Outliers

The 1.5 × IQR fences

In 5 of 20 recent exams
Paper 1 or 2
2 to 3 marks

How it is set in the exam

Paper 1 (no calculator) and Paper 2 (calculator), 2–3 marks, often a "show that" inside a box-plot or summary-statistics question. Typical (5 of 20 exam versions, 3 of them on Paper 1 (no calculator); 6 logged parts).

Spot it in the exam

Exam phrases to spot (5)

“show that … is an outlier”
“show that … is not an outlier”
“find the value above which a data item would be an outlier”
“the largest value is not an outlier; find the least possible …”
“the smallest value that would not be an outlier”

In disguise (1)

"should this value be removed?": decide whether it is an outlier, then whether it is an error or genuine, from the context

Variants (3)

fence: upper or lower
asked: the fence, show that a value is an outlier, show that it is not, the boundary value itself, an unknown quartile or IQR
context: the first step of drawing a box plot (Box plots), where an outlier is drawn as a cross and the whisker stops at the most extreme value that is not an outlier

Looks like it, but isn't

a boundary for a discrete random variable, such as "fewer than $k$ " or "at most $k$ ": that is a translation of words into $X \le \ldots$ (Inequality words for a whole-number variable), with no quartiles and no fences.

Worked example

Show that a value is an outlier

Paper 2 (calculator)

Question

The daily rainfall, in mm, at a weather station on $15$ days in March was:

0.4, \ 1.2, \ 2.5, \ 0.8, \ 3.1, \ 1.9, \ 0.0, \ 2.2, \ 1.5, \ 0.6, \ 12.4, \ 2.8, \ 1.1, \ 0.9, \ 1.7

(a) Find the interquartile range. [2]
(b) Show that $12.4$ mm is an outlier. [3]
(c) The value $12.4$ mm was recorded on the day of a storm. State, with a reason, whether it should be removed from the data. [1]

Try it yourself first, then compare.

Now try a question

Questions of this type are on their way. Try another one meanwhile: write your working on paper or on screen and see where each mark went. Free, with no account.

Try a question

Steps

IQR
$= Q_3 - Q_1$ (from the GDC for a list on ).
The fence on the value's side:
$Q_3 + 1.5 \times \mathrm{IQR}$ for a large value, $Q_1 - 1.5 \times \mathrm{IQR}$ for a small one. Write the expression with the numbers in it.
Compare, with the inequality sign:
" $12.4 > 5.05$ , so $12.4$ is an outlier". In a "show that", this comparison is the reason mark.
Reverse questions:
"is not an outlier" means the value is on or inside the fence. Set up the inequality (or the equation at the boundary) and solve for the unknown quartile or IQR.
The boundary value
itself is not an outlier: don't move to the next whole number.

Common mistakes

Measuring from the median
instead of from the nearest quartile.
Calling the fence value an outlier,
or moving to the next whole number: .
A comparison without numbers
"it is bigger than the fence" earns no reason mark. Write $12.4 > 5.05$ .

More mistakes (2)

Removing every outlier
only errors are removed; a genuine extreme value stays.
The wrong fence
a small value is compared with $Q_1 - 1.5 \times \mathrm{IQR}$ , not with the upper fence.

Easy to confuse with

Box plots

Next typeCompare two distributions in context

Now try a question

Keep going

Easy to confuse with

More in Descriptive statistics: sampling, averages, spread and data displays