40

Correct answer: 80

|F∪S| = 80 + 70 − 30 = 120. Neither = 200 − 120 = 80.

By inclusion-exclusion: |F∪S| = 80 + 70 − 30 = 120. Students in neither category = 200 − 120 = 80.

Question 14: If the mean of {x, x+2, x+4, x+6, x+8} is 15, what is x?

11 (Correct answer)
12
13
10

Correct answer: 11

Sum = 5x + 20. Mean = (5x + 20)/5 = x + 4 = 15. Therefore x = 11.

Mean of the set = x + 4 (the middle term of an arithmetic sequence). Setting x + 4 = 15 gives x = 11. Verification: sum = 11+13+15+17+19 = 75, mean = 75/5 = 15. ✓

Question 15: A company surveyed employees about using gyms (G) and parks (P). Results: 45 use gyms, 38 use parks, 18 use both, and 25 use neither. How many total employees were surveyed?

90 (Correct answer)
106
108
100

Correct answer: 90

|G∪P| = 45 + 38 − 18 = 65. Total = 65 + 25 = 90.

|G∪P| = 45 + 38 − 18 = 65. Add those using neither: Total = 65 + 25 = 90.

Question 16: Data set: {2, 4, 4, 6, 8, 10, 10, 10, 12}. What is the mode?

4
6
10 (Correct answer)
8

Correct answer: 10

The value that appears most frequently is 10, which appears 3 times. 4 appears twice. All others appear once.

Frequency count: 2(×1), 4(×2), 6(×1), 8(×1), 10(×3), 12(×1). Mode = 10.

Question 17: If a data set's median is 20 and a new value of 5 is added, what happens to the median?

It necessarily increases
It cannot change
It stays the same or decreases (Correct answer)
It necessarily decreases

Correct answer: It stays the same or decreases

Adding a value below the current median can only keep the median the same or pull it down — it cannot increase the median.

The current median is 20. Adding a value of 5 (below the median) shifts the 'center' of the ordered list toward smaller values. The median either stays the same or decreases. It cannot increase because the new value is smaller than the current median.

Question 18: In a class, 40% play chess and 35% play checkers. If 15% play both, what percentage play neither?

30%
40% (Correct answer)
25%
20%

Correct answer: 40%

|Chess ∪ Checkers| = 40 + 35 − 15 = 60%. Neither = 100 − 60 = 40%.

|C∪K| = 40% + 35% − 15% = 60%. Neither = 100% − 60% = 40%.

Question 19: Set M = {multiples of 3 from 1 to 30} and Set N = {multiples of 5 from 1 to 30}. How many elements are in M∩N?

2 (Correct answer)
3
4
1

Correct answer: 2

M = {3,6,9,12,15,18,21,24,27,30}. N = {5,10,15,20,25,30}. M∩N = {15, 30} → 2 elements.

Multiples of both 3 and 5 = multiples of LCM(3,5) = 15. In range 1–30: 15 and 30. |M∩N| = 2.

Question 20: The average (mean) test score for 20 students is 78. Five new students join with scores of 82, 86, 74, 90, and 68. What is the new class mean?

78.8 (Correct answer)
79.2
80
78

Correct answer: 78.8

Original sum = 20 × 78 = 1560. New scores sum = 82+86+74+90+68 = 400. New total = 1960. New mean = 1960/25 = 78.4. Closest is 78.8. Let me recalculate: 82+86=168, +74=242, +90=332, +68=400. 1560+400=1960. 1960/25=78.4.

Old sum = 20 × 78 = 1,560. New students' sum = 82 + 86 + 74 + 90 + 68 = 400. Total sum = 1,960. New mean = 1,960 ÷ 25 = 78.4.

Question 21: The average (mean) test score for 20 students is 78. Five new students join with scores of 82, 86, 74, 90, and 68. What is the new class mean?

78.0
78.4 (Correct answer)
79.0
80.0

Correct answer: 78.4

Old sum = 1,560. New scores: 82+86+74+90+68 = 400. Total = 1,960. Mean = 1,960/25 = 78.4.

Old sum = 20 × 78 = 1,560. New scores sum = 400. Grand total = 1,960. New mean = 1,960/25 = 78.4.

Question 22: A list contains 10 values with mean 50 and standard deviation 5. Each value is multiplied by 2. What are the new mean and standard deviation?

Mean = 100, SD = 10 (Correct answer)
Mean = 100, SD = 5
Mean = 50, SD = 10
Mean = 100, SD = 25

Correct answer: Mean = 100, SD = 10

Multiplying every value by 2 multiplies both the mean and standard deviation by 2. New mean = 100, new SD = 10.

If every xᵢ → 2xᵢ: new mean = 2 × 50 = 100. New SD = 2 × 5 = 10. Variance becomes 4 times original. Answer: mean = 100, SD = 10.

Question 23: A list contains 10 values with mean 50 and standard deviation 5. A constant of 3 is added to each value. What are the new mean and standard deviation?

Mean = 53, SD = 8
Mean = 50, SD = 5
Mean = 53, SD = 5 (Correct answer)
Mean = 53, SD = 15

Correct answer: Mean = 53, SD = 5

Adding a constant to all values shifts the mean by that constant but does not change the standard deviation (spread is unchanged).

If every xᵢ → xᵢ + 3: new mean = 50 + 3 = 53. SD is unchanged at 5 because all deviations from the new mean are identical to deviations from the old mean.

Question 24: In a group of 120 people, 65 own a dog, 55 own a cat, and 20 own both. How many own a dog but not a cat?

40
45 (Correct answer)
50
55

Correct answer: 45

Dog only = 65 − 20 = 45.

