Quartiles, percentiles and conventions
Quartiles split the ordered data into sections. Different agreed rules can give different quartiles in a small dataset.
Sort first. Q1 is the 25th percentile, Q2 the median, and Q3 the 75th percentile. The five-number summary is minimum, Q1, median, Q3 and maximum. IQR = Q3 - Q1 and describes the width of the middle half.
In Lab 2's eight-number exercise, use median of halves: find the median of the lower four and the upper four. For our practice list: Q1 = (40+45)/2 = 42.5; Q3 = (65+70)/2 = 67.5; IQR = 25 points. Five-number summary: 35, 42.5, 55, 67.5, 75.
For the larger Lab 2 dataset, pandas uses linear interpolation. On the same eight values, it gives Q1 = 43.75 and Q3 = 66.25. These differ because the conventions locate quartiles differently in finite samples, not because one analyst necessarily made an arithmetic error. State the convention.
A percentile describes rank, not ability or causation. With ties, the exact empirical rank convention matters; the lab explicitly uses the fraction at or below the target value.
Sources: W2 PDF pp. 2-60; L2 Sections 1-8. The personalised Lab 2 has 240 synthetic students, unlike the 1,200-student lecture dataset. Do not mix their numbers.
The lecture's robust summaries are tested by changing one extreme value. The lab also makes the quartile convention explicit.
Practice problems
Supplementary study questions on the lecture concepts — not verbatim exam questions. Each answer counts toward spaced review.
1. Conceptual understanding
Definitions and distinctions.
Why can two software packages report different quartiles?
For 35,40,45,50,60,65,70,75, what is Q1 using median of halves?
2. Application
Apply the concept to a case or calculation.
Q1=12 and Q3=20. What is IQR?
What must happen before using median-of-halves?
3. Interpretation and limitations
Evaluate assumptions and justify conclusions.
Which set of quantities is the five-number summary?
A lab defines percentile as the fraction at or below a value. Why mention “at or below”?
Explain it in writing
Self-checked, not marked. Write a few sentences, then compare with the example answer. Your writing is saved as you type.
Show an example answer
42.5 uses the median of the lower half. 43.75 comes from linear interpolation. They use different finite-sample quartile conventions.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?
Show an example answer
It describes the value's position relative to the observed values under a stated tie/rank convention. It does not prove ability, causation or distance from the mean.
Check: did you answer the question, use evidence or working, and avoid claiming more than the evidence supports?