= Centroid date of a mass-normalized nail kernel
{title2=$\mu_s=\int s\,w(s)\,ds$}
The \b[<centroid date of a mass-normalized nail kernel>] is the <weighted mean> of the reference-crossing times represented by the analyzed nail material. If $w(s)\ge0$ is normalized so that $\int w(s)\,ds=1$, its centroid is
$$
\mu_s=\int s\,w(s)\,ds.
$$
At collection time $t$, the corresponding mean material age is $\overline a=t-\mu_s$. This summarizes a distribution of material times; it is not automatically the date of an exposure or the instant of formation in the <nail matrix>.
For an original pooled-sample example, suppose equal structural <mass> comes from two material-age intervals, 0–2 and 18–20 days before collection. Age density is 0.25 per day in each interval and zero between them. Their interval means are 1 and 19 days, so the mean age is 10 days. Yet no sampled material has age 10 days. A single uniform kernel over 0–20 days has the same mean age, with a very different distribution.
The age <variance> makes part of the difference explicit. A uniform interval of width 2 days has variance $2^2/12=1/3\,\mathrm{day^2}$. The separated mixture therefore has
$$
\sigma_a^2=\frac13+\frac{(1-10)^2+(19-10)^2}{2}
=\frac{244}{3}\,\mathrm{day^2},
$$
whereas a uniform 20-day interval has variance $20^2/12=100/3\,\mathrm{day^2}$. Equal centroids do not imply equal temporal resolution.
\Image[/nail-kernel-centroid-ambiguity.png]
{title=Two original material-age kernels have the same mean age but different spread and sampled intervals}
{description=The lower kernel pools two separated segments of equal mass. Its mean age lies in a gap containing no sampled material. These are hypothetical reference-crossing ages, not observed exposure histories.}
The centroid has a precise averaging property. For any representative time $r$,
$$
\int (s-r)^2w(s)\,ds=\int(s-\mu_s)^2w(s)\,ds+(r-\mu_s)^2.
$$
It minimizes the mass-weighted squared timing discrepancy. Also, if retained <concentration> is affine in $s$, its kernel average equals its value at $\mu_s$. This equality generally fails for a nonlinear history: a quadratic $c(s)=c_0+k(s-\mu_s)^2$ averages to $c_0+k\sigma_s^2$, not $c_0$.
A primary growth experiment followed marks near the proximal nail fold over measured intervals.<Growth rate of human fingernails and toenails in healthy American young adults by S. Yaemsiri and others>[Abstract, Methods]{ref} Such displacement observations help constrain the material clock. They do not establish the full kernel, marker retention or one unique exposure date. Report the reference, distributional width and pooling assumptions alongside a centroid date.
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