SME
Grade 9 Math · Problem type
Box plot from a data list
Use this for “Points in nine games: 12, 18, 9, 22, 15, 30, 14, 17, 20. Make a box plot.”
Chart
9 values in order: 9, 12, 14, 15, 17, 18, 20, 22, 30.
The median is the middle value, 17.
The middle half of the data runs from 13 to 21, a width of 8.
Half the values are at or below the median, 17.
This is an example. Type your own values to start a new problem.
n
Number of values
example
x₁
Value 1
example
x₂
Value 2
example
x₃
Value 3
example
x₄
Value 4
example
x₅
Value 5
example
x₆
Value 6
example
x₇
Value 7
example
x₈
Value 8
example
x₉
Value 9
example
min
Least
worked out (not typed)
Q₁
First quartile
worked out (not typed)
M
Median
worked out (not typed)
Q₃
Third quartile
worked out (not typed)
max
Greatest
worked out (not typed)
IQR
Interquartile range
worked out (not typed)
Formulas
min
=
least
of
the
n
values
9 = least of the 9 values
M
=
median
of
the
n
values
17 = median of the 9 values
Q₁
=
first
quartile
of
the
n
values
13 = first quartile of the 9 values
Q₃
=
third
quartile
of
the
n
values
21 = third quartile of the 9 values
max
=
greatest
of
the
n
values
30 = greatest of the 9 values
IQR
=
Q₃
−
Q₁
8 = 21 − 13
Assumptions
•
Put the values in order first; the median splits them into a lower and an upper half.
•
Q₁ and Q₃ are the medians of the two halves, the median itself left out when n is odd.
•
The box runs from Q₁ to Q₃: the middle half of the data.
Step-by-step
WE KNOW
n = 9, x₁ = 12, x₂ = 18, x₃ = 9, x₄ = 22, x₅ = 15, x₆ = 30, x₇ = 14, x₈ = 17, x₉ = 20
FIND
least (min), median (M), first quartile (Q₁), third quartile (Q₃), greatest (max), interquartile range (IQR)
Step 1 · Find least (min)
Use min = least of the n values
The smallest value, first in order.
min
=
least
of
x₁
,
x₂
,
x₃
,
x₄
,
x₅
,
x₆
,
x₇
,
x₈
,
x₉
min = least of 12, 18, 9, 22, 15, 30, 14, 17, 20
In order: 9, 12, 14, 15, 17, 18, 20, 22, 30
min = 9
Step 2 · Find median (M)
Use M = median of the n values
The middle value in order (halfway between the middle two).
M
=
median
of
x₁
,
x₂
,
x₃
,
x₄
,
x₅
,
x₆
,
x₇
,
x₈
,
x₉
M
=
median
of
12,
18,
9,
22,
15,
30,
14,
17,
20
4 values on each side of the middle one: 17
M
=
17
Step 3 · Find first quartile (Q₁)
Use Q₁ = first quartile of the n values
The median of the lower half.
Q₁
=
first
quartile
of
x₁
,
x₂
,
x₃
,
x₄
,
x₅
,
x₆
,
x₇
,
x₈
,
x₉
Q₁
=
first
quartile
of
12,
18,
9,
22,
15,
30,
14,
17,
20
Lower half: 9, 12, 14, 15
Q₁
=
13
Step 4 · Find third quartile (Q₃)
Use Q₃ = third quartile of the n values
The median of the upper half.
Q₃
=
third
quartile
of
x₁
,
x₂
,
x₃
,
x₄
,
x₅
,
x₆
,
x₇
,
x₈
,
x₉
Q₃
=
third
quartile
of
12,
18,
9,
22,
15,
30,
14,
17,
20
Upper half: 18, 20, 22, 30
Q₃
=
21
Step 5 · Find greatest (max)
Use max = greatest of the n values
The largest value, last in order.
max
=
greatest
of
x₁
,
x₂
,
x₃
,
x₄
,
x₅
,
x₆
,
x₇
,
x₈
,
x₉
max = greatest of 12, 18, 9, 22, 15, 30, 14, 17, 20
max = 30
Step 6 · Find interquartile range (IQR)
Use IQR = Q₃ − Q₁
The width of the box: the middle half.
IQR = 21 − 13
IQR = 8
Check
Put the numbers back in. Both sides should match.
9 = least of 12, 18, 9, 22, 15, 30, 14, 17, 20 ✓
17 = median of 12, 18, 9, 22, 15, 30, 14, 17, 20 ✓
13 = first quartile of 12, 18, 9, 22, 15, 30, 14, 17, 20 ✓
21 = third quartile of 12, 18, 9, 22, 15, 30, 14, 17, 20 ✓
30 = greatest of 12, 18, 9, 22, 15, 30, 14, 17, 20 ✓
8 = 21 − 13 ✓
Topic check
See what you know: 5 questions on this topic
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