= Normal-location influence curves
For $F=N(\mu,\sigma^2)$, put $z=(x-\mu)/\sigma$. The <influence function of the mean>, <influence function of the sample median>, and <influence function of a Huber location estimator> with fixed $s=\sigma$ are respectively
$$
\sigma z,\qquad \sigma\sqrt{\frac\pi2}\operatorname{sgn}(z),\qquad
\frac{\sigma\psi_k(z)}{2\Phi(k)-1}.
$$
The first grows without bound, the second jumps at the <median>, and the third clips continuously. For $k=1.345$, $2\Phi(k)-1\simeq0.821375$ and the normalized Huber tail is about $1.63750\sigma$. This illustrates the <optimal bounded influence function for normal location> without asserting that this cutoff is optimal for every contamination model.
\Image[/normal-location-influence-curves.png]
{title=Mean, median, and normalized Huber influence curves for two normal location-scale models}
Both panels are theoretical <normal distributions>, with no sampled or observed data. Doubling $\sigma$ doubles the vertical <influence function> scale and the horizontal clipping distance. Changing $\mu$ translates the curves. The isolated dot marks the actual <median> derivative at the center; connecting its two tails by a vertical segment would misrepresent that value. Bounded influence does not mean every small residual has less effect than on the <mean>: the central Huber slope is $1/(2\Phi(k)-1)>1$, a price paid for reducing sensitivity in the tails.
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