Grade 9 › G9U08 Statistics: Spread

Grade 9 › G9U08 Statistics: Spread › L1
GRADE 9 · ALBERTAOUTCOME AB-G9-9ST1

L1 · The five-number summary

Learning goal

“I can build the five-number summary of a distribution - minimum, Q1, median, Q3, maximum - and use it to describe how spread out the data are.”

Distribution — a set of data values, thought of as one whole group whose shape and spread can be described.
Five-number summary — five landmarks that summarize a distribution: minimum, Q1, median, Q3, maximum.
Lower quartile (Q1) — the median of the lower half of the ordered data. About one quarter of the values lie at or below it.
Upper quartile (Q3) — the median of the upper half of the ordered data. About three quarters of the values lie at or below it.
Worked example

Quiz scores of 9 students: 62, 71, 68, 75, 80, 66, 71, 90, 58. Step 1: Always order first: 58, 62, 66, 68, 71, 71, 75, 80, 90. Step 2: The median is the middle value, the 5th of 9: median = 71. Step 3: Split at the median. The median itself belongs to NEITHER half. Lower half: 58, 62, 66, 68. Upper half: 71, 75, 80, 90. Step 4: Q1 is the median of the lower half. Four values, so average the middle two: Q1 = (62 + 66) / 2 = 64. Q3 is the median of the upper half: Q3 = (75 + 80) / 2 = 77.5. Step 5: Read off the ends: minimum 58, maximum 90, range = 90 - 58 = 32. Five-number summary: 58, 64, 71, 77.5, 90. Say it plainly: sort, find the middle, then find the middle of each half. The five numbers are a map of the whole distribution.

Check yourself — explain it

Why is the median 71 NOT included in either half when finding Q1 and Q3?

Paired problem — example still visible

Lap times (seconds) for 7 swimmers: 34, 29, 41, 36, 31, 45, 38. Build the five-number summary. -> Ordered: 29, 31, 34, 36, 38, 41, 45. Median 36. Lower half 29, 31, 34 -> Q1 = 31. Upper half 38, 41, 45 -> Q3 = 41. Summary: 29, 31, 36, 41, 45. Range 16.

Faded completion — last step is yours

"Data: 12, 15, 18, 21, 24, 27, 30, 33, 36 (nine values, already ordered). Median = 24. Lower half: 12, 15, 18, 21, so Q1 = (15 + 18) / 2 = ___. Upper half: 27, 30, 33, 36, so Q3 = ___." -> Q1 = 16.5; Q3 = 31.5

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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