Question 1

The ages of 6 employees are: 24, 28, 24, 32, 24, and 30 years. What is the mode of their ages?

Accepted Answer

Step 1: List all values: 24, 28, 24, 32, 24, 30. Step 2: Count the frequency of each age. The age 24 appears 3 times; 28, 32, and 30 each appear once. Step 3: Mode is the value with the highest frequency. Therefore, the mode is 24 years.

Question 2

The median of the dataset {7, 9, 5, 11, 8} is:

Accepted Answer

Step 1: Arrange the data in ascending order: 5, 7, 8, 9, 11. Step 2: Since there are 5 values (odd count), the median is the middle value at position (5+1)/2 = 3. Step 3: The 3rd value is 8. Therefore, the median is 8.

Question 3

The marks obtained by 5 students in a test are: 12, 18, 12, 15, and 23. What is the mode of their marks?

Accepted Answer

Step 1: Arrange the data: 12, 12, 15, 18, 23. Step 2: Identify the frequency of each value. The value 12 appears twice, while 15, 18, and 23 each appear once. Step 3: Mode is the value with the highest frequency. Therefore, the mode is 12.

Question 4

The mean of four numbers is 20. If three of the numbers are 16, 18, and 22, what is the fourth number?

Accepted Answer

Step 1: Use the formula: Mean = (Sum of all values) / (Number of values). Step 2: Given Mean = 20 and count = 4, so Sum = 20 × 4 = 80. Step 3: Sum of three known numbers = 16 + 18 + 22 = 56. Step 4: Fourth number = 80 - 56 = 24. Therefore, the fourth number is 24.

Question 5

The median of the dataset {14, 22, 18, 26, 30, 16, 20} is:

Accepted Answer

Step 1: Arrange the data in ascending order: 14, 16, 18, 20, 22, 26, 30. Step 2: Count the values: there are 7 values (odd count). Step 3: The median is at position (7+1)/2 = 4. Step 4: The 4th value is 20. Therefore, the median is 20.

Question 6

The mean of 8 numbers is 24. When two numbers are removed, the mean of the remaining 6 numbers becomes 22. What is the sum of the two removed numbers?

Accepted Answer

Step 1: Find the sum of all 8 numbers using the formula: Sum = Mean × Count. Sum of 8 numbers = 24 × 8 = 192. Step 2: Find the sum of the remaining 6 numbers. Sum of 6 numbers = 22 × 6 = 132. Step 3: Calculate the sum of the two removed numbers. Sum of removed numbers = 192 - 132 = 60. Therefore, the answer is 60.

Question 7

In a dataset of 10 numbers, the mean is 35 and the median is 32. If the largest number is replaced with 65 (previously it was 50), what is the new mean?

Accepted Answer

Step 1: Original sum = Mean × Count = 35 × 10 = 350. Step 2: The largest number (50) is replaced with 65. Step 3: Change in sum = 65 − 50 = 15. Step 4: New sum = 350 + 15 = 365. Step 5: New mean = 365 ÷ 10 = 36.5.

Question 8

A teacher records test scores for 6 students: {45, 52, 58, 65, 72, 88}. What is the median score?

Accepted Answer

Step 1: The dataset has 6 numbers (even count). Step 2: For even-count datasets, median is the average of the two middle values. Step 3: The two middle values are the 3rd and 4th values: 58 and 65. Step 4: Median = (58 + 65) ÷ 2 = 123 ÷ 2 = 61.5.

Question 9

The mean of 8 numbers is 24. If one number is removed, the mean of the remaining 7 numbers becomes 22. What is the removed number?

Accepted Answer

Step 1: Sum of 8 numbers = 8 × 24 = 192. Step 2: Sum of 7 numbers = 7 × 22 = 154. Step 3: Removed number = 192 − 154 = 38.

Question 10

A dataset has mode 15, and the frequency of 15 is 6. If all other values appear exactly 3 times each, and there are 4 other distinct values, how many total observations are in the dataset?

