L2 · Interquartile range and the 1.5 x IQR outlier rule
“I can find the interquartile range of a distribution, and I can use the 1.5 x IQR rule to decide - and justify - whether a value is an outlier.”
Quiz scores: 12, 15, 14, 18, 16, 20, 13, 17, 19, 42. Step 1: Order and summarize: 12, 13, 14, 15, 16, 17, 18, 19, 20, 42. Median = (16 + 17) / 2 = 16.5. Lower half 12, 13, 14, 15, 16 -> Q1 = 14. Upper half 17, 18, 19, 20, 42 -> Q3 = 19. Step 2: IQR = Q3 - Q1 = 19 - 14 = 5. The middle half of the class is packed within 5 points. Step 3: The rule allows 1.5 x IQR = 1.5 x 5 = 7.5 points of slack beyond each quartile. Upper limit = 19 + 7.5 = 26.5. Lower limit = 14 - 7.5 = 6.5. Step 4: Test the suspects: 42 > 26.5, so 42 is an outlier. No value is below 6.5, so there is no low outlier. Say it plainly: the limits are built from the middle of the data, so one wild value cannot move them much - that is the whole point of the rule.
Why is the outlier test built from Q1, Q3 and the IQR instead of from the minimum and maximum?
Test the value 55 in the data 30, 33, 36, 39, 42, 45, 48, 51, 55 (already ordered). -> Median 42; lower half 30, 33, 36, 39 -> Q1 = (33 + 36) / 2 = 34.5; upper half 45, 48, 51, 55 -> Q3 = (48 + 51) / 2 = 49.5; IQR = 15; upper limit = 49.5 + 22.5 = 72. 55 < 72, so 55 is NOT an outlier - it is just the maximum.
"A data set has Q1 = 20 and Q3 = 28. IQR = 8, so 1.5 x IQR = 12. The lower limit is 20 - 12 = ___ and the upper limit is ___." -> lower limit 8; upper limit 40
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.