Two-Section Test Score Comparison
Two sections of a class take the same quiz. Section A has 10 students with scores: 62, 68, 71, 74, 75, 77, 80, 83, 88, 92. Section B has 10 students with scores: 55, 60, 66, 70, 74, 78, 82, 86, 90, 99. A teacher glances at the lists and says the sections performed about the same because both lists 'look centered in the 70s.' A colleague wants a more careful comparison of center and spread before deciding which section needs review sessions.
1. Compute the mean and median for each section. Do the means and medians tell the same story? Explain any discrepancy.
Read the worked answer
Section A sum = 62+68+71+74+75+77+80+83+88+92 = 770, mean = 77.0. With 10 values, median = average of 5th and 6th = (75 + 77)/2 = 76.0.
Section B sum = 55+60+66+70+74+78+82+86+90+99 = 760, mean = 76.0. Median = (74 + 78)/2 = 76.0.
The means differ by only 1 point, and the medians are identical at 76. But Section B's mean is pulled down slightly by the low value 55 while its high value 99 pulls it back up; the distribution is more spread out. The means and medians roughly agree here, but the 'looks centered in the 70s' observation masks real differences in spread.
2. Compute the range and the interquartile range for each section, and explain which measure better supports the claim that the sections differ in consistency.
Read the worked answer
Range A = 92 − 62 = 30. Range B = 99 − 55 = 44.
For IQR, split each sorted list into lower and upper halves of 5 values each.
- A lower half: 62, 68, 71, 74, 75 → Q1 = 71. Upper half: 77, 80, 83, 88, 92 → Q3 = 83. IQR = 83 − 71 = 12.
- B lower half: 55, 60, 66, 70, 74 → Q1 = 66. Upper half: 78, 82, 86, 90, 99 → Q3 = 86. IQR = 86 − 66 = 20.
Both range (44 vs 30) and IQR (20 vs 12) show Section B is more spread out. The IQR is the better measure here because it ignores the extreme 55 and 99, yet still shows B is less consistent - so the difference is not merely an artifact of two outliers.