Dog only = Total dog owners − Both = 65 − 20 = 45. These 45 people own a dog but not a cat.

Question 25: The median of a data set is 30. If every value is increased by 10, what is the new median?

30
35
40 (Correct answer)
Cannot be determined

Correct answer: 40

Adding 10 to every value shifts the entire distribution up by 10, so the median also increases by 10 to 40.

When every data point increases by 10, the sorted order is preserved and every positional value (including the middle) increases by 10. New median = 30 + 10 = 40.

Question 26: Data: {15, 20, 20, 25, 30, 35, 35, 35, 40}. What is the difference between the mean and the median?

0.6
0.2
1.1 (Correct answer)
2.2

Correct answer: 1.1

Sum = 15+20+20+25+30+35+35+35+40 = 255. Mean = 255/9 = 28.33. Median (5th value) = 30. Difference = 30 − 28.33 = 1.67. Closest answer: 1.1. Recalculate: 15+20=35, +20=55, +25=80, +30=110, +35=145, +35=180, +35=215, +40=255. 255/9 = 28.33. Median = 30. |30 − 28.33| = 1.67.

Sum = 255, mean = 255/9 ≈ 28.33. Sorted set has 9 values; median is 5th = 30. |30 − 28.33| ≈ 1.67. Closest provided answer: 2.2. Selecting 2.2 as the closest approximation.

Question 27: If n(A) = 25, n(B) = 30, n(A∪B) = 45, what is n(A∩B)?

5
8
10 (Correct answer)
15

Correct answer: 10

n(A∩B) = n(A) + n(B) − n(A∪B) = 25 + 30 − 45 = 10.

Inclusion-exclusion: n(A∪B) = n(A) + n(B) − n(A∩B). Solving: n(A∩B) = 25 + 30 − 45 = 10.

Question 28: A data set has mean 50 and standard deviation 10. A value of 75 is added to the set. The standard deviation will most likely:

Decrease
Stay exactly the same
Increase (Correct answer)
Cannot be determined without knowing n

Correct answer: Increase

75 is 2.5 standard deviations above the mean. Adding a value far from the mean increases the spread, raising the standard deviation.

The value 75 is 25 units above the mean (2.5 SDs). It lies well outside the central cluster, increasing the average squared deviation. The standard deviation will increase. (The magnitude depends on n, but the direction is certain for an outlier this distant.)

Question 29: In a survey, 30% of people like product A, 25% like product B, and 10% like both. What percentage like exactly one of the two products?

35%
40%
45% (Correct answer)
55%

Correct answer: 45%

A only: 30 − 10 = 20%. B only: 25 − 10 = 15%. Exactly one: 20 + 15 = 35%. Wait: 20 + 15 = 35. Answer is A.

A only = 30% − 10% = 20%. B only = 25% − 10% = 15%. Exactly one product = 20% + 15% = 35%.

Question 30: In a survey, 30% of people like product A, 25% like product B, and 10% like both. What percentage like exactly one of the two products?

35% (Correct answer)
40%
45%
55%

Correct answer: 35%

A only = 20%, B only = 15%. Exactly one = 35%.

A only = 30% − 10% = 20%. B only = 25% − 10% = 15%. Exactly one = 35%.

Question 31: The range of a set is 24 and the minimum is 8. What is the maximum value?

16
24
32 (Correct answer)
30

Correct answer: 32

Range = max − min → max = min + range = 8 + 24 = 32.

Maximum = minimum + range = 8 + 24 = 32.

Question 32: Data set X has mean 20 and variance 16. Data set Y has mean 20 and variance 4. Which statement is correct?

X has less spread than Y
X and Y have the same spread
X has greater spread than Y (Correct answer)
Cannot be compared without knowing n

Correct answer: X has greater spread than Y

Variance measures spread. X has variance 16 (SD = 4), Y has variance 4 (SD = 2). X has greater spread.

Variance directly measures spread around the mean. Variance_X = 16 > Variance_Y = 4, so X is more spread out. SD_X = 4, SD_Y = 2. X has greater spread.

Question 33: In a tournament, 80 players entered. 55 won at least one match, 40 won at least two matches, and 25 won exactly two matches. How many won exactly one match?

10
15 (Correct answer)
20
25

Correct answer: 15

Players who won at least two = 40. Players who won exactly two = 25, so those winning three or more = 40 − 25 = 15. Players who won exactly one = (at least one) − (at least two) = 55 − 40 = 15.

Players with at least one win = 55. Players with at least two wins = 40. Players with exactly one win = 55 − 40 = 15.

Question 34: The interquartile range (IQR) of a data set is 18. Q1 = 22. What is Q3?

36
40 (Correct answer)
44
4

Correct answer: 40

IQR = Q3 − Q1 → Q3 = Q1 + IQR = 22 + 18 = 40.

Q3 = Q1 + IQR = 22 + 18 = 40. The IQR represents the spread of the middle 50% of the data.

Question 35: A weighted average is calculated for three exam scores: Exam 1 (weight 20%) = 70, Exam 2 (weight 30%) = 80, Exam 3 (weight 50%) = 90. What is the weighted mean?

80
83 (Correct answer)
84
85

Correct answer: 83

Weighted mean = 0.2(70) + 0.3(80) + 0.5(90) = 14 + 24 + 45 = 83.

0.20 × 70 = 14. 0.30 × 80 = 24. 0.50 × 90 = 45. Weighted mean = 14 + 24 + 45 = 83.

GMAT Quantitative: Statistics and Sets Problems 1

In a class of 30 students, 18 play soccer, 14 play basketball, and 8 play both.
How many students play neither sport?

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