Descriptive Statistics

Five-Number Summary

Compute the five-number summary: minimum, Q1, median, Q3, and maximum of a data set.

Example

Sort the data and compute min, quartiles, median, and max.

highlighted = computed this step

Step 1 — Data

List the data values.

posx11024314425126871686\begin{array}{cc}\text{pos} & \text{x} \\ 1 & 10 \\ 2 & 4 \\ 3 & 14 \\ 4 & 2 \\ 5 & 12 \\ 6 & 8 \\ 7 & 16 \\ 8 & 6\end{array}

Step 2 — Sort

Sort the data first.

posx12243648510612714816\begin{array}{cc}\text{pos} & \text{x} \\ 1 & \hlmath{2} \\ 2 & \hlmath{4} \\ 3 & \hlmath{6} \\ 4 & \hlmath{8} \\ 5 & \hlmath{10} \\ 6 & \hlmath{12} \\ 7 & \hlmath{14} \\ 8 & \hlmath{16}\end{array}
Sorted data on a number line246810121416

Step 3 — Min and max

Read the minimum and maximum.

min=2max=16min= \hl{2} \quad max= \hl{16}
min 2 and max 16246810121416min2max16

Step 4 — Median

Average the two middle values.

median=(8+10)/2=9median=( 8 + 10 )/ 2 = \hl{9}
median 9 splits the data246810121416min2median9max16

Step 5 — Q1

Average the middle of the lower half.

Q1=(4+6)/2=5Q_1= ( 4 + 6 )/ 2 = \hl{5}
Q1 5 marks the lower half246810121416min2Q15median9max16

Step 6 — Q3

Average the middle of the upper half.

Q3=(12+14)/2=13Q_3= ( 12 + 14 )/ 2 = \hl{13}
Q3 13 marks the upper half246810121416min2Q15median9Q313max16

Step 7 — Summary

State the five-number summary.

(2,5,9,13,16)( \hl{2} , \hl{5} , \hl{9} , \hl{13} , \hl{16} )
Five-number summary on one line246810121416min2Q15median9Q313max16
five-number-summary Five-number summary: Min, Q1, Median, Q3, Max. Steps: 1. Sort the data. 2. Identify Min and Max. 3. Find Median (middle value or average of two middles). 4. Q1 = median of the lower half. 5. Q3 = median of the upper half.