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Statistics

Five-Number Summary

Lesson

The five-number summarydescribes a data set with five carefully chosen values — enough to capture the shape without listing everything. It’s the basis of the box plot.

Five-number summary=(min⁡, Q1, median, Q3, max⁡)\text{Five-number summary} = (\min,\ Q_1,\ \text{median},\ Q_3,\ \max)

Each piece you already know:

  • min⁡\min and max⁡\max — smallest and largest values.
  • Median — the middle value of the sorted data.
  • Q1Q_1 and Q3Q_3 — medians of the lower and upper halves (excluding the overall median when the count is odd).

The procedure:

  1. Sort the data.
  2. Read off the min and max from the ends.
  3. Find the median.
  4. Split into lower and upper halves; find each half’s median to get Q1Q_1 and Q3Q_3.

Worked example

Data: 2, 5, 8, 11, 14, 17, 20, 23, 26 (already sorted).

Min: 2. Max: 26.

Nine values → median is the 5th: 1414.

Lower half: 2,5,8,112, 5, 8, 11. Q1=(5+8)/2=6.5Q_1 = (5 + 8)/2 = 6.5.

Upper half: 17,20,23,2617, 20, 23, 26. Q3=(20+23)/2=21.5Q_3 = (20 + 23)/2 = 21.5.

(2, 6.5, 14, 21.5, 26)(2,\ 6.5,\ 14,\ 21.5,\ 26)

How to type your answer

List all five values, separated by commas — in increasing order (min first, max last). No parentheses. Examples: 3,4,7,10,11, 2,6.5,14,21.5,26, -5,-2,3,11,15.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Five-number summary of 3, 5, 7, 9, 11\text{Five-number summary of } 3,\ 5,\ 7,\ 9,\ 11

Problem 2

Five-number summary of 1, 3, 5, 7, 9\text{Five-number summary of } 1,\ 3,\ 5,\ 7,\ 9

Problem 3

Five-number summary of 2, 4, 6, 8, 10, 12, 14\text{Five-number summary of } 2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14

Problem 4

Five-number summary of 5, 10, 15, 20, 25, 30, 35\text{Five-number summary of } 5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35

Practice

Standard problems matching the lesson.

Problem 5

Five-number summary of 2, 5, 8, 11, 14, 17, 20, 23, 26\text{Five-number summary of } 2,\ 5,\ 8,\ 11,\ 14,\ 17,\ 20,\ 23,\ 26

Problem 6

Five-number summary of 15, 20, 25, 30, 35, 40, 45\text{Five-number summary of } 15,\ 20,\ 25,\ 30,\ 35,\ 40,\ 45

Problem 7

Five-number summary of 1, 4, 7, 10, 13, 16, 19, 22, 25\text{Five-number summary of } 1,\ 4,\ 7,\ 10,\ 13,\ 16,\ 19,\ 22,\ 25

Problem 8

Five-number summary of 6, 8, 10, 12, 14\text{Five-number summary of } 6,\ 8,\ 10,\ 12,\ 14

Problem 9

Five-number summary of 12, 17, 22, 27, 32, 37\text{Five-number summary of } 12,\ 17,\ 22,\ 27,\ 32,\ 37

Problem 10

Five-number summary of −5, −2, 0, 3, 7, 11, 15\text{Five-number summary of } -5,\ -2,\ 0,\ 3,\ 7,\ 11,\ 15

Problem 11

Five-number summary of 100, 200, 300, 400, 500, 600, 700, 800\text{Five-number summary of } 100,\ 200,\ 300,\ 400,\ 500,\ 600,\ 700,\ 800

Problem 12

Five-number summary of 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5\text{Five-number summary of } 1.5,\ 2.5,\ 3.5,\ 4.5,\ 5.5,\ 6.5,\ 7.5

Problem 13

Five-number summary of 0, 2, 4, 6, 8, 10\text{Five-number summary of } 0,\ 2,\ 4,\ 6,\ 8,\ 10

Problem 14

Five-number summary of 8, 11, 14, 17, 20, 23, 26, 29\text{Five-number summary of } 8,\ 11,\ 14,\ 17,\ 20,\ 23,\ 26,\ 29

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

Five-number summary of 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43\text{Five-number summary of } 3,\ 7,\ 11,\ 15,\ 19,\ 23,\ 27,\ 31,\ 35,\ 39,\ 43

Problem 16

Five-number summary of 10, 14, 18, 22, 26, 30, 34, 38, 42, 46\text{Five-number summary of } 10,\ 14,\ 18,\ 22,\ 26,\ 30,\ 34,\ 38,\ 42,\ 46

Problem 17

Five-number summary of 5, 9, 12, 14, 18, 20, 24\text{Five-number summary of } 5,\ 9,\ 12,\ 14,\ 18,\ 20,\ 24

Problem 18

Five-number summary of −10, −5, 0, 5, 10, 15, 20\text{Five-number summary of } -10,\ -5,\ 0,\ 5,\ 10,\ 15,\ 20

Problem 19

Five-number summary of 2.1, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5\text{Five-number summary of } 2.1,\ 2.5,\ 3.0,\ 3.5,\ 4.0,\ 4.5,\ 5.0,\ 5.5

Problem 20

Five-number summary of 1, 1, 2, 3, 5, 8, 13, 21, 34\text{Five-number summary of } 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34

Problem 21

Five-number summary of 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80\text{Five-number summary of } 25,\ 30,\ 35,\ 40,\ 45,\ 50,\ 55,\ 60,\ 65,\ 70,\ 75,\ 80

Problem 22

Five-number summary of 7, 8, 8, 10, 10, 10, 12, 12, 14, 15\text{Five-number summary of } 7,\ 8,\ 8,\ 10,\ 10,\ 10,\ 12,\ 12,\ 14,\ 15

Practice

Standard problems matching the lesson.

Problem 23

Bowling scores: 110, 130, 140, 155, 170, 180, 200. Give the five-number summary.

Problem 24

Test scores: 65, 70, 72, 75, 78, 80, 82, 85, 88, 92. Give the five-number summary.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Daily rainfall (mm): 0, 2, 5, 8, 12, 15, 18, 22, 30, 45. Give the five-number summary.

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