SME
Grade 9 Math · Problem type
Outliers and the 1.5 × IQR fences
Use this for “The five-number summary is 12, 20, 24, 28, 45. Is 45 an outlier?”
Chart
Interquartile range: 28 − 20 = 8
1.5 × 8 = 12
Lower fence 20 − 12 = 8
Upper fence 28 + 12 = 40
45 is past a fence: an outlier.
This is an example. Type your own values to start a new problem.
min
Least
example
Q₁
First quartile
example
M
Median
example
Q₃
Third quartile
example
max
Greatest
example
IQR
Interquartile range
worked out (not typed)
L
Lower fence
worked out (not typed)
U
Upper fence
worked out (not typed)
Formulas
IQR
=
Q₃
−
Q₁
8 = 28 − 20
L
=
Q₁
−
1.5
×
IQR
8 = 20 − 1.5 × 8
U
=
Q₃
+
1.5
×
IQR
40 = 28 + 1.5 × 8
Assumptions
•
The fences sit 1.5 × IQR below Q₁ and 1.5 × IQR above Q₃.
•
A value past a fence is an outlier.
•
An outlier moves the mean and the range a lot, the median and the IQR hardly at all.
Step-by-step
WE KNOW
min = 12, Q₁ = 20, M = 24, Q₃ = 28, max = 45
FIND
interquartile range (IQR), lower fence (L), upper fence (U)
Step 1 · Find interquartile range (IQR)
Use IQR = Q₃ − Q₁
The width of the box: the middle half.
IQR = 28 − 20
IQR = 8
Step 2 · Find lower fence (L)
Use L = Q₁ − 1.5 × IQR
Go 1.5 box-widths below the first quartile.
L
=
20
−
1.5
×
8
L
=
20
−
12
L
=
8
Step 3 · Find upper fence (U)
Use U = Q₃ + 1.5 × IQR
Go 1.5 box-widths above the third quartile.
U
=
28
+
1.5
×
8
U
=
28
+
12
U
=
40
→
outlier:
45
Check
Put the numbers back in. Both sides should match.
8 = 28 − 20 ✓
8 = 20 − 1.5 × 8 ✓
40 = 28 + 1.5 × 8 ✓
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