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.

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

Each piece you already know:

$\min$ and $\max$ — smallest and largest values.
Median — the middle value of the sorted data.
$Q_1$ and $Q_3$ — medians of the lower and upper halves (excluding the overall median when the count is odd).

The procedure:

Sort the data.
Read off the min and max from the ends.
Find the median.
Split into lower and upper halves; find each half’s median to get $Q_1$ and $Q_3$ .

Worked example

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

Min: 2. Max: 26.

Nine values → median is the 5th: $14$ .

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

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

(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

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

Problem 2

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

Problem 3

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

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

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

Problem 11

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

Problem 12

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

Problem 13

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

Problem 14

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

Challenge

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

Problem 15

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

Problem 16

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

Problem 17

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

Problem 18

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

Problem 19

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

Problem 20

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

Problem 21

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

Problem 22

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