Central Tendency — Advanced Practice
1. The table shows exam scores and their frequencies:
What is the mean?
Solution:
Weighted sum: 2×60 + 3×80 + 1×100 = 120 + 240 + 100 = 460.
Total students: 2 + 3 + 1 = 6.
Mean = 460 ÷ 6 ≈ 76.67.
2. A final course score is calculated as: 40% from assignment average + 60% from exam.
A student earned an assignment average of 80 and an exam score of 70. What is the final score?
Solution:
Final = 0.4 × 80 + 0.6 × 70 = 32 + 42 = 74.
3. Group A has 10 students with a mean score of 70. Group B has 20 students with a mean score of 80. What is the mean of all 30 students together?
Solution:
Group A total: 10×70 = 700. Group B total: 20×80 = 1600.
Combined: 700 + 1600 = 2300. Mean = 2300 ÷ 30 ≈ 76.67.
4. The mean of four numbers is 12. Three of them are 10, 15, and 9. What is the fourth number?
Solution:
Total sum = 12 × 4 = 48. Known sum: 10 + 15 + 9 = 34. Fourth number = 48 − 34 = 14.
5. The class mean is 70. A 5-point bonus is added to every score. What will the new mean be?
Solution:
Adding the same constant to every value increases the mean by that amount. New mean = 70 + 5 = 75. The median and mode also increase by 5.
6. The mean student weight is 50 kg. Convert units from kg to grams (× 1000). What is the new mean in grams?
Solution:
Multiplying all values by 1000 multiplies the mean by 1000. New mean = 50 × 1000 = 50,000 grams.
7. Data: 10, 12, 13, 14, 80 — what is true?
Solution:
80 is an extreme value that greatly increases the mean, but the median (middle value after sorting) stays at 13. The median is "protected" against extreme values.
8. Frequency table: score 60 — 2 students, score 70 — 3 students, score 80 — 1 student. What is the median?
Solution:
Expanded list: 60, 60, 70, 70, 70, 80. With 6 students, the 3rd and 4th values are both 70. Median = 70.
9. Score 50 appeared 4 times, 60 appeared 2 times, 70 appeared 4 times. What is the mode?
Solution:
Both 50 and 70 have frequency 4 — the highest frequency. Mode = 50 and 70 (bimodal distribution).
10. Final course score: 30% from Exam A, 70% from Exam B.
A student scored 60 in Exam A and 90 in Exam B. What is the final score?
Solution:
0.3 × 60 = 18; 0.7 × 90 = 63. Total = 18 + 63 = 81.
11. A score table divided into class intervals:
- 50–60: 2 students
- 60–70: 3 students
- 70–80: 5 students
Assuming each student scores at the midpoint of their interval, what is the approximate mean?
Solution:
Midpoints: 55, 65, 75.
Weighted sum: 2×55 + 3×65 + 5×75 = 110 + 195 + 375 = 680.
Total students: 10. Mean = 680 ÷ 10 = 68 ≈ 69.
12. What is the median of: 4, 9, 2, 7, 3, 10?
Solution:
Sort: 2, 3, 4, 7, 9, 10. With 6 values (even count), the two middle values are 4 and 7. Median = (4 + 7) ÷ 2 = 5.5.
13. Data: 5, 7, 9, 11, 13 — the mean is 9. Replace 5 with 15. What is the new mean?
Solution:
Original sum = 45. New sum: 15+7+9+11+13 = 55. New mean = 55 ÷ 5 = 11. Adding 10 to one value increases the total by 10, so the mean rises by 10÷5 = 2.
14. The class mean for 20 students is 75. Five new students join, each scoring 80. What is the new mean?
Solution:
Original total: 20 × 75 = 1500. New students: 5 × 80 = 400. Combined total: 1900. New mean = 1900 ÷ 25 = 76.
15. Most employees in a company earn between 8,000 and 12,000, but the CEO earns 80,000. Which central measure better describes the "typical employee salary"?
Solution:
The CEO's salary is an extreme outlier that inflates the mean. The median describes the middle employee's earnings — a better representation of the typical salary.
16. A survey asked students how many courses they take each semester:
1 course — 5 students; 2 courses — 12 students; 3 courses — 8 students; 4 courses — 5 students.
What is the modal number of courses?
Solution:
The value with the highest frequency is 2 courses (12 students). Mode = 2.
17. A constant of 10 is added to every value in a dataset. What happens to the mean, median, and mode?
Solution:
Shifting all data points by the same constant shifts all measures of central tendency by that constant. Mean, median, and mode all increase by 10.
18. Every value in a dataset is multiplied by 3. What happens to the mean, median, and mode?
Solution:
Multiplying all values by a constant multiplies all measures of central tendency by that constant. Mean, median, and mode are all multiplied by 3.
19. 40% of students have a mean of 70 and 60% have a mean of 90. What is the overall class mean?
Solution:
Weighted mean: 0.4 × 70 + 0.6 × 90 = 28 + 54 = 82.
20. The median of a dataset is 50. What can be concluded with certainty?
Solution:
The median is the positional midpoint: at least half the values are ≤ 50 and at least half are ≥ 50. This says nothing about the mean or mode.