Values that stand apart
An outlier is a value far from the rest of the data. Outliers may be:
- Errors: a typo (a height of 1,750 cm instead of 175 cm), a broken sensor.
- Genuine but unusual: a record-breaking performance, a billionaire’s income.
Each kind needs a different response.
Method 1: The IQR rule
- Find Q1, Q3 and the interquartile range (IQR = Q3 − Q1).
- Calculate the fences:
- Lower fence: Q1 − 1.5 × IQR
- Upper fence: Q3 + 1.5 × IQR
- Values outside the fences are outliers.
Example: 12, 14, 15, 15, 16, 17, 18, 19, 20, 45
- Q1 = 15, Q3 = 19, IQR = 4
- Fences: 15 − 6 = 9 and 19 + 6 = 25
- 45 is an outlier.
On a box plot, outliers are shown as separate dots.
Method 2: z-scores
For roughly normal data, values with |z-score| > 3 (sometimes > 2) are unusual.
Example: Mean 60, σ = 5. A value of 78 has z = 3.6 → outlier.
Effects of outliers
| Statistic | Effect of an outlier |
|---|---|
| Mean | pulled toward the outlier |
| Median | barely affected (resistant) |
| Range | greatly increased |
| IQR | resistant |
| Standard deviation | increased |
Example: Salaries $40k, $42k, $45k, $47k, $500k
- Mean = $134.8k; median = $45k. The median better represents a typical salary.
Outliers in scatter plots
In two-variable data, an outlier can be an influential point: it may pull a line of best fit and change the correlation a lot. Try the analysis with and without it.
What to do with outliers
- Check for errors. Fix or remove mistakes.
- Investigate. Is there a real reason?
- Report honestly. Don’t delete genuine data just because it’s inconvenient. You can report results with and without the outlier.
- Use resistant statistics (median, IQR) when outliers are present.
Worked examples
Example 1: Q1 = 20, Q3 = 30. Upper fence? 30 + 15 = 45.
Example 2: A class’s test scores have one 2% because a student was absent. Error or genuine? Probably not a real measure of learning; investigate before including.
Example 3: Which is better for house prices with a few mansions: mean or median? Median.
Common mistakes
- Deleting outliers automatically.
- Using the mean with skewed data.
- Forgetting to recompute fences for the correct data set.
Try it yourself
- Data: 5, 6, 6, 7, 8, 8, 9, 30. Q1 = 6, Q3 = 8.5. Is 30 an outlier?
- Mean 100, σ = 15. Is 148 unusual by the |z| > 3 rule?
- Which measure of centre is resistant?
- Give one legitimate reason to remove an outlier.
Answers: 1) Yes: upper fence = 8.5 + 3.75 = 12.25. 2) Yes: z = 3.2. 3) the median 4) It’s a confirmed recording error.
Quick recap
- An outlier is far from the rest of the data.
- Detect with IQR fences or z-scores.
- Outliers distort the mean; the median and IQR are resistant.