Outlier
An 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, an observation that lies an abnormal distance from other values. A common detection method flags a value as 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. Other detection methods include 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
Test scores $78$, $80$, $82$, $84$, $86$, and $3$ have an outlier at $3$, far below all the others, which would drag the average down significantly. With $\text{IQR} = Q_3 - Q_1 = 90 - 70 = 20$, the lower fence is $70 - 30 = 40$ and upper fence is $90 + 30 = 120$, so a score of $35$ or $130$ would be flagged as an outlier. Cook's distance $D_i = \frac{1}{p}\sum \frac{(\hat{y}_j - \hat{y}_{j(i)})^2}{\text{MSE}}$ (summed over all $n$ observations) uses a common threshold of $D_i > 4/n$ or $D_i > 1$ to flag influential points in regression.
Key Insight
Outliers can be mistakes (a data entry error) or real, important values (a record-breaking athlete), so always investigate them before deciding what to do. The mean is affected by outliers while the median is not, so the median is often a better measure of center when outliers are present, and box plots display outliers as individual points beyond the whiskers. 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.