Comparing Two Tutoring Branches with Mean, Median and IQR
Two branches of a tutoring centre each enter nine students for a mock mathematics examination. The scores are: Branch A: 52, 58, 61, 64, 67, 70, 73, 76, 82. Branch B: 45, 55, 60, 65, 68, 72, 78, 85, 95. The centre's advertisement claims that Branch B is the stronger branch because its mean score is higher. A parent comparing the two branches is unsure, because Branch B also has one very high score and one low score. The parent wants to know which branch is more consistent, which branch has the higher median, and how the comparison changes if every student in Branch A receives 3 extra marks after a marking review. Assume the scores are final and no other changes are made.
1. Calculate the mean and median for each branch. Does the mean alone justify the advertisement?
Read the worked answer
Branch A sum = 52+58+61+64+67+70+73+76+82 = 603, so mean = 603/9 = 67. With 9 ordered scores, the median is the 5th score = 67.
Branch B sum = 45+55+60+65+68+72+78+85+95 = 623, so mean = 623/9 = 69.22 (to 2 d.p.). The median is the 5th score = 68.
The mean for B is higher by 2.22 marks, but the median is higher by only 1 mark. The mean is pulled upward by the 95 and downward by the 45; the median is resistant to these extremes. The advertisement's claim based only on the mean is therefore incomplete: it does not show how consistently students perform or where the bulk of scores lie.
2. Calculate the interquartile range for each branch and interpret the result. Which branch would you recommend to a parent whose child is around the median?
Read the worked answer
For 9 ordered values, Q1 is the median of the first four values and Q3 is the median of the last four.
Branch A: lower half 52,58,61,64 gives Q1=(58+61)/2=59.5; upper half 70,73,76,82 gives Q3=(73+76)/2=74.5; IQR=74.5-59.5=15.
Branch B: lower half 45,55,60,65 gives Q1=(55+60)/2=57.5; upper half 72,78,85,95 gives Q3=(78+85)/2=81.5; IQR=81.5-57.5=24.
Branch A has a smaller IQR, so the middle 50% of its scores is more tightly clustered; Branch B is more spread out. For a child around the median, Branch A offers more predictable performance, whereas Branch B offers a slightly higher median and a much higher top score but also more variability. The recommendation depends on the parent's priority: consistency favours A; chance at a high score favours B.
3. After every Branch A score is increased by 3, state the new mean, median and interquartile range of Branch A, and explain how this affects the comparison with Branch B.
Read the worked answer
Adding 3 to every score shifts the whole distribution by 3.
New Branch A mean = 67+3=70; new median = 67+3=70. The IQR is unchanged at 15 because both quartiles also increase by 3: Q1=62.5, Q3=77.5, and the difference is 15. The spread of Branch A is therefore unchanged.
Compared with Branch B, Branch A now has a higher mean (70 vs 69.22) and a higher median (70 vs 68), while still having the smaller IQR (15 vs 24). An advertisement based solely on the mean would now favour A, showing how sensitive a mean-only comparison is to a uniform mark adjustment.