Cook's distance

Cook's distance measures how much all fitted regression coefficients change when one observation is deleted, combining leverage and residual size. Values exceeding 1, or the more conservative 4/n, flag potentially influential observations.

ByHoang TruongUpdated July 3, 2026

FrameworkInfluential observations

See it move

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Cook's distance measures how much every fitted coefficient would shift if one observation were dropped, combining that point's leverage with the size of its residual into a single influence score. A value of Dᵢ above 1 — or the stricter 4 divided by n — flags a point worth investigating rather than deleting automatically, since it may reveal a genuine pattern the model is missing.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicRegression Diagnostics & ProblemsAdvanced

The formula

LaTeX

D_i = \frac{(\hat{\mathbf{y}}_{(-i)} - \hat{\mathbf{y}})^\top(\hat{\mathbf{y}}_{(-i)} - \hat{\mathbf{y}})}{p \cdot \text{MSE}}

Variables

$D_{i}$: Cook's distance for observation i (dimensionless)
$\overset{y}{^}_{(- i)}$: n-vector of fitted values from the model estimated without observation i (units of y)
$\overset{y}{^}$: n-vector of fitted values from the full model (units of y)
$p$: number of parameters in the model including the intercept (dimensionless)
$M S E$: mean squared error from the full model ((units of y)²)

Observations with Dᵢ > 1, or the more conservative threshold Dᵢ > 4/n, are flagged as potentially influential. High influence combines high leverage with a large residual.