Accepted Answer

Step 1: Mode 15 appears 6 times. Step 2: There are 4 other distinct values, each appearing 3 times. Step 3: Total observations = 6 + (4 × 3) = 6 + 12 = 18.

Question 11

The mean of a dataset is 50. If each observation is increased by 20%, what is the new mean?

Accepted Answer

Step 1: When each observation is multiplied by a constant k, the mean is also multiplied by k. Step 2: Increase by 20% means multiply by 1.20. Step 3: New mean = 50 × 1.20 = 60.

Question 12

In a dataset of 5 numbers, the median is 18 and the mode is 12. If four of the numbers are 12, 12, 18, and 24, what is the fifth number?

Accepted Answer

Step 1: Arrange known values: 12, 12, 18, 24, and unknown x. Step 2: For median of 5 numbers, the 3rd value when sorted must be 18. Step 3: If x ≤ 12, sorted order is x, 12, 12, 18, 24 → median = 12 (not 18). Step 4: If 12 < x < 18, sorted order is 12, 12, x, 18, 24 → median = x (not 18). Step 5: If x ≥ 18, sorted order is 12, 12, 18, x, 24 or 12, 12, 18, 24, x → median = 18 ✓. Step 6: Mode is 12 (appears twice), so x ≠ 12, 18, 24. Step 7: Therefore x ≥ 18 and x ∉ {12, 18, 24}. The only valid value is x = 20.

Question 13

A dataset has 12 values with mean 25. Another dataset has 8 values with mean 35. When both datasets are combined, what is the mean of the combined dataset?

Accepted Answer

Step 1: Sum of first dataset = 25 × 12 = 300. Step 2: Sum of second dataset = 35 × 8 = 280. Step 3: Combined sum = 300 + 280 = 580. Step 4: Combined count = 12 + 8 = 20. Step 5: Combined mean = 580 ÷ 20 = 29.

Question 14

In a dataset of 9 observations, the mean is 32. One observation is removed, and the new mean becomes 30. What was the value of the removed observation?

Accepted Answer

Step 1: Original sum = Mean × Count = 32 × 9 = 288. Step 2: New sum (after removal) = New mean × New count = 30 × 8 = 240. Step 3: Removed value = Original sum - New sum = 288 - 240 = 48.

Question 15

A dataset contains 8 observations. The mean is 24. If two new observations of values 40 and 32 are added to the dataset, what is the new mean?

Accepted Answer

Step 1: Original sum = Mean × Count = 24 × 8 = 192. Step 2: New sum = 192 + 40 + 32 = 264. Step 3: New count = 8 + 2 = 10. Step 4: New mean = 264 ÷ 10 = 26.4.

Question 16

A frequency distribution has 10 values. The mode is 15 (appears 4 times), and the remaining 6 values are distinct and all different from 15. If the sum of all 10 values is 140, what is the mean?

Accepted Answer

Step 1: Mode = 15 appears 4 times. Contribution = 15 × 4 = 60. Step 2: Sum of remaining 6 distinct values = 140 - 60 = 80. Step 3: Mean = Total sum / Count = 140 / 10 = 14.

Question 17

The mean of 6 numbers is 15. If one number is replaced by 27, the new mean becomes 18. What was the original number that was replaced?

Accepted Answer

Step 1: Mean of 6 numbers is 15, so sum of original 6 numbers = 6 × 15 = 90. Step 2: After replacing one number with 27, new mean is 18, so new sum = 6 × 18 = 108. Step 3: Increase in sum = 108 - 90 = 18. Step 4: This increase equals (27 - original number). So, 27 - original number = 18. Step 5: Original number = 27 - 18 = 9.

