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Find the fences and identify outliers

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A value looks far from the other observations. What should you do before calling it an outlier under the 1.5 × IQR rule?

Year 11, VCE Units 1 & 2, Chapter 2, 4 difficulty levels, Calculator allowedBuild a worksheet

Outliers and fences

The 1.5 × IQR rule flags observations for investigation. State the quartile convention, keep the measurement units and test both tails

\[\begin{gathered} IQR=Q_3-Q_1\\ L=Q_1-1.5\times IQR\\ U=Q_3+ 1.5\times IQR\end{gathered}\]
  1. Strict inequalities Only x<L or x>U is flagged. Equality is retained
  2. Observed whisker endpoints A modified whisker ends at the most extreme non-outlier observation. A fence is a calculated threshold
  3. Context matters An outlier may be genuine. Investigate rather than automatically deleting it

Worked example

Recorded times are 6, 10, 12, 14, 16, 18 and 40 minutes. Find the fences and the modified upper whisker

\[Q_1=10,\quad Q_3=18\]
  1. \(\displaystyle IQR=18-10=8\)Omit the median 14. The lower half is 6, 10, 12, with median 10; the upper half is 16, 18, 40, with median 18.
  2. \(\displaystyle L=10-1.5\times8=-2\)The lower fence is −2 minutes
  3. \(\displaystyle U=18+ 1.5\times8=30\)The upper fence is 30 minutes
  4. \(\displaystyle 40>30\)40 is the only outlier; the modified upper whisker ends at 18 minutes

Common mistakes

Measuring from the median
✕\(\displaystyle L=14-1.5\times8=2\)✓\(\displaystyle L=10-1.5\times8=-2\)
Measure from Q1, not the median. Q1 is 10 in the worked example
Drawing to a fence
✕\(\displaystyle \text{upper whisker}=30\)✓\(\displaystyle \text{upper whisker}=18\)
30 is the threshold; 18 is the greatest recorded non-outlier
Counting equality
✕\(\displaystyle x=U\Rightarrow\text{outlier}\)✓\(\displaystyle x>U\Rightarrow\text{upper outlier}\)
A value exactly on either fence is not flagged

Difficulty levels

introCalculate both fences from a summary
standardIdentify outliers and modified whiskers using sufficient observed extreme values
challengeCalculate from raw data; handle both tails, multiple outliers and exact boundaries
examDistinguish correct fences and outlier values from common errors

Questions about this topic

Can a fence be negative?+

Yes. It is a calculated threshold, not necessarily a possible observation

Is a value on a fence an outlier?+

No. The 1.5 × IQR rule uses strict inequalities

Do I remove outliers before finding quartiles?+

No. First use the full recorded sample. Removing a valid observation changes the data and needs justification

Must an outlier be an error?+

No. A genuine unusual observation may be important. Check its context and the recording process

Does a whisker always end at a fence?+

No. It ends at the most extreme observed non-outlier. It equals a fence only if an observation lies there

Can the five-number summary locate every modified whisker?+

No. If an extreme is an outlier, you may need additional observations to locate the next non-outlier

Can several outliers occur on the same side?+

Yes. Compare all relevant extreme observations with the fences, not only the minimum and maximum

What if the IQR is zero?+

Both fences equal the common quartile. Values unequal to it are flagged by this rule; investigate the ties and context

Should I round before comparing with a fence?+

Keep the precision of the supplied measurements and calculations. Premature rounding can change whether a value is strictly beyond a boundary

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions

In this chapter

Classify a variable as nominal, ordinal, discrete or continuousDistinguish interval and ratio measurement scalesBuild a frequency and percentage frequency tableBuild a bar chart from a frequency table2A Data types and categorical displays: Read values from a frequency table or bar chart2B Reading categorical displays: Read values from a frequency table or bar chartReport on the distribution of a categorical variableCompare categorical distributions across two groupsGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramDescribe the shape of a distributionConstruct a dot plot from raw dataConstruct an ordered stem plot from raw dataChoose an appropriate display for a variableFind the mean and the median, and choose between themFind measures of centre from a frequency displayWork backwards from a mean or medianFind the range, the quartiles and the IQRCalculate the standard deviation from the formulaApply the 68–95–99.7% ruleProduce a five-number summary and construct a boxplotFind the fences and identify outliersRead values and percentages from a boxplotCompare a numerical variable across groups