L4 · Comparing distributions by spread: clinic
“I can compare the spread of two distributions - two samples of the same population, or the same data with and without an outlier - using range and IQR, and I can explain which measure of spread an outlier damages and which one resists it.”
Two classes write the same quiz (out of 20). Their box plots, on the same axis, are described: Class A: minimum 8, Q1 11, median 13, Q3 15, maximum 18. Class B: minimum 2, Q1 8, median 13, Q3 18, maximum 20. Step 1: Compare centres first: the medians are identical (13). As typical scores, the classes match. Step 2: Compare the middles: IQR of A = 15 - 11 = 4; IQR of B = 18 - 8 = 10. B's middle half is spread over 10 points; A's over only 4. Step 3: Compare end to end: range of A = 10; range of B = 18. B stretches further at both ends too. Conclusion: same typical score, but Class A's results are far more consistent. In Class B, a score of 13 is the median, yet half the class is scattered from 8 to 18 - the median "typifies" B's scores much less well. Spread is what tells you how much trust a centre deserves. Say it plainly: compare in a fixed order - centre, then IQR, then range - and never judge spread by the median.
Two samples have the same median. What extra information do you need before saying one is more consistent?
Two groups time a commute (minutes). Group 1 summary: 18, 24, 29, 33, 40. Group 2 summary: 15, 26, 30, 34, 44. Compare the spreads. -> Medians 29 and 30 (nearly equal). IQRs: Group 1 = 33 - 24 = 9; Group 2 = 34 - 26 = 8 - Group 2's middle half is slightly tighter. Ranges: Group 1 = 22; Group 2 = 29 - Group 2 stretches further overall. Verdict: middles about equal (Group 2 marginally tighter); overall, Group 1 is less spread because its extremes stay closer in.
"Samples X and Y have IQRs 7 and 7 and ranges 15 and 26. Their medians are equal. X and Y have equally spread middle halves, but overall ___ is more spread out, because ___." -> Y; its range is larger - its extreme values stretch further from the centre
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.