Question 18

The median of the dataset {4, 8, 2, 10, 6, 12, 14} is:

Accepted Answer

Step 1: Arrange data in ascending order: 2, 4, 6, 8, 10, 12, 14. Step 2: Count the number of values: n = 7 (odd number). Step 3: Median position = (n+1)/2 = (7+1)/2 = 4th position. Step 4: The 4th value in the ordered list is 8. Therefore, median = 8.

Question 19

Five numbers have a mean of 24. Four of the numbers are 18, 22, 26, and 28. What is the fifth number?

Accepted Answer

Step 1: Mean of 5 numbers is 24, so sum of all 5 numbers = 5 × 24 = 120. Step 2: Sum of four known numbers = 18 + 22 + 26 + 28 = 94. Step 3: Fifth number = 120 - 94 = 26.

Question 20

A dataset has 8 values. When arranged in order, the 4th and 5th values are 35 and 39 respectively. What is the median?

Accepted Answer

Step 1: For an even number of values (n=8), median = (value at position n/2 + value at position (n/2)+1) / 2. Step 2: Median = (4th value + 5th value) / 2 = (35 + 39) / 2. Step 3: Median = 74 / 2 = 37. Therefore, median = 37.

Question 21

The marks obtained by 5 students in a test are: 12, 18, 12, 15, and 23. What is the mode of their marks?

Accepted Answer

Step 1: Arrange the data: 12, 12, 15, 18, 23. Step 2: Identify the frequency of each value — 12 appears twice, all others appear once. Step 3: Mode is the value with highest frequency. Therefore, mode = 12.

Question 22

The median of the dataset {7, 11, 5, 9, 13} is ______.

Accepted Answer

Step 1: Arrange in ascending order: 5, 7, 9, 11, 13. Step 2: Count the number of values = 5 (odd number). Step 3: Median is the middle value = 3rd value = 9. Therefore, median = 9.

Question 23

The mean of 8 observations is 20. If two observations are removed, the mean of the remaining 6 observations is 18. What is the sum of the two removed observations?

Accepted Answer

Step 1: Sum of 8 observations = 8 × 20 = 160. Step 2: Sum of remaining 6 observations = 6 × 18 = 108. Step 3: Sum of two removed observations = 160 - 108 = 52.

Question 24

The median of the dataset {7, 9, 5, 11, 3} is:

Accepted Answer

Step 1: Arrange data in ascending order: 3, 5, 7, 9, 11. Step 2: Count the number of values: n = 5 (odd number). Step 3: Median position = (n+1)/2 = (5+1)/2 = 3rd position. Step 4: The 3rd value in the ordered list is 7. Therefore, median = 7.

Question 25

The mean of five numbers is 18. Four of the numbers are 12, 16, 20, and 24. What is the fifth number?

Accepted Answer

Step 1: Sum of five numbers = 5 × 18 = 90. Step 2: Sum of four known numbers = 12 + 16 + 20 + 24 = 72. Step 3: Fifth number = 90 − 72 = 18.

Question 26

The mean of 8 numbers is 24. If one number is replaced by 40, the new mean becomes 27. What was the original number that was replaced?

Accepted Answer

Step 1: Sum of original 8 numbers = 8 × 24 = 192. Step 2: Sum of new 8 numbers = 8 × 27 = 216. Step 3: Difference = 216 − 192 = 24. Step 4: Original number = 40 − 24 = 16.

Question 27

The mean of 6 numbers is 18. If the mean of the first 4 numbers is 16, what is the mean of the last 2 numbers?

Accepted Answer

Step 1: Sum of all 6 numbers = 6 × 18 = 108. Step 2: Sum of first 4 numbers = 4 × 16 = 64. Step 3: Sum of last 2 numbers = 108 − 64 = 44. Step 4: Mean of last 2 numbers = 44 ÷ 2 = 22.

Question 28

A dataset has 9 observations: 12, 15, 18, 12, 20, 25, 12, 30, 22. What is the mode of this dataset?

