Outliers (Deep) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Outliers (Deep).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An outlier is a data value that lies unusually far from most other values, potentially indicating measurement error, a rare event, or an important exception.

The weird one that doesn't fit. Is it a mistake, or something interesting?

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Outliers can be errors to remove OR important discoveries to investigate.

Common stuck point: Don't automatically remove outliers—first ask WHY they're there.

Sense of Study hint: Calculate Q1 - 1.5*IQR and Q3 + 1.5*IQR as fences. Any value outside these fences is an outlier. Then investigate why.

Worked Examples

Example 1

medium
Data: \{12, 15, 14, 13, 16, 14, 15, 85\}. Use the 1.5 \times IQR rule to determine if 85 is an outlier, and discuss whether it should be removed.

Solution

  1. 1
    Sort data: \{12, 13, 14, 14, 15, 15, 16, 85\}; n=8
  2. 2
    Q_1 = \frac{13+14}{2} = 13.5; Q_3 = \frac{15+16}{2} = 15.5
  3. 3
    IQR = 15.5 - 13.5 = 2; Upper fence = 15.5 + 1.5(2) = 18.5
  4. 4
    85 > 18.5, so 85 is flagged as an outlier
  5. 5
    Decision: investigate before removing — 85 could be a data entry error (e.g., 15 mis-typed as 85) or a genuine extreme value (e.g., a special event)

Answer

85 is an outlier (exceeds fence of 18.5). Investigate cause before removing.
The 1.5×IQR rule identifies potential outliers but does not determine whether to remove them. Outliers might be data errors (should remove), legitimate rare events (keep), or indicators of a different subgroup (analyze separately).

Example 2

hard
Calculate the effect of an outlier (value 200) on the mean and median for \{10, 12, 11, 13, 12, 200\}, comparing to the data without the outlier \{10, 12, 11, 13, 12\}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A data set has Q_1 = 50, Q_3 = 70. Determine if each value is an outlier: (a) 85, (b) 15, (c) 105.

Example 2

hard
A researcher finds that removing one outlier changes the correlation from 0.45 to 0.82. Discuss whether the outlier should be removed and what this dramatic change reveals.
← Back to Outliers (Deep) Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

variabilityinterquartile range