Question

1. the mean mark on a quiz was 7. the median mark was 6 and the mode was 5. fifteen students wrote the quiz. the lowest mark was 1 and the highest mark was 10. a) write a possible set of marks. b) one student, who was absent, wrote the quiz later. her mark was 20. how does this mark affect the mean, the median, and the mode? explain.

Explanation:

Step1: Define set constraints

We need 15 marks where:

• Mean = 7: Total sum = $15 \times 7 = 105$
• Median = 6: 8th value (sorted) is 6
• Mode = 5: 5 appears most frequently
• Min = 1, Max = 10

Step2: Build the sorted set

Start with min, mode, median, max:
1, 5, 5, 5, x, x, x, 6, y, y, y, y, y, y, 10
First, set enough 5s to be mode (at least 3, more than any other value). Now calculate remaining sum:
Sum of known values: $1 + 5+5+5 + 6 + 10 = 32$
Remaining sum needed: $105 - 32 = 73$ for 7 values (3 before median, 6 after median)
Assign values to meet constraints:
Before median (3 values, ≤6, not more than 3 of any non-5 value): 2, 3, 4
After median (6 values, ≥6, ≤10, no value appears 3+ times): 6,7,8,9,10,10
Verify sum: $2+3+4+6+7+8+9+10+10 = 59$? Correct total check: $1+2+3+4+5+5+5+6+6+7+8+9+10+10+10 = 105$

Step3: Analyze new mark impact (part b)

Original set: 15 values, sum=105, median=6, mode=5
New mark=20, total values=16

Substep3a: New mean calculation

New total sum = $105 + 20 = 125$
New mean = $\frac{125}{16} = 7.8125$

Substep3b: New median calculation

Sorted set now has 16 values, median = average of 8th and 9th values. Original 8th=6, 9th=6. Adding 20 (largest) keeps 8th=6,9th=6. New median = $\frac{6+6}{2}=6$

Substep3c: New mode check

5 still appears 3 times, no other value appears more often. Mode remains 5.

Answer:

a) A possible sorted set of marks: 1, 2, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10
b)

• Mean: Increases from 7 to 7.8125, since the new mark (20) is higher than the original mean, pulling the average up.
• Median: Stays 6, because adding a value larger than all original values does not change the average of the 8th and 9th values in the now 16-value sorted set, which are still both 6.
• Mode: Remains 5, because 5 still appears more frequently than any other mark, and the new mark (20) only appears once.