Accepted Answer

Step 1: Arrange and count frequency of each value. Step 2: 12 appears 3 times, 15 appears 1 time, 18 appears 1 time, 20 appears 1 time, 25 appears 1 time, 30 appears 1 time, 22 appears 1 time. Step 3: Mode is the value with highest frequency = 12.

Question 29

The median of five numbers is 16. If the numbers in ascending order are 10, 12, x, 20, 25, what is the value of x?

Accepted Answer

Step 1: For 5 numbers in ascending order, the median is the 3rd number (middle position). Step 2: The 3rd number is x. Step 3: Given median = 16, therefore x = 16.

Question 30

Seven students scored: 45, 52, 48, 55, 52, 60, 52 marks. The mean score is 52. What is the difference between the mode and the median of these scores?

Accepted Answer

Step 1: Arrange in order: 45, 48, 52, 52, 52, 55, 60. Step 2: Mode = 52 (appears 3 times, highest frequency). Step 3: Median = 4th value (middle of 7) = 52. Step 4: Difference = 52 − 52 = 0.

Question 31

The mean of 8 numbers is 24. If one number is removed, the mean of the remaining 7 numbers becomes 22. What is the removed number?

Accepted Answer

Step 1: Sum of 8 numbers = 8 × 24 = 192. Step 2: Sum of 7 numbers = 7 × 22 = 154. Step 3: Removed number = 192 − 154 = 38.

Question 32

In a dataset of 10 values arranged in ascending order: 5, 8, 12, 15, 18, 22, 25, 28, 32, 40. What is the median?

Accepted Answer

Step 1: Count of values = 10 (even number). Step 2: For even count, median = (5th value + 6th value) / 2. Step 3: Median = (18 + 22) / 2 = 40 / 2 = 20.

Question 33

A dataset has the following frequency distribution: Value 10 appears 4 times, Value 15 appears 6 times, Value 20 appears 5 times. What is the mode?

Accepted Answer

Step 1: Count frequencies: 10 (frequency 4), 15 (frequency 6), 20 (frequency 5). Step 2: Mode is the value with highest frequency. Step 3: Value 15 has the highest frequency of 6. Therefore, mode = 15.

Question 34

A dataset has 9 values arranged in ascending order: 5, 8, 11, 14, x, 20, 23, 26, 29. If the mean equals the median, what is the value of x?

Accepted Answer

Step 1: With 9 values, median is the 5th value = x. Step 2: Mean = (5 + 8 + 11 + 14 + x + 20 + 23 + 26 + 29) / 9 = (136 + x) / 9. Step 3: Set mean = median: (136 + x) / 9 = x. Step 4: 136 + x = 9x → 136 = 8x → x = 17.

Question 35

A dataset contains 12 observations. When arranged in ascending order, the median is 45. If the 6th observation is 42 and the 7th observation is 48, what is the sum of the 6th and 7th observations?

Accepted Answer

Step 1: For 12 observations (even number), the median is the average of the 6th and 7th values when arranged in order. Step 2: Median = (6th observation + 7th observation) ÷ 2 = 45. Step 3: Therefore, 6th observation + 7th observation = 45 × 2 = 90. Step 4: Verify: (42 + 48) ÷ 2 = 90 ÷ 2 = 45 ✓. The sum is 90.

Question 36

In a frequency distribution, the mode is 35 and the median is 40. If the mean is 42, and we know that the relationship 3(Mean − Median) = Mean − Mode holds approximately, what is the value of (Mode + Median − Mean)?

Accepted Answer

Step 1: Verify the empirical relationship: 3(Mean − Median) = Mean − Mode. Step 2: Substitute values: 3(42 − 40) = 42 − 35 → 3(2) = 7 → 6 ≠ 7. The relationship is approximate, not exact. Step 3: Calculate Mode + Median − Mean = 35 + 40 − 42 = 33. Step 4: This tests whether students blindly apply the formula or compute directly from given values.

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