Outlier
Statistics & ProbabilityAn outlier is a data value that is significantly different from the rest of the dataset, appearing far from the other values.
Formula
\text{Outlier if value} < Q_1 - 1.5 \cdot \text{IQR} \quad \text{or} \quad \text{value} > Q_3 + 1.5 \cdot \text{IQR}
Definition
An outlier is a data value that is very different from the rest, standing far apart from the other values in the dataset.
Example
Test scores: $78$, $80$, $82$, $84$, $86$, and $3$. The score of $3$ is an outlier, far below all the others. It would drag the average down significantly.
Key Insight
Outliers can be mistakes (a data entry error) or real, important values (a record-breaking athlete). Always investigate outliers before deciding what to do with them.
Definition
An outlier is an observation that lies an abnormal distance from other values. A common detection method: a value is an outlier if it falls below $Q_1 - 1.5 \cdot \text{IQR}$ or above $Q_3 + 1.5 \cdot \text{IQR}$. Outliers can signal data errors, unusual events, or important discoveries.
Example
$\text{IQR} = Q_3 - Q_1 = 90 - 70 = 20$. Lower fence $= 70 - 30 = 40$. Upper fence $= 90 + 30 = 120$. A score of $35$ would be flagged as an outlier (below $40$); a score of $130$ would also be flagged.
Key Insight
The mean is affected by outliers; the median is not. When outliers are present, the median is often a better measure of center. Box plots display outliers as individual points beyond the whiskers.
Definition
Outlier detection methods include the IQR rule, Grubbs' test, the z-score threshold ($|z| > 3$), and the generalized extreme studentized deviate (ESD) test for multiple outliers. In regression, influential observations are detected by Cook's distance and leverage (hat matrix diagonal), which measure impact on fitted values and coefficient estimates respectively.
Example
Cook's distance $D_i = \frac{1}{p}\sum \frac{(\hat{y}_j - \hat{y}_{j(i)})^2}{\text{MSE}}$, where the sum is over all $n$ observations. A common threshold is $D_i > 4/n$ or $D_i > 1$ to flag influential points in regression.
Key Insight
Robust regression methods (M-estimators, least trimmed squares) down-weight or exclude outliers automatically, providing estimates that resist distortion from extreme observations without requiring manual outlier removal.